HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs ...€¦ · ior of nonlinear sources in the...

American Institute of Mathematical Sciences

Asymptotic Behavior of Dynam

ical Systems in Fluid M

echanics

HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations


AIMS on Applied Mathematics Vol.6

Giovanni Alberti Fabio Ancona Stefano Bianchini Gianluca Crippa Camillo De Lellis Andrea Marson Corrado Mascia (Eds.)

Giovanni AlbertiFabio AnconaStefano BianchiniGianluca CrippaCamillo De LellisAndrea MarsonCorrado Mascia (Eds.)

HCDTE Lecture Notes. Part I.

Nonlinear Hyperbolic PDEs,

Dispersive and Transport Equations

American Institute of Mathematical Sciences

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Editor in Chief: Benedetto Piccoli (USA)

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AMS 2010 Classifications: 35-xx Partial Differential Equations

ISBN-10: 1-60133-014-6; ISBN-13: 978-1-60133-014-7

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Foreword

This is the first of two volumes containing the lecture notes of some of thecourses given during the intensive trimester HCDTE, Nonlinear HyperboliCPDEs, Dispersive and Transport Equations: analysis and control, held atSISSA, Trieste (Italy) from May 16th to July 22nd, 2011, in the framework ofthe activities funded by the ERC Starting Grant “ConLaws”, Hyperbolic Sys-tems of Conservation Laws: singular limits, properties of solutions and controlproblems1.

The lectures covered a number of different topics within the fields of hyper-bolic equations, fluid dynamic, dispersive and transport equations, measuretheory and control and they were primarily intended for PhD students andyoung researchers at the beginning of their career. With the aim of makingpossible the access to the contents of the courses to a wider public, we proposedto the lecturers to re-organize the material in a self-contained manuscriptand we received many earnest acceptances, that collected together form thesetexts.

The first volume collects three contributions. It starts with the lectures,authored by I. Chueshov and I. Lasiecka, on the analysis of stability and con-trol of long-time behavior of a class of nonlinear partial differential equationsof hyperbolic type, including a number of significant specific models. The fo-cus is toward the possibility of reducing the infinite-dimensional dynamics toa finite dimensional one in the large-time regime, having in mind applicationsin control theory.

Then, we pass to the lecture notes by J.-F. Coulombel which concern thestability analysis of finite difference schemes for initial-boundary value prob-lems for hyperbolic equations. Particular attention is paid to cases where thestability is characterized by the von Neumann condition and to the funda-mental role played by the uniform Kreiss-Lopatinskii condition, illustrated bymeans of specific examples such as Lax-Friedrichs and leap-frog schemes.

1 The detailed program and the slides of the HCDTE Trimester can be found onthe website http://events.math.unipd.it/hcdte.

VI Foreword

The third and final part of the first volume, by S. Liu and R. Triggiani,describes results on boundary control and boundary inverse problems for non-homogeneous, second-order hyperbolic equations, showing that the applica-tion of sharp Carleman estimates is capable to merge both control theoryand inverse problems issues. Particular attention is dedicated to the inverseproblem of determining the interior damping and potential coefficients of amixed, second-order hyperbolic equation with non-homogeneous Neumann orDirichlet boundary datum.

The second forthcoming volume gathers five papers.S. Daneri and A. Figalli present some models for the motion of homoge-

neous incompressible fluids in bounded domains. Initially, results on existenceand uniqueness of different types of solutions are discussed; then the attentionis drawn to the formulation of a variational model directly related with theusual Euler equations for incompressible fluids.

Next, we encounter the contribution of A. Pratelli and S. Puglisi whichaims to give an overview on the problem of the approximation of homeomor-phisms in the plane. Different feasible concepts of approximation are explored:the one requiring the smoothness of the regularizations, and the one based onpiecewise affine approximants.

The analysis of periodic Schrodinger equations is the topic of the chapterauthored by G. Staffilani, looking at the problem as an infinite dimensionalHamiltonian system. The presentation collects facts on dispersive equationsand different types of nonlinear Schrodinger equations, covering also the no-tion of Gibbs measures, its definition and invariance.

Then, we pass to the essay by L. Szeklyhidi which concerns the use ofresults motivated by the problem of determining isometric embeddings, suchas the Nash-Kuiper theorem, to construct very large sets of weak solutions formodels in fluid dynamics, as the incompressible Euler equations, by means ofsuitable variants of convex integration.

Finally, the existence of global finite energy solutions to the isentropic Eu-ler equations are discussed by M. Westdickenberg in the final chapter, startingfrom the first result proved by DiPerna in 1983, based on the use of compen-sated compactness method.

We are grateful to whom contributed to the accomplishment of these twovolumes: to the authors for dedicating additional time to organize and revisethe material used for the lectures, to the referees for careful readings andinvaluable suggestions which helped to improve the material, to all the peoplewho attended the original courses during the intensive period, for their interestand participation. Finally, a special thanks is due to the American Institutefor Mathematical Sciences, which accepted our editorial proposal.

Giovanni Alberti, Fabio Ancona,Stefano Bianchini, Gianluca Crippa,

Camillo De Lellis, Andrea Marson,Corrado Mascia

Contents

1 Well-posedness and long time behavior in nonlineardissipative hyperbolic-like evolutions with critical exponentsIgor Chueshov, Irena Lasiecka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Description of the PDE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Well-posedness and generation of continuous flows . . . . . . . . . . . . . . . 81.4 General tools for studying attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5 Long time behavior for canonical models . . . . . . . . . . . . . . . . . . . . . . . . 481.6 Other models covered by the methods presented . . . . . . . . . . . . . . . . . . 82References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2 Stability of finite difference schemes for hyperbolic initialboundary value problemsJean-Francois Coulombel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.2 Fully discretized hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.3 Fully discrete initial boundary value problems: strong stability . . . . . 1222.4 Characterization of strong stability: proof of the main results . . . . . . 1542.5 Fully discrete initial boundary value problems: semigroup stability . . 1972.6 A partial conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213A Other examples of discretizations for the Cauchy problem . . . . . . . . . 214References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3 Boundary control and boundary inverse theory fornon-homogeneous second-order hyperbolic equations: Acommon Carleman estimates approachShitao Liu, Roberto Triggiani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2273.1 Preparatory material: Carleman estimates, interior and boundary

regularity of mixed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.2 Control theory results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2383.3 Inverse theory results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

VIII Contents

3.4 Inverse problems for second-order hyperbolic equations withnon-homogeneous Neumann boundary data: Global uniquenessand Lipschitz stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

3.5 Inverse problems for second-order hyperbolic equations withnon-homogeneous Dirichlet boundary data: Global uniqueness andLipschitz stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

3.6 Inverse problems for a system of strongly coupled wave equationswith Neumann boundary data: Global uniqueness and Lipschitzstability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3.7 Recovery damping and source coefficients in one shot by means ofa single boundary measurement. The Dirichlet case. . . . . . . . . . . . . . . 312

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

1

Well-posedness and long time behavior innonlinear dissipative hyperbolic-like evolutionswith critical exponents

Igor Chueshov1 and Irena Lasiecka2

1 Department of Mechanics and Mathematics, Karazin Kharkov NationalUniversity, Kharkov, 61022, Ukraine [email protected]

2 Department of Mathematics, University of Virginia, Charlottesville, VA 22901;Systems Research Institute, Polish Academy of Sciences, [email protected]

1.1 Introduction

These lectures are devoted to the analysis of stability and control of longtime behavior of PDE models described by nonlinear evolutions of hyperbolictype. Specific examples of the models under consideration include: (i) nonlin-ear systems of dynamic elasticity: von Karman systems, Berger’s equations,Kirchhoff - Boussinesq equations, nonlinear waves (ii) nonlinear flow - struc-ture and fluid - structure interactions, (iii) and nonlinear thermo-elasticity. Agoal to accomplish is to reduce the asymptotic behavior of the dynamics toa tractable finite dimensional and possibly smooth sets. This type of resultsbeside having interest in its own within the realm of dynamical systems, arefundamental for control theory where finite dimensional control theory canbe used in order to forge a desired outcome for the dynamics evolving in theattractor.

A characteristic feature of the models under consideration is criticality orsuper-criticality of sources (with respect to Sobolev’s embeddings) along withsuper-criticality of damping mechanisms which may be also geometrically con-strained. This means the actuation takes place on a “small” sub-region only.Super-criticality of the damping is often a consequence of the “rough” behav-ior of nonlinear sources in the equation. Controlling supercritical potentialenergy may require a calibrated nonlinear damping that is also supercritical.On the other hand super-linearity of the potential energy provides beneficialeffect on the long time boundedness of semigroups. From this point of view,the nonlinearity does help controlling the system but, at the same time, italso does raise a long list of mathematical issues starting with a fundamentalquestion of uniqueness and continuous dependence of solutions with respect

2 Igor Chueshov and Irena Lasiecka

to the given (finite energy) data. It is known that solutions to these problemscan not be handled by standard nonlinear analysis-PDE techniques.

The aim of these lectures is to present several methods of nonlinear PDEwhich include cancelations, harmonic analysis and geometric methods whichenable to handle criticality and also super-criticality in both sources and thedamping. It turns out that if carefully analyzed the nonlinearity can be taken“advantage of” in order to produce implementable control algorithms.

Another aspects that will be considered is the understanding of controlmechanisms which are geometrically constrained. Here one would like to useminimal sensing and minimal actuating (geometrically) in order to achieve theprescribed goal. This is indeed possible, however analytical methods used aremore subtle. The final task boils down to showing that appropriately dampedsystem is “quasi-stable” in the sense that any two trajectory approach eachother exponentially fast up to modulo a compact term which can grow intime. Showing this property- formulated as quasi-stability estimate -is the keyand technically demanding issue that requires suitable tools. These include:weighted energy inequalities, compensated compactness, Carleman’s estimatesand some elements of microlocal analysis.

The lecture are organized as follows.

• We start in section 1.2 with description of main PDE models such as waveand plate equations with both interior and boundary damping. Instead ofstriving for the most general formulation, we provide canonical models inthe simplest possible form which however retains the main features of theproblems studied.

• Section 1.3 deals with well-posedness of weak solutions corresponding tothese models.

• Section 1.4 describes general and abstract tools used for proving existenceof attractors and also properties of attractors such as structure, dimen-sionality and smoothness. Here we emphasize methods which allow to dealwithin non-compact environment typical when one deals with hyperboliclike dynamics and critical sources. Specifically a simple but very usefulmethod of energy relations due to J. Ball (see [Bal04] and also [MRW98])is presented (allowing to deal with supercritical sources), a version of”compensated compactness method” introduced first by A. Khanmamedov[Kha06] (allowing to deal with some critical sources) and a method basedon ”quasi-stability” estimate originated in the authors work [CL04a] (andfurther developed in [CL08a]) which gives in one shot several properties ofthe attractor such as smoothness, finite-dimensionality and also limitingproperties for the family of attractors.

• Section 1.5 demonstrates how the abstract methods can be applied to theproblems of interest. Clearly, the trust of the arguments is in derivationof appropriate inequalities. While we provide the basic insight into thearguments, the details of calculations are often referred to the literature.This way the interested reader will be able to find a complete justification

1 Dissipative hyperbolic-like evolutions 3

of the claims made. We should also emphasize that presentation of ourresults is focused on main features of the dynamics and not necessarily on afull generality. However, subsections such as Generalizations or Extensionsprovide information on possible generalizations with a well documentedliterature and citations.

• Section 1.6 is devoted to other models such as: structurally dampedplates, fluid-structures, fluid-flow interactions, thermoelastic interactions,Midlin Timoshenko beams and plates, Quantum Zakharov system andShrodinger-Boussinesq equations. These equations exemplify models wheregeneral abstract tools for the treatment of long behavior, presented in sec-tion 4 apply. Due to space limitations, the analysis here is brief with detailsdeferred to the literature.

• Each section concludes with examples of further generalizations -extensionsand also with a list of open problems.

Basic notations: Let Ω ⊂ Rd, d = 2, 3, be a bounded domain with asmooth boundary Γ = ∂Ω. We denote by Hs(Ω) L2-based Sobolev space ofthe order s endowed with the norm ‖u‖s,Ω ≡ ‖u‖Hs(Ω) and the scalar product(u, v)s,Ω . As usual for the closure of C∞0 (Ω) in Hs(Ω) we use the notationHs

0(Ω). Below we use the notations:

((u, v)) ≡∫Ω

uvdΩ, ||u||2 ≡ ((u, u)), << u, v >>≡∫Γ

uvdΓ,

QT ≡ Ω × (0, T ), ΣT ≡ Γ × (0, T ).

We also denote by C(0, T ;Y ) a space of strongly continuous functions on theinterval [0, T ] with values in a Banach space Y . In the case when we deal withweakly continuous functions we use the notation Cw(0, T ;Y ).

1.2 Description of the PDE models

We consider second order (in time) PDE models of evolutions with nonlinear(velocity type) feedback control represented by monotone, continuous func-tions, and critical-supercritical sources defined on Ω ⊂ Rd, d = 2, 3. This classof models is representative of many physical phenomena occurring in physics,engineering and life sciences that involve wave/sound propagation, mechanicalvibrations, oscillations of membranes, plates and shells. The underlined equa-tions are prototypes of dynamics which do not exhibit any internal smoothingor damping (unlike parabolic equations) . This has strong implications on reg-ularity of solutions and their long time behavior. There is no natural sourceof compactness or dissipation- two main ingredients when addressing longtime behavior of the orbits. This makes the subject challenging from both thephysical and mathematical point of view.

In what follows, we shall focus on canonical models in the aforementionedclass: wave and plate equations with added dissipation occurring either in the


interior of the domain or on the boundary. Three examples given below arebenchmark models which admit vast array of generalizations, some of whichare discussed in Section 1.6. We begin with the models exhibiting the interiordamping first.

1.2.1 Waves and plates with nonlinear interior dampingand critical-supercritical sources

(A) Wave equation with nonlinear damping-source interaction. In abounded domain Ω ⊂ R3 we consider the following wave equation with theDirichlet boundary conditions:

wtt −∆w+ a(x)g(wt) = f(w) in Ω × (0, T ); w = 0 on Γ × (0, T ), (1.1)

where T > 0 may be finite or ∞. We suppose that the damping has thestructure g(s) = g1s + |s|m−1s for some m ≥ 1 and assume the followingtypical behavior for the source f(w) ∼ −|w|p−1w, where either 1 ≤ p ≤ 3 or

else 3 < p ≤ min

5, 6mm+1

. The nonnegative function a ∈ C1(Ω) represents

the support and intensity of the damping. The associated energy function hasthe form:

E (t) ≡ 1

2||wt(t)||2 +

1

2||∇w(t)||2 −

∫Ω

f(w(t))dx, (1.2)

where f denotes the antiderivative of f , i.e. f ′ = f . The energy balance relationfor this model has the form

E (t) +

∫ t

s

((ag(wt), wt)))dτ = E (s). (1.3)

When p ≤ 3 the source in the wave equation is up to critical. This is due toSobolev’s embedding H1(Ω) ⊂ L6(Ω) so that f(w) ∈ L2(Ω) for w of finiteenergy. In this case local (in time) well-posedness holds with no restrictions form. Globality of solutions is guaranteed by suitable a priori bounds resultingfrom either a structure of the source f(w) or from the interplay with thedamping, see Theorem 2 below.

Long time behavior will be analyzed under the condition of weak degen-eracy of the damping coefficient a(x) in the so called “double critical case”,i.e. p ≤ 3, m ≤ 5. We use the compensated compactness criterion in The-orem 14 for proving existence of an attractor and quasi-stability inequality(1.66) required by Definition 11 in order to establish both smoothness andfinite dimensionality of global attractor.

When 3 < p ≤ 5 the damping required for uniqueness of solutions needs tobe superlinear. With sufficiently superlinear damping we show that the cor-responding dynamical system is well posed on the finite energy phase space.However, the issue of long time behavior, in that case, is still an open problem.


The known frameworks for studying supercritical sources fail due to nonlin-earity of the damping. More details will be given later. The ranges of p ∈ [5, 6)are considered in [BL08a].

We also refer to [CV02, Chapter XIV] where the case of a linear damp-ing and a supercrtical force is considered from point of view of trajectoryattractors.

(B) Von Karman plate equation with nonlinear interior damping.Let Ω ⊂ R2 and α ∈ [0, 1]. We denote Mα ≡ I−α∆ and consider the equation

Mαwtt +∆2w + a(x)[g(wt)− α divG(∇wt)

]= [F (w), w] + P (w) (1.4)

in Ω × (0,∞) with the clamped boundary conditions:

w =∂

∂nw = 0 on Γ × (0,∞), (1.5)

where the Airy stress function F (w) solves the elliptic problem

∆2F (w) = −[w,w], in Ω, with F =∂

∂nF = 0 on Γ, (1.6)

and the von Karman bracket [u, v] is given by

[u, v] = ∂2x1u · ∂2x2

v + ∂2x2u · ∂2x1

v − 2 · ∂2x1x2u · ∂2x1x2

v. (1.7)

The damping functions g : R 7→ R+ and G : R2 7→ R2+ have the following form

g(s) = g1s+ |s|m−1s and G(s, σ) = G1 · (s;σ) + (|s|m−1s;σm−1σ) (1.8)

where g1 and G1 are nonnegative constants.The source term P is assumed locally Lipschitz operator acting from

H2(Ω) into L2(Ω) when α = 0 and from H2(Ω) into H−1(Ω) when α > 0.The associated energy function and the energy balance relation have the

form

E (t) ≡ 1

2

(||wt(t)||2 + α||∇wt(t)||2

)+

1

2||∆w(t)||2 +

1

4||∆F (w)||2 (1.9)

and

E (t) +

∫ t

s

D(wt)dτ = E (s) +

∫ t

s

((P (w), wt))dτ, (1.10)

where the interior damping form D(wt) is given by

D(wt) = ((ag(wt), wt)) + α((aG(∇wt),∇wt)). (1.11)

The case when α > 0 is subcritical with respect to the Airy source. Thewell-posedness and existence of attractors in this case is more standard [CL10].When α = 0 the Airy stress source is critical. Wellposedness will be achieved


by displaying hidden regularity (see Lemma 2 below) for Airy’s stress function.Existence of attractors along with their smoothness will be shown by meansof quasi-stability inequality (see (1.66) in Definition 11) which can be provedas long as P is subcritical.

(C) Kirchhoff-Boussinesq plate with interior damping. With notationsas in the case of the von Karman plate we consider

Mαwtt+∆2w+a(x)[g(wt)−α divG(∇wt)

]= div

[|∇w|2∇w

]+P (w) (1.12)

with the clamped boundary conditions (1.5). The damping functions g : R 7→R+, G : R2 7→ R2

+ and the source P are the same as above. The associatedenergy function has the form

E (t) ≡ 1

2

(||wt(t)||2 + α||∇wt(t)||2

)+

1

2||∆w(t)||2 +

1

4

∫Ω

|∇w(x, t)|4dx.

(1.13)The corresponding energy balance relation is given by (1.10).

The well-posedness for the case α > 0 is standard. This is due to the factthat div |∇w|2∇w ∈ H−1(Ω) for finite energy solutions w. The case α = 0 issubtle. Its analysis requires special consideration and depends on linearity ofthe damping. Similarly, long time behavior dealt with by using Ball’s methodpresented in an abstract version in Theorem 15 also requires linear damping.

1.2.2 Nonlinear waves and plates with geometrically constraineddamping and critical-supercritical sources

This class contains models in the situation when the support of the dampinga(x) is strictly contained in Ω. This is to say supp a(x) ⊂ Ω0 ⊂ Ω. In addi-tion, we considered singular case of interior localized damping which is theboundary damping. These models are described next.

(A) Wave equation with nonlinear boundary damping-source in-teraction. In a bounded domain Ω ⊂ R3 we consider the following waveequation

wtt −∆w + a(x)g(wt) = f(w) in Ω × (0, T ), (1.14)

with nonlinear dynamical boundary conditions of Neumann type:

∂

∂nw + g0(wt) = h(w) in Γ × (0, T ). (1.15)

Concerning the internal damping g and source f terms our hypotheses are thesame as in Section 1.2.1(A). The internal damping coefficient a(x) satisfiesrelations

a ∈ C1(Ω), a(x) ≥ 0 in Ω with supp a ⊂ Ω0 ⊂⊂ Ω


The boundary damping has the form g0(s) = g2s + |s|q−1s with q ≥ 1 andg2 ≥ 0. The boundary source has the behavior h(w) ∼ −|w|k−1w, where

1 ≤ k ≤ max

3, 4qq+1

.

The associated energy function is given by

E (t) ≡ 1

2||wt(t)||2 +

1

2||∇w(t)||2 −

∫Ω

f(w(t))dx−∫Γ

h(w(t))dx (1.16)

where as above f and h denote the antiderivative of f and h. The energybalance relation in this case has the form

E (t) +

∫ t

s

[((ag(wt), wt))+ << g0(wt), wt >>

]dτ = E (s). (1.17)

Local and global well-posedness result will be presented for the boundary-source and damping model. One could consider larger range of boundary andinterior sources k ∈ [1, 4), p ∈ [1, 6) [BL08a]. In this case, however, potentialenergy corresponding to the sources is not well defined. Well-posedness re-sult requires higher integrability of initial data [BL08a, BL10]. More generalstructure of sources and damping can be considered. However, the requiredpolynomial bounds are critical.

In the triple critical case p = 3, q = 3,m = 5 with k = 2 the existence ofa global attractor is shown by taking advantage of compensated compactnessresult in Theorem 14 (see [CL07a]). With the additional hypotheses imposedon the damping smoothness finite-dimensionality of attractor is establishedby proving that the system is quasi-stable (in the sense of Definition 11).

In the case when only boundary damping is active (or internal dampingis localized) existence and smoothness of attractor require additional growthcondition restrictions imposed on the damping. This is to say, we need toassume p ≤ 3 and q ≤ 1 [CLT08, CLT09].

(B) Von Karman plate equation with nonlinear boundary damping.In a bounded domain Ω ⊂ R2 we consider the equation

Mαwtt +∆2w + a(x)[g(wt)− α divG(∇wt)

]= [F (w), w] + P (w) (1.18)

with the hinged dissipative boundary conditions:

w = 0, ∆w = −g0(∂

∂nwt), on Γ × (0, T ) (1.19)

Here we use the same notations as in Section 1.2.1(B). In particular, Airy’sstress function F (w) solves (1.6) and the internal damping functions g and Ghave the form (1.8). Concerning the boundary damping we assume that g0(s) ∼g2s+ |s|q−1s, q ≥ 1. The associated energy function E has the same form as inSection 1.2.1(B), see (1.9). The corresponding energy balance relation readsas follows


E (t) +

∫ t

s

[D(wt)+ << g0

(∂wt∂n

),∂wt∂n

>>

]dτ = E (s) +

∫ t

s

((P (w), wt))dτ,

(1.20)where the internal damping term D(wt) is given by (1.11).

Below we show the existence of a continuous semiflow corresponding to(1.18) and (1.19). The existence and smoothness of an attractor with bound-ary damping alone is shown under additional hypotheses restricting the growthof the boundary damping. In the case of boundary damping one could con-sider damping affecting other boundary conditions such as free and simplysupported, see [CL10].

(C) Kirchhoff -Boussinesq plate with boundary damping. With thesame notations as above in a domain Ω ⊂ R2 we consider the equation

Mαwtt+∆2w+a(x)[g(wt)−α divG(∇wt)

]= div

[|∇w|2∇w

]+P (w) (1.21)

with the hinged dissipative boundary conditions (1.19). The internal dampingfunctions g and G satisfy (1.8). The boundary damping is the same as theprevious case, i.e., g0(s) ∼ g2s + |s|q−1s, q ≥ 1. The source P is same asabove. The associated energy function has the form (1.13). The energy balancerelation is given by (1.20).

When α > 0 the wellposedness of finite energy solutions with internaldamping follows from the observation that the model can be represented asa locally Lipschitz perturbation of monotone operator. This approach is alsoadaptable to boundary damping, like in the case of von Karman plate. Inthe case α = 0 the problem is much more delicate. While existence of finiteenergy solutions can be established with general form of the interior-boundarydamping, the uniqueness of solutions requires the linearity of the damping.In addition, continuous dependence on the data depends on time reversibilityof dynamics which, in turn, does not allow for boundary damping. In view ofthe above most of the questions asked are open in the presence of boundarydamping.

1.3 Well-posedness and generation of continuous flows

In this section we present several methods which enable to deal with Hadamardwell-posedness of PDE equations with supercritical Sobolev exponents. By su-percritical, we mean sources that may not belong to finite kinetic energy spacefor trajectories of finite energy. While existence of finite energy solutions isoften handled by various variants of Galerkin method, it is the uniqueness andcontinuous dependence on the data that is problematic. There is no unifiedtheory for the treatment of such problems, however there exist methods thatare applicable to classes of these problems.

In this section we concentrate on three models described in Section 1.2.1,where each of the model displays different characteristics and associated dif-ficulties. In order to cope with this, different methods need to be applied. We


provide a show case for the following methods which are critical for provingHadamard well-posedness in the examples cited:

• Interaction of superlinear sources via the damping.Illustration: wave equation.

• Cancelations-harmonic analysis and microlocal analysis methods.Illustration: Von Karman equations.

• Sharp control of Sobolev’s embeddings and duality scaling.Illustration: Kirchoff-Boussinesq plate.

1. In the first example the equation can be viewed as a perturbation of amonotone operator. However, the superlinearity of the source destroys locallyLipschitz character of the semilinear equation. In order to offset the difficulty,superlinear damping g(wt) is used. The interplay between the superlinearityof potential and kinetic energy lies in the heart of the problem. The presenceof this damping becomes critical in establishing well defined dynamical systemevolving on a finite energy phase space.2. In the second example, classical regularity results for von Karman nonlin-earity, when α = 0, fail to show that the perturbation of monotone operatoris locally Lipschitz. However, in this case, more subtle methods based on har-monic analysis and compensated compactness allow to show that the Airy’sstress function has hidden regularity property. This estimate allows to provethat, contrary to the original prediction, the semilinear term is locally Lips-chitz. This allows to prove, again, that the resulting dynamical system is wellposed on a finite energy space.3. In the third example, the source of the difficulties is restoring forcediv[|∇w|2∇w

]which fails to be locally Lipschitz with respect to finite energy

solutions in the case α = 0. By using method relying on logarithmic con-trol of Sobolev’s embeddings and topological shift of the energy (”Sedenko’smethod”) we are able to prove that the resulting system with linear dampingis well posed on finite energy space. Critical role in the argument is played bytime reversibility of the dynamics and linearity of the damping. This excludessuperlinear damping and boundary dissipation in the case α = 0.

Conclusion: These three canonical examples present three different methodson how to deal with the loss of local Lipschitz property and still be able toobtain well-posed flows defined on finite energy phase space. The methods pre-sented are transcendental and applicable to other dynamics displaying similarproperties. Some of the examples are given in Section 1.6.

1.3.1 Wave equation with a nonlinear interior damping-model in(1.1)

The statements of the results

With reference to the model (1.1) with a(x) ≥ a0 > 0 in Ω the followingassumption is assumed throughout.


Assumption 1 1. A scalar function g(s) is assumed to be of the form g(s) =g1s + g2(s), where g1 ∈ R and g2(s) is continuous and nondecreasing onR with g2(0) = 0

2. The source f(s) is either represented by a C1 function and such that|f ′(s)| ≤ C(1+|s|p−1) with 1 ≤ p ≤ 3, or else we have that f ∈ C2(R) and(i) there exist mg2 ,Mg2 > 0 such that the damping function g2 satisfiesthe inequality

mg2 |s|m+1 ≤ g2(s)s ≤Mg2 |s|m+1, |s| ≥ 1. (1.22)

for some m > 1, (ii) |f ′′(s)| ≤ C(1+ |s|p−2) with p satisfying the followingcompatibility growth condition

3 ≤ p < 6 and p ≤ 6m

m+ 1(1.23)

While in the case when the source exponent p ≤ 3 we adopt classical semigroupdefinition of the generalized solution (see, e.g., [CL08a]), for supercritical casesp > 3 we provide the following variational definition.

Definition 1 (Weak solution). Let (1.22) and (1.23) be in force. By a weaksolution of (1.1), defined on some interval (0, T ) with initial data (w0;w1)from H1(Ω)× L2(Ω), we mean a function u ∈ Cw(0, T ;H1(Ω)), such that

1. ut ∈ Lm+1((0, T )×Ω) ∩ Cw(0, T ;L2(Ω)).2. For all φ ∈ C(0, T ;H1

0 (Ω))∩C1(0, T ;L2(Ω))∩Lm+1([0, T ]×Ω), we havethat∫ T

0

∫Ω

(−utφt +∇u∇φ) dΩdt+

∫ T

0

∫Ω

g(ut)φ dΩdt

= −∫Ω

utφdΩ∣∣∣T0

+

∫ T

0

∫Ω

f(u)φ dΩdt. (1.24)

3. limt→0

(u(t)− u0, φ)1,Ω = 0 and limt→0

((ut(t)− u1, φ)) = 0 for all φ ∈ H10 (Ω).

Theorem 2. Let Assumption 1 be in force. We assume that initial data(w0;w1) possess the properties

w(0) = w0 ∈ H10 (Ω), wt(0) = w1 ∈ L2(Ω)

and for p > 5, w0 ∈ Lr(Ω) with r = 32 (p− 1).

Then

• There exist a unique, local (in time) solution w(t) of finite energy. This isto say: there exists T > 0 such that

w ∈ C(0, T ;H10 (Ω)), wt ∈ C(0, T ;L2(Ω)),

where w is a generalized solution when p ≤ 3 and w is a weak solutionwhen condition (1.22) holds with some m ≥ 1 (which is the case whenp ≥ 3).


• In this case 3 < p ≤ 5 weak solutions satisfy the energy relation in (1.3).• If p ≤ 3 and (1.22) holds with some m, then generalized solution satisfies

also the variational form (1.24) with the test functions φ from the class

C(0, T ;H10 (Ω)) ∩ C1(0, T ;L2(Ω)) ∩ L∞(QT ).

Moreover, the said solutions are continuously dependent on the initial data.• When p ≤ m the obtained solutions are global, i.e., T = ∞. The same

holds under dissipativity condition:

lim inf|s|→∞

−f(s)

s> −λ1 (1.25)

where λ1 is the first eigenvalue of −∆ with the zero Dirichlet data.• When p > 3, p > m and f(s) = |s|p−1s local solution has finite time

blow-up for negative energy initial data.

Remark 1. In the case when p ∈ [1, 3] one constructs solutions as the limits ofstrong semigroup solutions [CL07a]. These are referred to as generalized solu-tions [CL07a, CL08a]. It can be shown that generalized semigroup solutionsdo satisfy the variational form in question and variational weak solutions areunique [CL08a]. In order to justify this, it suffices to consider only the non-linear damping term g(s) with g1 = 0, since the latter provides a boundedlinear perturbation-thus it does not affect the limit passage. The energy in-equality provides L1 bound for g(wn). This allows to apply Dunford-Pettiscompactness criterion (see later) in order to transit with the weak limit in L1,and thus obtaining variational form of solutions (1.24) with appropriate testfunction as specified in the third part of Theorem 2.

Sketch of the proof

Without loss of generality we may assume a(x) = 1 and g1 = 0, so g2(s) = g(s)(these parameters have no impact on the proof).

Case 1: Critical and subcritical sources and arbitrary monotonedamping. In this case the proof is standard and follows from monotone op-erator theory. Setting the second order equation as a first order system

Wt +A(W ) = F (W ), t > 0, W (0) = (w0;w1) (1.26)

on the space H ≡ H10 (Ω)× L2(Ω) where W = (w;wt) and

A ≡(

0 −I−∆ g(·)

), F (W ) ≡

(0

f(w)

).

allows to the use of monotone operator theory (see, e.g., [Bar76, Sho97] for thebasic theory). Indeed, A is maximally monotone and F (W ) is locally Lipschitz


on H. This last statement is due to the restriction p ≤ 3 and Sobolev’sembedding H1(Ω) ⊂ L6(Ω). General theorem (see [CEL02] and also [CL10,Chapter 2]) allows to conclude local unique existence of semigroup solutions.

The arguments leading to variational characterization of weak solutionsare given in the Appendix [CL07a]. It can be based on weak L1 compactnesscriteria due to Danford and Pettis (see [DS58, Section IV.8]) which states thatthe set M ⊂ L1(Q) is relatively compact with respect to weak topology if andonly if this set is bounded and uniformly absolutely continuous, i.e.,

∀ ε > 0 ∃ δ > 0 : mes (E) ≤ δ ⇒ supf∈M

∣∣∣∣∫E

fdQ

∣∣∣∣ ≤ ε.We want to show that M = g(wn) is weakly L1(Q) compact for a sequenceof strong solutions which also converges almost everywhere. For this usingdissipation inequality

∫Qg(wn)wndQ ≤ C which (by splitting the integration

into |wn| ≤ R and |wn| ≥ R) implies that∫E

|g(wn)|dQ ≤ 1

R

∫Q

g(wn)wndQ+ gRmeas (E), ∀R > 0.

Case 2: supercritical source. In this case superlinearity of the damping isexploited. The key role is played by energy inequality obtained first for smoothsolutions corresponding to truncated problem where f is approximated by lo-cally Lipschitz function fK defined in [BL10]. Locally Lipschitz theory (Case 1above) allows to obtain the energy estimate

E(t) +

∫ t

0

((g(wt), wt))ds ≤ E(0) +

∫ t

0

((fK(w), wt))ds (1.27)

with E(t) ≡ 12 (||wt(t)||2 + ||∇w(t)||2). Exploiting the growth conditions im-

posed on g and f yields

E(t) + ‖wt‖m+1Lm+1(Qt)

≤ C[E(0) + ‖fK(w)‖

m+1m

Lm+1m

(Qt)

]and Sobolev’s embeddings H1(Ω) ⊂ L6(Ω) along with the conditionp(m+ 1) ≤ 6m implies

E(t) + ‖wt‖m+1Lm+1(Qt)

≤ C[E(0) + L

(t+

∫ t

0

‖w‖p(m+1)m

1,Ω ds

)].

The above estimate followed by a rather technical limit argument (with K →∞) [BL10] allows to establish local existence of weak solutions.

Energy identity. This is an important step of the argument. By using finitedifference approximation for the velocity wt(t) one shows that weak solutionssatisfy not only energy inequality but also energy identity. The detailed argu-ment is given in the proof of Lemma 2.3 [BL08a].


Uniqueness of the solutions is more subtle. We need to show that anysolution from the existence class, i.e. satisfying

w ∈ Cw(0, T ;H10 (Ω) ∩ C1

w(0, T ;L2(Ω)), wt ∈ Lm+1(QT )

is uniquely determined by the initial data. To this end let’s consider the dif-ference of two solutions: z ≡ w−u, where both w and u are finite energy weaksolutions specified as above with the same initial data (w0;w1). Denoting by

Ez(t) ≡ ||∇z||2 + ||zt||2

owing to energy inequality satisfied for weak solutions and using bounds im-posed on the source function f(s) one obtains after some calculations:

Ez(t) ≤∫ t

0

((f(w)− f(u), zt))ds ≡ Rf (u,w, z, t) (1.28)

The following lemma is critical:

Lemma 1. Let p < 5. Then ∀ε > 0 and ∀(w0;w1) ∈ H10 (Ω) × L2(Ω) such

that ‖w0‖1,Ω + ‖w1‖0,Ω ≤ R, there exist a constant Cε(R, T ) > 0 such that

|Rf (T )| ≤ εEz(T ) + Cε(R, T )[TEz(T )

+

∫ T

0

(1 + ‖ut(t)‖Lm+1(Ω) + ‖vt(t)‖Lm+1(Ω)

)Ez(t) dt

].

This lemma is specialized to the case when p < 5. For p ∈ [5, 6) there isanother term in the inequality above which accounts for the fact that thepotential energy term may not be in L1. In order to focus presentation weomit this term and refer the reader to the source [BL08a].

Proof. Using integration by parts we have that

Rf =((f(w)− f(u), z))∣∣∣t0−∫ t

0

((f ′(w)wt − f ′(u)ut, z))dτ

=((f(w)− f(u), z))∣∣∣t0−∫ t

0

((f ′(w)zt + (f ′(w)− f ′(u))ut, z))dτ

=

[((f(w)− f(u), z))− 1

2((f ′(w), z2))

]t0

+1

2

∫ t

0

((f ′′(w)wtz, z))dτ −∫ t

0

(((f ′(w)− f ′(u))ut, z))dτ

≤C∫Ω

z2(t)[1 + |u(t)|p−1 + |w(t)|p−1]dx

+ C

∫ t

0

∫Ω

z2(t)[1 + |u(t)|p−2 + |w(t)|p−2][|ut|+ |wt|]dxdτ.


It is convenient to use the following elementary estimate valid with any finiteenergy element z such that z(0) = 0:

||z(t)||2 =

∫Ω

∣∣∣ ∫ t

0

zt(s)ds∣∣∣2dx ≤ t∫ t

0

||zt(s)||2ds ≤ 2t

∫ t

0

Ez(s)ds. (1.29)

Estimate for∫ T0

∫Ωz2(t)|ut(t)| dxdt. We use Holder’s Inequality with p = 3

and q = 3/2 and obtain:∫ T

0

∫Ω

z2(t)|ut(t)| dQ ≤∫ T

0

‖z(t)‖2L6(Ω) · ‖ut(t)‖L3/2(Ω) dt. (1.30)

This leads to∫ T

0

∫Ω

z2(t)[|wt(t)|+ |ut(t)|] dQ ≤ C(R)

∫ T

0

Ez(t)dt. (1.31)

Estimate for∫Ω|u(T )|p−1z2(T ) dx. First, we write∫

Ω

|u(T )|p−1z2(T ) dx ≤ ||z(T )||2 +

∫Ω∩|u(T )|>1

|u(T )|p−1z2(T ) dx. (1.32)

The first term is estimated in (1.29). The argument for the second term is givenbelow. We consider here the supercritical case 3 < p < 5. The correspondingestimate for p ∈ [5, 6) is given in [BL08a].

Since |u(T )| > 1, there exists ε0 > 0 such that |u(T )|p−1 ≤ |u(T )|4−ε0 . Wechoose ε < ε0

4 and apply Holder’s inequality with q = 31+2ε and q = 3

2(1−ε) . We

then use Sobolev’s embeddings, the fact that (4− ε0) 32(1−ε) ≤ 6, interpolation

and Young’s inequality to obtain∫Ω∩|u(T )|>1

|u(T )|p−1z2(T ) dx

≤(∫

Ω

|z(T )|6

1+2ε dx) 1+2ε

3(∫

Ω

|u(T )|3(4−ε0)

2(1−ε) dx) 2(1−ε)

3

≤ C‖z(T )‖2H1−ε‖u(T )‖(4−ε0)L 3(4−ε0)2(1−ε)

(Ω) ≤ εEz(T ) + Cε(R)||z(T )||2.

(1.33)

Combining (1.32) with (1.29) and (1.33), we obtain the final estimate in thiscase: ∫

Ω

|u(T )|p−1z2(T ) dx ≤ εEz(T ) + Cε(R)T

∫ T

0

Ez(t) dt. (1.34)

Estimate for∫ T0

∫Ω|u(t)|p−2|ut(t)|z2(t) dQ. Since for |u(t)| ≤ 1, we obtain

the term estimated in the previous case, it is sufficient to look at

1 Dissipative hyperbolic-like evolutions 15∫ T

0

∫Ω

|u(t)|p−2|ut(t)|z2(t) dQ with Ω = Ω ∩ |u(t)| > 1.

We start by applying Holder’s inequality with q = 3 and q = 3/2:∫ T

0

∫Ω

|u(t)|p−2|ut(t)|z2(t)dxdt

≤∫ T

0

‖z(t)‖2L6(Ω)

[ ∫Ω

|u(t)|3(p−2)

2 |ut(t)|32 dx

] 23

dt.

(1.35)

We use Holder’s inequality again, with q = 4p−2 and q = 4

6−p , and notice that6

6−p ≤ m+ 1⇔ p ≤ 6mm+1 . Hence (1.35) becomes:∫ T

0

∫Ω

|u(t)|p−2|ut(t)|z2(t)dxdt ≤ C∫ T

0

Ez(t)‖u(t)‖p−2L6(Ω)‖ut(t)‖L 66−p

(Ω)dt

≤ C∫ T

0

Ez(t)‖u(t)‖p−2L6(Ω)‖ut(t)‖Lm+1(Ω)dt.

(1.36)

Since ‖u(t)‖1,Ω ≤ CR,T we obtain:∫ T

0

∫Ω

|u(t)|p−2|ut(t)|z2(t) dQ ≤ CR,T∫ T

0

Ez(t)‖ut(t)‖Lm+1(Ω) dt. (1.37)

Similarly, following the strategy used in the previous step, we obtain theestimate∫ T

0

∫Ωu

|u(t)|p−2|wt(t)|z2(t) dQ ≤ CR,T∫ T

0

Ez(t)‖wt(t)‖Lm+1(Ω) dt,

∫ T

0

∫Ωw

|w(t)|p−2|ut(t)|z2(t) dQ ≤ CR,T∫ T

0

Ez(t)|ut(t)|Lm+1(Ω) dt,

with Ωw = Ω ∩ |w(t)| > 1. This completes the proof of Lemma 1.

Completion of the proof: In (1.28), we use Lemma 1 to obtain:

Ez(T ) ≤ εEz(T ) + Cε(R, T )[TEz(T )

+

∫ T

0

(1 + ‖ut(t)‖Lm+1(Ω) + ‖vt(t)‖Lm+1(Ω)

)Ez(t) dt

] (1.38)

for every ε > 0. In (1.38), we choose ε and T such that ε+ Cε(R, T )T < 1/2(for uniqueness it is enough to look at a small interval for T , since the processcan be reiterated) and apply Gronwall’s inequality to obtain that Ez(T ) = 0for all T ≤ Tmax, where [0, Tmax] is a common existence interval for bothsolutions w and u.


The above argument completes the proof of uniqueness of finite energysolutions. This result along with energy equality leads to Hadamard well-posedness. Details are in [BL08a, BL10].

Finite-time blow up of solutions is established in [BL08b]. This is accom-plished by showing that appropriately constructed anti-Lyapunov function[STV03, Vit99] blows up in a finite time.

Generalizations-Extensions

• In the subcritical and critical case additional regularity of weak solutionsequipped with more regular initial data can be established. Quantitativestatements are in [CL07a].

• The wave model discussed is equipped with zero Dirichlet data. However,the same result holds for Neumann or Robin boundary conditions.

• In the subcritical and critical case, p ≤ 3, the same results hold withlocalized damping. This is to say when supp a(x) ⊂ Ω0 ⊂ Ω.

• The analysis of wellposedness for the same model in the case when solu-tions are confined to a potential well are given in [BRT11].

Open question: Obtain the same result with partially localized dampingsupp a(x) ⊂ Ω0 ⊂⊂ Ω in the supercritical case p > 3.

1.3.2 Von Karman equation with interior damping - model (1.4)

The statement of the results

With reference to the model (1.4) the following assumption is assumedthroughout.

Assumption 3 1. Scalar function g(s) is assumed to be continuous andmonotone on R with g(0) = 0.

2. In the case α > 0 we assume that the damping G has the form G(s, σ) =(g1(s); g2(σ)), where gi(s), i = 1, 2, are continuous and monotone on Rwith gi(0) = 0. Moreover, they are of polynomial growth, i.e., |gi(s)| ≤C(1 + |s|q−1) for some q ≥ 1.

3. The source P (w) is assumed locally Lipschitz from H2(Ω) into [Hα(Ω)]′,where

Hα(Ω) =

L2(Ω) in the case α = 0;H1

0 (Ω) in the case α > 0.(1.39)

Below we also use the following refinement of the hypothesis concerning thesource P .

Assumption 4 We assume that P (w) = −P0(w) + P1(w), where

1. P0(w) is a Frechet derivative of the functional Π0(w) and the propertyholds: there exist a, b ∈ R and ε > 0 such that

Π0(w) + a‖w‖22−ε,Ω ≥ b, ((P0(w), w)) + a‖w‖22−ε,Ω ≥ b, ∀w ∈ H20 (Ω);


2. there exists K > 0 and ε > 0 such that ‖P1(w)‖ ≤ K‖w‖2−ε,Ω for allw ∈ H2

0 (Ω).

Remark 2. A specific choice of interest in applications is P (w) = [F0, w] − p,where [·, ·] denotes the von Karman bracket (see (1.7)), F0 ∈ H3+δ(Ω) ∩H1

0 (Ω), δ > 0, and p ∈ L2(Ω). The term F0 models in-plane forces in theplate and p is a transversal force. The given source complies with all thehypotheses stated. In the case α > 0 the hypotheses concerning P (w), F0 andp can be relaxed. However we do not pursue this generalizations and refer to[CL10].

In addition to the concept of generalized (semigroup) solutions [CL08a,CL10] we can also define weak solutions.

Definition 2 (Weak solution). By a weak solution of (1.4), defined on someinterval [0, T ], with initial data (u0;u1) we mean a function

u ∈ Cw(0, T ;H20 (Ω)), ut ∈ Cw(0, T ; Hα(Ω))

such that

1. For all φ ∈ H20 (Ω),

((ut(t), φ)) + α((∇ut(t),∇φ)) +

∫ t

0

((∆u,∆φ)) dt (1.40)

+

∫ t

0

[((g(ut), φ)) + α((G(∇ut(t)),∇φ))

]dt

=((u1, φ)) + α((∇u1,∇φ)) +

∫ t

0

((P (u) + [F (u), u], φ)) dt.

2. limt→0

(u(t)− u0, φ)2,Ω = 0.

Here as above Cw(0, T ;Y ) denotes a space of weakly continuous functions withvalues in a Banach space Y .

Theorem 5. Under the assumption 3 for all initial data

w(0) = w0 ∈ H20 (Ω), wt(0) = w1 ∈Hα(Ω)

(with Hα(Ω) defined by (1.39)) there exist a unique, local (in time) generalized(semigroup) solution of finite energy, i.e., there exists T > 0 such that

w ∈ C(0, T ;H20 (Ω)), wt ∈ C(0, T ; Hα(Ω)).

Moreover,

• If, in addition in the case α = 0 the damping g(s) is of some polynomialgrowth, generalized solution becomes weak solution. A weak solution is alsocontinuously dependent of the initial data.

• The solutions are global provided Assumption 4 holds. In this case a boundfor the energy of solution is independent of time horizon.


Sketch of the proof

We concentrate on a more challenging case when α = 0. The case α > 0 canbe found in [CL10, Chapter 3].

For the proof of this result the support of the damping described by a(x)plays no role. Thus, without loss of generality we may assume that a(x) = 1.

The following two lemmas describing properties of von Karman bracketscan be found in [CL10]. The first Lemma is critical for the proof of well-posedness and the second Lemma implies boundedness of solutions that isindependent on time horizon.

Lemma 2. Let ∆−2 denotes the map defined by z ≡ ∆−2f iff

∆2z = f in Ω and f =∂

∂nf = 0 on Γ.

Then‖∆−2[u, v]‖W 2,∞(Ω) ≤ C‖u‖2,Ω‖v‖2,Ω ,

where the von Karman bracket [u, v] is defined by (1.7). In particular, for theAiry stress function F we have the estimate ‖F (w)‖W 2,∞(Ω) ≤ C‖w‖22,Ω.

Notice that standard result [Lio69] gives

‖∆−2[u, v]‖3−ε,Ω ≤ C‖u‖2,Ω‖v‖2,Ω

which does not imply the result stated in the lemma.The second lemma [CL10] has to do with a control of low frequencies by

nonlinear term.

Lemma 3. Let u ∈ H2(Ω)∩H10 (Ω). Then for every ε > 0 there exists Mε > 0

such that||u||2 ≤ ε

(||∆u||2 + ||∆F (u)||2

)+Mε.

Equipped with these two lemma, the proof of global well-posedness followsstandard by now procedure:

Step 1: Establish maximal monotonicity of the operator

A ≡(

0 −I∆2 g(·)

)on the space H ≡ H2

0 (Ω)× L2(Ω) with

D(A) = (u; v) ∈ H20 (Ω)× L2(Ω) : ∆2u+ g(v) ∈ L2(Ω)

This follows from a standard argument in monotone operator theory [Bar76,Sho97].

Step 2: Consider nonlinear term as a perturbation of A:


F (W ) ≡(

0[F (w), w] + P (w)

)

Step 3: Show that F (W ) is locally Lipschitz on H. This can be done withthe help of Lemma 2 which implies the following estimate

||[F (u), u]− [F (w), w]|| ≤ C(1 + ‖u‖22,Ω + ‖w‖22,Ω)‖u− w‖2,Ω . (1.41)

The above steps lead to well-defined semigroup solutions, defined locallyin time, which are unique and satisfy local Hadamard well-posedness.

The final step is global well-posedness for which a priori bounds are handy.The needed a priori bound results from the property of von Karman bracketdescribed in Lemma 3. Indeed, in order to claim global well-posedness, itsuffices to notice that the relation

(([F (w), w] + P (w), wt)) = − d

dt

[1

2||∆F (w)||2 +Π0(w)

]+ ((P1(w), wt))

implies the a priori bounds for the energy function E (t) given by (1.9) withα = 0 via energy inequality and Gronwall’s lemma.

Stationary solutions. For description of the structure of the global attrac-tor we also need consider stationary solutions to problem (1.4). We define astationary weak solution as a function w ∈ H2

0 (Ω) satisfying in variationalsense the following equations

∆2w = [F (w), w] + P (w) in Ω, w =∂

∂nw = 0 on Γ, (1.42)

where the Airy stress function F (w) solves the elliptic problem (1.6).

Proposition 1. Let Assumption 4 be in force. In addition assume that P isweakly continuous mapping from H2

0 (Ω) into H−2(Ω). Then problem (1.42)has a solution. Moreover, the set N∗ of all stationary solutions is compact inH2

0 (Ω).

For the proof of this proposition we refer to [CR80], see also [CL10, Cia00].We note problem (1.42) may have several solutions [CR80]. However in ageneric situation the set N∗ is finite (see a discussion in [CL10, Chapter 1]).


1. By assuming more regular initial data one obtains regular solutions. Pre-cise quantitative statement of this is given in [CL10]. Regular solutionsare global in time.

2. Related results hold when α > 0. The analysis here is simpler and thereis no need for sharp results in Lemma 2, see [CL10].


3. One can consider other boundary conditions such as simply supported orfree or combination thereof, [CL10].

4. Damping can be partially supported in Ω. This has no effect on the ar-guments.

5. A related wellposedness result holds for non-conservative models with ad-ditional energy level terms that are non-conservative (for instance ∇w ·Ψfor some smooth vector Ψ).

Open question

Consider the full von Karman system that consists of system of 2D elasticitycoupled with von Karman equations. This model accounts for in-plane ac-celerations. As such, there is no decomposition using Airy’s stress function.The nonlinearities entering are super-critical (even in the rotational case whenα > 0) and hidden regularity of Airy’s stress function plays no longer any role.For such model with α > 0 one can still prove existence and uniqueness ofsolutions, by appealing to “Sedenko’s method” (see [Sed91]). Even more, fullHadamard well-posedness and energy identity can also be proved in that case[Las98, KL02]. However, the problem is entirely open in the non-rotationalcase α = 0. In this latter case only existence of weak solutions, obtained byGalerkin method, is known.

1.3.3 Kirchhoff-Boussinesq model with interior damping - modelin (1.12)

As before we concentrate on the most demanding case when α = 0. For thecase α > 0 we refer to [CL08a]. The main challenge of this model is thepresence of restorative force term div

[|∇w|2∇w

]which is not in L2(Ω) for

finite energy solutions w. This is due to the failure of Sobolev’s embeddingH1 ⊂ L∞ in two dimensions. We also assume, without loss of generality forthe well-posedness, that a(x) ≡ 1.

The statement of the results

With reference to the model (1.12) with α = 0 the following assumption isassumed throughout.

Assumption 6 1. g(s) is continuous and monotone on R with g(0) = 0.In addition g(s) is of polynomial growth at infinity and also a(x) ≡ 1(without loss of generality for the well-posedness).

2. The source P (w) is assumed locally Lipschitz from H2(Ω) into L2(Ω).

Below we deal with weak solutions.


Definition 3 (Weak solution). By a weak solution of (1.12), with α = 0and a(x) ≡ 1, defined on some interval (0, T ), with initial data (u0;u1) wemean a function u ∈ Cw(0, T ;H2

0 (Ω)) such that ut ∈ Cw(0, T ;L2(Ω)) withg(ut) ∈ L1(QT ) and

1. For all φ ∈ H20 (Ω),∫

Ω

ut(t)φdΩ +

∫ t

0

∫Ω

(∆u∆φ) dΩdt+

∫ t

0

∫Ω

g(ut)φ dΩdt

=

∫Ω

u1φdΩ +

∫ t

0

∫Ω

[P (u)φ− |∇w|2(∇w,∇φ)

]dΩdt. (1.43)

2. limt→0

(u(t)− u0, φ)2,Ω = 0.

Theorem 7. Let α = 0. Under the assumption 6 for all initial data

w(0) = w0 ∈ H20 (Ω), wt(0) = w1 ∈ L2(Ω)

there exist a local (in time) weak solution of finite energy. This is to say: thereexists T > 0 such that

w ∈ Cw(0, T ;H20 (Ω)), wt ∈ Cw(0, T ;L2(Ω)). (1.44)

Moreover

• Under the additional assumption that g(s) is linear the said solution isunique. In addition energy identity is satisfied for all weak solutions whichare also Hadamard well-posed (locally).

• The solutions are global under the following dissipativity condition: forevery δ > 0 there exists Cδ > 0 such that∫ t

0

((P (w(s)), wt(s)))ds ≤ δE (t) + Cδ

[E (0) +

∫ t

0

E (s)ds

](1.45)

for any function w(t) possessing the properties in (1.44).

Remark 3. A specific choice of interest in applications is Boussinesq sourcegiven by P (w) = ∆[w2]. This source complies with all the hypotheses stated.Indeed,

((∆w2, wt)) =((∆w,d

dtw2)) + 2((|∇w|2, wt))

=d

dt((∆w,w2))− ((∆wt, w

2)) + 2((|∇w|2, wt)),

which then gives

((∆w2, wt)) = − d

dt((|∇w|2, w)) + ((|∇w|2, wt)). (1.46)


Since

‖w(t)‖2 ≤ ‖w(0)‖2 + 2

∫ t

0

‖w(τ)‖‖wt(τ)‖dτ ≤ C[E (0) +

∫ t

0

E (τ)dτ

],

relation (1.46) quickly implies the conclusion desired in (1.45).

1.3.4 Sketch of the proof

As mentioned before the challenge in the proof of Theorem 7 is to handle thelack of local Lipschitz condition satisfied by the restorative force.

We explain the main steps. Full details are given in [CL11, CL06b].

Step 1 (Existence): It follows via Faedo-Galerkin method. This step isstandard.

Step 2 (Uniqueness): For this we use Sedenko’s method which based onwriting the difference of two solutions in a split form as projection and copro-jection on some finite dimensional space and estimating.

We start with some preliminary facts.We introduce the operator A in L2(Ω) by the formula A u = ∆2u with

the domain D(A ) = H4(Ω) ∩ H20 (Ω). The operator A is a strictly positive

self-adjoint operator with the compact resolvent. Let ek be the orthonormalbasis in L2(Ω) of eigenvectors of the operator A and λk be the correspond-ing eigenvalues:

A ek = λkek, k = 1, 2, ....; 0 ≤ λ1 ≤ λ2 ≤ . . .

W also note that for every s ∈ [0, 1/2] we have D(A s) = H4s0 (Ω), s 6=

1/8, 3/8. Moreover, the corresponding Sobolev norms are equivalent to thegraph norms of the corresponding fractional powers of A , i.e.

c1‖A su‖ ≤ ‖u‖4s ≤ c2‖A su‖, u ∈ D(A s), (1.47)

for all admissible s ∈ [0, 1/2].The following assertion is critical for the proof.

Lemma 4 ([BC98]). Let PN be the projector in L2(Ω) onto the spacespanned by e1, e2, ..., eN and f(x) ∈ D(A 1/4). Then there exists N0 > 0such that

maxx∈Ω|(PNf)(x)| ≤ C · log(1 + λN )1/2 ‖ f ‖1 (1.48)

for all N ≥ N0. The constant C does not depend on N .

Remark 4. For the first time a relation similar to (1.48) was used for unique-ness in some shell models in [Sed91]. Latter the same method was applied forcoupled 2D Schrodinger and wave equations [CS05, CS12a], for the inertial2D Cahn-Hilliard equation [GSZ09] and also for some models of fluid-shellinteraction [CR13b]. One can show (see [CL10, Appendix A] and the discus-sion therein) that (1.48) is equivalent to some Bresis–Gallouet [BG80] typeinequality.


Assume that α = 0, a(x) ≡ 1 and g(s) = ks in (1.4). Let w1(t) and w2(t) betwo weak solutions of the original problem (1.12) with the same initial dataand w(t) = w1(t) − w2(t). Then wN (t) = PNw(t) is a solution of the linear,but nonhomogenous problem

wtt + kwt +∆2w = (PNM)(t), x ∈ Ω, t > 0, (1.49)

with the boundary and initial conditions

w|∂Ω =∂w

∂n

∣∣∣∂Ω

= 0, w|t=0 = 0, ∂tw|t=0 = 0.

Here

M(t) = div[|∇w1(t)|2∇w1(t)− |∇w2(t)|2∇w2(t)

]+ [P (w1(t))− P (w2(t))] .

Multiplying the equation (1.49) by A −1/2wt and integrating we obtain that

||A −1/4PNwt(t)||2+||A 1/4PNw(t)||2 ≤ C∫ t

0

||A −1/4M(τ)||||A −1/4wt(τ)||dτ

for all t ∈ [0, T ]. From here after accounting for (1.47) we obtain

‖ wt(t) ‖2−1,Ω + ‖ w(t) ‖21≤ C ·∫ t

0

‖M(τ) ‖−1,Ω · ‖ wt(τ) ‖−1,Ω dτ.

In particular this implies that

‖ w(t) ‖1,Ω≤ C ·∫ t

0

‖M(τ) ‖−1,Ω dτ. (1.50)

The inequality above is the basis for further estimates. Below we also use theestimate which follows from the definition of weak solutions:

supt∈[0,T ]

‖w1(t)‖2,Ω + ‖w2(t)‖2,Ω ≤ R, (1.51)

where R > 0 is a constant. Using Lemma 4 we can estimate the quantity‖M(t)‖−1 in the following way:

‖M(t)‖−1 ≤ C1 · log(1 + λN )· ‖ w(t) ‖1 +C2 · λ−s/4N+1 (1.52)

for some 0 < s < 1 and with the constants C1 and C2 depending on R from(1.51) (for details we refer to [CL06b]). Therefore it follows from (1.50) and(1.52) that ψ(t) = ‖w(t)‖1 satisfies the inequality

ψ(t) ≤ C1 · log(1 + λN )

∫ t

0

ψ(τ)dτ + C2 · T · λ−s/4N+1 , t ∈ [0, T ],

for some 0 < s < 1. Thus using Gronwall’s lemma we conclude that


ψ(t) ≤ C2 · T · λ−s/4N+1 · (1 + λN )C1t, t ∈ [0, T ].

If we let N → ∞, then for 0 ≤ t < t0 ≡ s · (4C1)−1 we obtain ψ ≡ 0. Thusw1(t) ≡ w2(t) for 0 ≤ t < t0. Now we can reiterate the procedure in orderto conclude that w1(t) ≡ w2(t) for all 0 ≤ t ≤ T , where T is the time ofexistence. This completes the proof of uniqueness.

Step 3 (Energy identity): Energy inequality relies on a standard weaklower-semicontinuity argument. Since the system is time reversible, one ob-tains energy inequality for the backward problem. Combining the two inequal-ities: forward and backward, leads to energy identity (1.10) with E given by(1.13) satisfied by weak solutions. Details are given in [CL11].

Step 4: Equipped with uniqueness and energy identity standard argumentfurnishes Hadamard well-posedness.

Step 5: The last step is to extend local (in time) solutions to the global ones.This is based on a priori bounds postulated by nonlinearities in (1.45).


1. Higher regularity of solutions can be proved by assuming more regularinitial data. Quantitative statements are given in [CL11].

2. Rotational models, when α > 0, can be considered without extra difficulty.In fact, in this subcritical case one obtains full Hadamard wellposednessalso in a presence of nonlinear damping subject to the same assumptionsas in the von Karman case. Some details can be found in [CL08a].

3. We can also consider different boundary conditions such as hinged andfree. Free boundary conditions are most challenging due to intrinsic non-linearity on the boundary, [CL06b].

4. More general structures of restoring forces can be also considered, see[CL06a, CL06b, CL08a, CL11].

5. The support of the damping may be localized to a small (or even empty)subset of Ω0 ⊂ Ω. This will not affect a finite time behavior of solutions.

Open problem: Uniqueness of weak solutions with nonlinear damping.

1.3.5 Wave equation with boundary source and damping - modelin (1.14)

With reference to the model (1.14) and (1.15), where, for simplicity, we takeg ≡ 0 and f ≡ 0, the following assumption is assumed throughout.

Assumption 8 1. Scalar function g0(s) is assumed to be continuous andmonotone on R with g0(0) = 0.


2. The source h(s) is represented by a C2 function such that |h′′(s)| ≤ C(1+|s|k−2), where 2 ≤ k < 4, and the following growth condition is imposedon the damping g0(s) with the constants mg0 ,Mg0 > 0:

mg0 |s|q+1 ≤ g0(s)s ≤Mg0 |s|q+1, |s| ≥ 1, with q ≥ k

4− k≥ 1. (1.53)

When the damping is sublinear i.e., q ∈ (0, 1), then the the source isrequired to satisfy |h′(s)| ≤ C(1 + |s|k−1), 1 ≤ k < 4 and the condition in(1.53) should be satisfied for this q ∈ (0, 1).

Definition 4 (Weak solution). By weak solution of problem (1.14) and(1.15) with g ≡ 0 and f ≡ 0, defined on some interval (0, T ) with ini-tial data (u0;u1), we mean a function u ∈ Cw(0, T ;H1(Ω)) such that ut ∈Cw(0, T ;L2(Ω)) and

1. ut ∈ Lq+1(ΣT ), where ΣT = [0, T ]× Γ ,2. For all φ ∈ C(0, T,H1(Ω)) ∩ C1(0, T ;L2(Ω)) ∩ Lq+1(ΣT ),∫ T

0

∫Ω

(−utφt +∇u∇φ) dΩdt+

∫ T

0

∫Γ

g0(ut)φ dΩdt

= −∫Ω

utφdΩ∣∣∣T0

+

∫ T

0

∫Γ

h(u)φ dΩdt. (1.54)

3. limt→0

(u(t)− u0, φ)1,Ω = 0 and limt→0

((ut(t)− u1, φ)) = 0 for all φ ∈ H1(Ω).

Theorem 9. Let f ≡ 0, g ≡ 0 and Assumption 8 be in force. Let initial data(w0;w1) be such that

w(0) = w0 ∈ H1(Ω), wt(0) = w1 ∈ L2(Ω),

and also w0|Γ ∈ L2k−2(Γ ) when k > 3. Then there exist a unique, local (intime) weak solution of finite energy. This is to say: there exists T > 0 suchthat

w ∈ C(0, T ;H1(Ω)), wt ∈ C(0, T ;L2(Ω)).

Moreover,

• When k ≤ 3 the energy identity holds for weak solutions and weak solutionis continuously dependent on the initial data.

• When k ≤ q the obtained solutions are global, i.e., T =∞. The same holdsunder dissipativity condition: −h(s)s ≥ 0.

• When 1 < k ≤ 3, k > q and h(s) = |s|k−1s local solution blows up in afinite time for negative energy initial data.

Remark 5. In the case when h(s) = αs, no growth conditions imposed ong0 are required. In fact, in that case the obtained solution is the semigroupgeneralized solution.


When |h′′(s)| ≤ c the variational form of the solution can be obtained withmore relaxed hypotheses imposed on the high frequencies of the damping. Forinstance, we can assume

lim inf|s|→∞

g0(s)

s> 0, |g0(s)| ≤ c[1 + |s|3] (1.55)

Under the above condition one can show that the generalized solution satisfiesalso variational form (1.54) with the test functions φ ∈ C(0, T ;H1(Ω)) ∩C1(0, T ;L2(Ω)) ∩ C(ΣT ) [CL07a]

Reference to the proofs

Notice that the damping g0 is assumed active for all values of the parameterk > 1. This is unlike the interior source where only supercritical values ofthe source require presence of the damping. In the boundary case, however,Lopatinski condition is not satisfied for the Neumann problem and this neces-sitates the presence of the damping which, in some sense, forces Lopatinskicondition [Sak70]. This is manifested by the fact that H1(Q) solutions of waveequation with Neumann boundary conditions do not possess H1(Σ) boundaryregularity (unlike Dirichlet solutions). It is the presence of the damping which,in some sense, recovers certain amount of boundary regularity [LT00, Vit02a].

The proof of Theorem 9 follows similar conceptual lines as in the interiorcase. There are however few subtle differences due to the presence of undefinedtraces in the equation. In the case when |h′′(s)| ≤ C and ms2 ≤ g0(s)s ≤Ms4

the proof is given in [CL07a]. For the remaining values of the parametersHadamard wellposedness is proved in [BL08a, BL10]. The analysis of theboundary case is more demanding than in the interior case. The correspondingdifferences are clearly exposed in the structure of a counterpart to Lemma 1,which in the boundary case has a different form given in Lemma 4.2 in [BL08a].

The last statement in Theorem 9 regarding finite time blow up of energyis proved in [BL08b].


1. Additional regularity of solutions corresponding to more regular initialconditions when q ≤ 3 and |h′′(s)| ≤ C is given in [CL07a].

2. One could obtain a version of Theorem 9 that incorporates both dampings:internal and boundary. This is done in [BL08a, BL10].

3. Well-posedness theory for small data taken from potential well (k ≤ 3) isalso available [BRT11, Vit02b].

Open questions:

1. Well-posedness of finite energy solutions theory without the boundarydamping when k ≤ 2 (so that h(u) ∈ L2(Γ ) for finite energy solutions).


2. Interplay between boundary and interior damping. Is it possible to showwell-posedness of finite energy solutions for supercritical case p > 3 withboundary damping only (i.e., g = 0)?

3. The same question asked for finite time blow results. Does boundarysource alone lead to blow up of energy in the presence of a boundarydamping when q < k?

4. Boundedness of solutions when time goes to infinity and sources are gen-erating energy. Quantitative description of the behavior of global solutions- when p ≤ m and k ≤ q.

1.3.6 Von Karman equation with boundary damping - model(1.18)

With reference to the model (1.18) and (1.19) the following assumption isassumed throughout.

Assumption 10 1. Scalar function g(s) and g0(s) are assumed to be con-tinuous and monotone on R with 0 at 0. The rotational damping G (thecase α > 0) satisfies Assumption 3(2).

2. The source P (w) is assumed locally Lipschitz from H2(Ω) into H−1(Ω)when α > 0 and from H2(Ω) into L2(Ω) when α = 0 (see Remark 2concerning a specific choice of interest in applications).

We concentrate on a more challenging case when α = 0. In addition to theconcept of generalized (semigroup) solutions we also define weak solutions.

Definition 5 (Weak solution). By a weak solution of (1.18) and (1.19) withα = 0 and with initial data (u0;u1), defined on some interval (0, T ), we meana function u ∈ Cw(0, T ;H2(Ω)∩H1

0 (Ω)) such that ut ∈ Cw(0, T ;L2(Ω)) withg(ut) ∈ L1(QT ), g0((∂/∂n)ut) ∈ L1(ΣT ) and

1. For all φ ∈ H2(Ω) ∩H10 (Ω),∫ t

0

∫Ω

(∆u∆φ) dΩdτ +

∫ t

0

∫Ω

g(ut)φ dΩdτ +

∫ t

0

∫Γ

g0(∂

∂nut)

∂

∂nφdΓdτ

= −∫Ω

(ut(t)− u1)φdΩ +

∫ t

0

∫Ω

(P (u) + [F (u), u])φ dΩdτ. (1.56)

2. limt→0

(u(t)− u0, φ)2,Ω = 0 and limt→0

((ut(t)− u1, φ)) = 0.

Theorem 11. Let α = 0. Under Assumption 10 for all initial data

w(0) = w0 ∈ H2(Ω) ∩H10 (Ω), wt(0) = w1 ∈ L2(Ω)

there exist a unique, local (in time) generalized (semigroup) solution w offinite energy, i.e.,

∃T > 0 : w ∈ C(0, T ;H2(Ω)), wt ∈ C(0, T ;L2(Ω)).

Moreover


• If, in addition, the damping g(s) and g0(s) are of some polynomial growthand (g0(x)− g0(y))(x− y) ≥ c|x− y|r for some r ≥ 1, then a generalizedsolution becomes also weak solution. Weak solution is also continuouslydependent on the initial data.

• The solutions are global under Assumption 4 concerning P (w) with theenergy bound independent of the time horizon.

Reference to the proof

Well-posedness of Von Karman plates with the boundary damping occurringin the moments have been proved in [CL10]. See also [JL99, HL95].

1.3.7 Generalizations

1. Additional regularity of finite-time solutions for more regular initial data,[CL10, HL95].

2. Similar problems can be formulated for plate equations with the dampingacting in shears and torques [Lag89]. Thus the boundary conditions underconsideration are ”free” with the feedback control given by g0(wt) actingon the highest third order boundary conditions [CL04b, CL07b].

3. Combination of interior and boundary damping can also be considered bycombining the methods.

4. Model with rotational inertial forces α > 0 subject to boundary damping,[CL10, HL95].

5. Regular solutions for infinite time interval can be obtained [CL10].

Open problem: Are polynomial bounds imposed on g(s) and g0(s) andstrong coercivity condition imposed on g0 necessary for obtaining weak so-lutions?

1.3.8 Kirhhoff-Boussinesq equation with boundarydamping-model (1.21)

Rotational case α > 0

In the case when rotational inertia are retained in the model, α > 0, thenonlinear terms are subcritical and the well-posedness theory can be carriedout along the same lines as in the case of von Karman equations. For in-stance, when considered (1.21) with g = 0, G = 0 and the following boundarydamping

w = 0, ∆w = −g0(∂

∂nwt) on Σ, (1.57)

where g0(s) is continuous and monotone, full Hadamard local well-posednesscan be established.


Theorem 12. Under the assumption 10 with g = 0 and α > 0 for all initialdata

w(0) = w0 ∈ H2(Ω) ∩H10 (Ω), wt(0) = w1 ∈ H1

0 (Ω)

there exist a unique, local (in time) generalized (semigroup) solution w offinite energy, i.e.,

∃T > 0 : w ∈ C(0, T ;H2(Ω) ∩H10 (Ω)), wt ∈ C(0, T ;H1

0 (Ω)).

Moreover

• If, in addition, the damping g0(s) are of some polynomial growth and(g0(x) − g0(y))(x − y) ≥ c|x − y|r for some r ≥ 1, then a generalizedsolution becomes also weak solution. Weak solution is also continuouslydependent on the initial data.

• The solutions are global under the dissipativity hypothesis in (1.45).

The proof of this theorem is along the same lines as in the case of von Karmanequations with boundary damping.

Non-rotational case: α = 0

As we have seen already before, this case is much more delicate even in thecase of interior damping. The nonlinear source term is supercritical. While thisdifficulty was overcome in the case of linear interior damping, the presenceof boundary damping brings new set of issues. This is both, at the level ofuniqueness and continuous dependence. More specifically, in the case g ≡ 0:

• Existence of finite energy solutions

w ∈ L∞(0, T ;H2(Ω) ∩H10 (Ω)), wt ∈ L∞(0, T ;L2(Ω))

defined variationally and locally in time with initial data

w(0) = w0 ∈ H2(Ω) ∩H10 (Ω), wt(0) = w1 ∈ L2(Ω).

can be proved by Galerkin method with a boundary damping g0(s) of somepolynomial growth and such that (g0(x) − g0(y))(x − y) ≥ c|x − y|r forsome r ≥ 1.

• Uniqueness of solutions can be established only in the case of linear damp-ing, i.e., for g0(s) = as. This can be done by adapting Sedenko’s method[Sed91] as in [Las98, KL02] where full Von Karman system was considered.The latter displays similar difficulties when dealing with well-posedness.

• Continuous dependence on the data can be proved only in a weak topology.The difficulty lies in the fact that time reversibility of the flow is lost withthe presence of boundary damping. This latter property was critical inproving energy identity for weak solutions in the case of internal lineardamping.



1. Existence and uniqueness of regular solutions in H4(Ω) × H2(Ω) (whenα = 0). This can be accomplished by taking advantage of higher topologiesfor the state, hence avoiding the problem of supercriticality. This methodhas been pursued in [CL11] in the case of interior damping. However, thearguments are applicable to the case of boundary damping as well.

2. Other types of boundary damping -such as ocurring in “free” [CL10]boundary conditions can also be considered. In the case rotational case,well-posedness results are complete - in line with Theorem 12. In the non-rotational case, comments presented above apply to this case as well (withthe same limitations regarding linearity of the damping). There is how-ever an additional difficulty resulting from the fact that free boundaryconditions provide intrinsically nonlinear contribution on the boundary.This, however, is of lower order and can be handled as in [CL06b, CL11].

3. Combination of internal damping and boundary damping can be treatedby combining the results available for each case separately.

4. Sources that are nonconservative also can be considered, but these areoutside the scope of these lectures, see [CL10].

Open problems:

1. Hadamard well-posedness with linear boundary damping in the case α = 0.Due to the loss of time reversibility, the arguments used for the internaldamping are no longer applicable.

2. Ultimately: uniqueness and Hadamard well-posedness in the presence ofnonlinear damping g0 in the case α = 0 and boundary conditions (1.57),where g0(s) is monotone and -say- of linear growth at infinity. The argu-ments used so far for the uniqueness of weak solutions rely critically onlinearity of the damping. Thus, the case described above appears to becompletely open.

1.4 General tools for studying attractors

In this section we describe several approaches to the study of long-time be-havior of hyperbolic-like systems described above. For the general discussionof long-time behavior of systems with dissipation we refer to the monographs[BV92, Chu99, Hal88, Lad91, SY02, Tem88].

1.4.1 Basic notions

By definition a dynamical system is a pair of objects (X,St) consisting of acomplete metric space X and a family of continuous mappings St : t ∈ R+of X into itself with the semigroup properties:


S0 = I, St+τ = StSτ .

We also assume that y(t) = Sty0 is continuous with respect to t for anyy0 ∈ X. Therewith X is called a phase space (or state space) and St is calledan evolution semigroup (or evolution operator).

Definition 6. Let (X,St) be a dynamical system.

• A closed set B ⊂ X is said to be absorbing for (X,St) iff for any boundedset D ⊂ X there exists t0(D) such that StD ⊂ B for all t ≥ t0(D).

• (X,St) is said to be (bounded, or ultimately) dissipative iff it possessesa bounded absorbing set B. If X is a Banach space, then a value R > 0 issaid to be a radius of dissipativity of (X,St) if B ⊂ x ∈ X : ‖x‖X ≤ R.

• (X,St) is said to be asymptotically smooth iff for any bounded set Dsuch that StD ⊂ D for t > 0, there exists a compact set K in the closureD of D, such that

limt→+∞

dXStD |K = 0, (1.58)

where dXA|B = supx∈A distX(x,B).

A set D ⊂ X is said to be forward (or positively) invariant iff StD ⊆ Dfor all t ≥ 0. It is backward (or negatively) invariant iff StD ⊇ D for allt ≥ 0. The set D is said to be invariant iff it is both forward and backwardinvariant; that is, StD = D for all t ≥ 0.

Let D ⊂ X. The setγtD ≡

⋃τ≥t

SτD

is called the tail (from the moment t) of the trajectories emanating from D.It is clear that γtD = γ0StD.

If D = v is a single point set, then γ+v := γ0D is said to be a positivesemitrajectory (or semiorbit) emanating from v. A continuous curve γ ≡u(t) : t ∈ R in X is said to be a full trajectory iff Stu(τ) = u(t + τ) forany τ ∈ R and t ≥ 0. Because St is not necessarily an invertible operator, afull trajectory may not exist. Semitrajectories are forward invariant sets. Fulltrajectories are invariant sets.

To describe the asymptotic behavior we use the concept of an ω-limit set.The set

ω(D) ≡⋂t>0

γtD =⋂t>0

⋃τ≥t

SτD

is called the ω-limit set of the trajectories emanating from D (the bar over aset means the closure). It is equivalent to saying that x ∈ ω(D) if and only ifthere exist sequences tn → +∞ and xn ∈ D such that Stnxn → x as n→∞.It is clear that ω-limit sets (if they exist) are forward invariant.


1.4.2 Criteria for asymptotic smoothness

The following assertion is a generalization of the Ceron-Lopes criteria (see[Hal88] and the references therein).

Theorem 13. Let (X,St) be a dynamical system on a Banach space X. As-sume that for any bounded positively invariant set B in X there exist T > 0,a continuous nondecreasing function g : R+ 7→ R+, and a pseudometric %TBon C(0, T ;X) such that

(i) g(0) = 0; g(s) < s, s > 0.(ii) The pseudometric %TB is precompact (with respect to the norm of X) in the

following sense. Any sequence xn ⊂ B has a subsequence xnk suchthat the sequence yk ⊂ C(0, T ;X) of elements yk(τ) = Sτxnk is Cauchywith respect to %TB.

(iii) The following estimate

‖ST y1 − ST y2‖ ≤ g(‖y1 − y2‖+ %TB(Sτy1, Sτy2)

)holds for every y1, y2 ∈ B, where we denote by Sτyi the element in thespace C(0, T ;X) given by function yi(τ) = Sτyi.

Then, (X,St) is asymptotically smooth dynamical system.

Note that a precompact pseudometric is evaluated on trajectories Sτ , ratherthan on initial conditions (as in the classical treatments, see, e.g., [Hal88]).This fact becomes quite useful when applying the criterion to hyperbolic-likedynamics.

Proof. We refer to [CL08a]. The main ingredient of the proof is the relation

α(STB) ≤ g(α(B)), (1.59)

where α(B) is the Kuratowski α-measure of noncompactness which is definedby the formula

α(B) = infδ : B has a finite cover of diameter < δ (1.60)

for every bounded set B of X. For properties of this metric characteristic werefer to [Hal88] or [SY02, Lemma 22.2].

The property in (1.59) implies that for every bounded forward invariantset B there exists T > 0 such that α(STB) < α(B) provided α(B) > 0. If forsome fixed T > 0 this property holds for every bounded set B such that STBis also bounded, then, by the definition (see [Hal88]), ST is a conditional α-condensing mapping. It is known [Hal88] that conditional α-condensing map-pings are asymptotically smooth. Therefore Theorem 13 can be considered asa generalization of the results presented in [Hal88].

The above criterion is rather general, however it requires “compactness”of the sources in the equation. There are two other criteria that avoid such arequirement of a priori compactness. These are:


• compensated compactness criterion -an idea introduced in [Kha06] andlater expanded in [CL08a];

• J. Ball’s energy method, see [Bal04] and [MRW98].

The corresponding results are presented below.

Compensated compactness method

Theorem 14. Let (X,St) be a dynamical system on a complete metric spaceX endowed with a metric d. Assume that for any bounded positively invariantset B in X and for any ε > 0 there exists T ≡ T (ε, B) such that

d(ST y1, ST y2) ≤ ε+ Ψε,B,T (y1, y2), yi ∈ B, (1.61)

where Ψε,B,T (y1, y2) is a functional defined on B ×B such that

lim infm→∞

lim infn→∞

Ψε,B,T (yn, ym) = 0 (1.62)

for every sequence yn from B. Then (X,St) is an asymptotically smoothdynamical system.

Note that in a “compact” situation, i.e., when the functional Ψ is se-quentially compact, the condition (1.62) is automatically satisfied. The abovecriterion applies in the case of critical nonlinearities.

Proof. The properties in (1.61) and (1.62) makes it possible to prove thatα(StB) → 0 as t → +∞, where α(B) is the Kuratowski α-measure definedby (1.60). The latter property implies the conclusion desired. For details werefer to [CL08a] or [CL10].

John Ball’s “energy” method

This second method applies even in the case of supercritical nonlinear terms,but there are other requirements which restrict applicability of this method.The main idea behind Ball’s method is to construct an appropriate energytype functional which can be then decomposed into exponentially decayingpart and compact part. While the idea of decomposition of semigroup intouniformly stable and compact part is behind almost all criteria leading toasymptotic smoothness, the “energy” method described below postulates sucha decomposition on functionals rather than operators (semigroups). This facthas far reaching consequences and allows application of the method in super-critical situations.

We follow presentation of the method given in [MRW98]. Another exposi-tion of this method in the case of the damped wave equation can be found in[Bal04].


Theorem 15. Let St be an evolution continuous semigroup of weakly andstrongly continuous operators. Assume that there exist functionals Φ, Ψ , L, Kon phase space such that the following equality

[Φ(Stu) + Ψ(Stu)] +

∫ t

s

L(Sτu)e−ω(t−τ)dτ

= [Φ(Ssu) + Ψ(Ssu)]e−ω(t−s) +

∫ t

s

K(Sτu)e−ω(t−τ)dτ, (1.63)

holds for any u ∈ X. The functionals Φ, Ψ , L, K are assumed to have thefollowing properties:

• Φ : X 7→ R+ is continuous, bounded and if Ujj is bounded in X, tj →+∞, StjUj U weakly in X, and lim supn→∞ Φ(StjUj) ≤ Φ(U), thenStjUj → U strongly in X.

• Ψ : X 7→ R is ’asymptotically weakly continuous’ in the sense that if Ujjis bounded in X, tj → +∞, StjUj U weakly in X, then Ψ(StjUj) →Ψ(U).

• K : X 7→ R is ’asymptotically weakly continuous’ in the sense that ifUjj is bounded in X, tj → +∞, StjUj U weakly in X, then K(SsU) ∈L1(0, t) and

limj→∞

∫ t

0

e−ω(t−s)K(Ss+tjUj)ds =

∫ t

0

e−ω(t−s)K(SsU)ds, ∀t > 0.

• L is ’asymptotically weakly lower semicontinuous’ in the sense that if Ujjis bounded in X , tj → +∞, StjUj U weakly in X, then L(SsU) ∈L1(0, t) and

lim infj→∞

∫ t

0

e−ω(t−s)L(Ss+tjUj)ds ≥∫ t

0

e−ω(t−s)L(SsU)ds, ∀t > 0.

Then St is asymptotically smooth.

Remark 6. 1. The decomposition of the functionals in (1.63) depends on va-lidity of energy identity satisfied for weak solutions. This, alone, is a severecondition which may be difficult to verify (in contrast to energy inequal-ity). In addition, the proof of equality in (1.63) hides behind the fact thatthe damping in the equation is very structured-typically linear.

2. The functional (typically convex) Φ plays role of the energy of linearizedsystem - a good topological measure for the solution. In uniformly convexspaces X the assumptions postulated by Φ are automatically satisfied. In-deed, this results from the fact that weak convergence and the convergenceof the norms to the same element imply strong convergence.

3. The hypotheses required from Ψ , K and L represent some compactnessproperty of part of the nonlinear energy describing the system. This is


often the case even for supercritical nonlinearities. In fact, these termsallow to deal with non-compact sources in the equation provided thatthe corresponding nonlinear part of the energy is sequentially compact.Typical application involves 3D wave equation, where sources |f(s)| ≤C(1 + |s|p) with p < 5 can be handled (in view of compactness of theembedding H1(Ω) ⊂ Lq(Ω), q < 6).

1.4.3 Global attractors

The main objects arising in the analysis of long-time behavior of infinite-dimensional dissipative dynamical systems are attractors. Their study allowsus to answer a number of fundamental questions on the properties of limitregimes that can arise in the systems under consideration. At present, there areseveral general approaches and methods that allow us to prove the existenceand finite-dimensionality of global attractors for a large class of dynamicalsystems generated by nonlinear partial differential equations (see, e.g., [BV92,Chu99, Hal88, Lad91, Tem88] and the references listed therein).

Definition 7. A bounded closed set A ⊂ X is said to be a global attractor ofthe dynamical system (X,St) iff the following properties hold.

(i) A is an invariant set; that is, StA = A for t ≥ 0.(ii) A is uniformly attracting; that is, for all bounded set D ⊂ X

limt→+∞

dXStD |A = 0,

where dXA|B = supx∈A distX(x,B) is the Hausdorff semidistance.

It turns out that dissipativity property along with asymptotic smoothnessimply an existence of global attractor. The corresponding result is standardby now and reported below (see [Hal88] and also [BV92, Lad91, Tem88]).

Theorem 16. Any dissipative asymptotically smooth system (X,St) in a Ba-nach space X possesses a unique compact global attractor A. This attractor isa connected set and can be described as a set of all bounded full trajectories.Moreover A = ω(B) for any bounded absorbing set B of (X,St).

In the case when the dynamical system has special property-referred to asgradient system- dissipativity property is not needed (in the explicit form) inorder to prove existence of a global attractor. This is a very handy propertyparticularly when the proof of dissipativity is technically involved.

Gradient systems

The study of the structure of the global attractors is an important problemfrom the point of view of applications. There are no universal approachessolving this problem. It is well known that even in finite-dimensional cases


an attractor can possess extremely complicated structure. However, some setsthat belong to the attractor can be easily pointed out. For example, everystationary point (Stx = x for all t > 0) belongs to the attractor of the system.One can shows that any bounded full trajectory also lies in the global attractor(see, e.g., [BV92] or [Tem88]).

We begin with the following definition.

Definition 8. Let N be the set of stationary points of the dynamical system(X,St):

N = v ∈ X : Stv = v for all t ≥ 0 .

We define the unstable manifold M u(N ) emanating from the set N as aset of all y ∈ X such that there exists a full trajectory γ = u(t) : t ∈ R withthe properties

u(0) = y and limt→−∞

distX(u(t),N ) = 0.

Now we introduce the notions of Lyapunov functions and gradient systems(see, e.g., [BV92, Chu99, Hal88, Lad91, Tem88] and the references therein).

Definition 9. Let Y ⊆ X be a forward invariant set of a dynamical system(X,St).

• The continuous functional Φ(y) defined on Y is said to be a Lyapunovfunction for the dynamical system (X,St) on Y iff the function t 7→Φ(Sty) is a nonincreasing function for any y ∈ Y .

• The Lyapunov function Φ(y) is said to be strict on Y iff the equationΦ(Sty) = Φ(y) for all t > 0 and for some y ∈ Y implies that Sty = y forall t > 0; that is, y is a stationary point of (X,St).

• The dynamical system (X,St) is said to be gradient iff there exists a strictLyapunov function for (X,St) on the whole phase space X.

A connection between gradient systems and existence of compact attractors isgiven below. The main result stating existence and properties of attractors forgradient systems is the following theorem (for the proof we refer to [CL08a,CL10] and the references therein; see also [Rau02, Theorem 4.6] for a similarassertion).

Theorem 17. Assume that (X,St) is a gradient asymptotically smooth dy-namical system. Assume its Lyapunov function Φ(x) is bounded from aboveon any bounded subset of X and the set ΦR = x : Φ(x) ≤ R is bounded forevery R. If the set N of stationary points of (X,St) is bounded, then (X,St)possesses a compact global attractor A = M u(N ). Moreover,

• The global attractor A consists of full trajectories γ = u(t) : t ∈ R suchthat

limt→−∞

distX(u(t),N ) = 0 and limt→+∞

distX(u(t),N ) = 0.


• For any x ∈ X we have

limt→+∞

distX(Stx,N ) = 0,

that is, any trajectory stabilizes to the set N of stationary points. In par-ticular, this means that the global minimal attractor Amin coincides withthe set of the stationary points, Amin = N .

• If N = z1, . . . , zn is a finite set, then A = ∪ni=1Mu(zi), where M u(zi)

is the unstable manifold of the stationary point zi, and also(i) The global attractor A consists of full trajectories γ = u(t) : t ∈ R

connecting pairs of stationary points: any u ∈ A belongs to some fulltrajectory γ and for any γ ⊂ A there exists a pair z, z∗ ⊂ N suchthat

u(t)→ z as t→ −∞ and u(t)→ z∗ as t→ +∞.

(ii) For any v ∈ X there exists a stationary point z such that Stv → z ast→ +∞.

Dimension of global attractor

Finite-dimensionality is an important property of global attractors that can beestablished for many dynamical systems, including those arising in significantapplications. There are several approaches that provide effective estimates forthe dimension of attractors of dynamical systems generated by PDEs (see,e.g., [BV92, Lad91, Tem88]). Here we present an approach that does notrequire C1-smoothness of the evolutionary operator (as in [BV92, Tem88]).The reason for this focus is that dynamical systems of hyperbolic-like naturedo not display smoothing effects, unlike parabolic equations. Therefore, theC1 smoothness of the flows is most often beyond question, particularly inproblems with a nonlinear dissipation. Instead, we present a method whichcan be applied to more general locally Lipschitz flows. This method generalizesLadyzhenskaya’s theorem (see, e.g., [Lad91]) on finite dimension of invariantsets. We also refer to [Pra02] for a closely related approach based on somekind of squeezing property. However, we wish to point out that the estimatesof the dimension based on the theorem below usually tend to be conservative.

Definition 10. Let M be a compact set in a metric space X.

• The fractal (box-counting) dimension dimf M of M is defined by

dimf M = lim supε→0

lnn(M, ε)

ln(1/ε),

where n(M, ε) is the minimal number of closed balls of the radius ε whichcover the set M .


We can also consider the Hausdorff dimension dimH to describe complexityand embeddings properties of compact sets. We do not give a formal defi-nition of this dimension characteristic (see, e.g., [Fal90] for some details andreferences) and we only note that (i) the Hausdorff dimension does not exceed(but is not equal, in general) the fractal one; (ii) fractal dimension is moreconvenient in calculations.

The following result which generalizes [Lad91] was proved in [CL04a], seealso [CL08a].

Theorem 18. Let H be a separable Banach space and M be a bounded closedset in H. Assume that there exists a mapping V : M 7→ H such that

(i) M ⊆ VM .(ii) V is Lipschitz on M ; that is, there exists L > 0 such that

‖V v1 − V v2‖ ≤ L‖v1 − v2‖, v1, v2 ∈M.

(iii) There exist compact seminorms n1(x) and n2(x) on H such that

‖V v1 − V v2‖ ≤ η‖v1 − v2‖+K · [n1(v1 − v2) + n2(V v1 − V v2)]

for any v1, v2 ∈M , where 0 < η < 1 and K > 0 are constants (a seminormn(x) on H is said to be compact iff n(xm)→ 0 for any sequence xm ⊂ Hsuch that xm → 0 weakly in H).

Then M is a compact set in H of a finite fractal dimension. Moreover, if His a Hilbert space and the seminorms n1 and n2 have the form ni(v) = ‖Piv‖,i = 1, 2, where P1 and P2 are finite-dimensional orthoprojectors, then

dimf M ≤ (dimP1 + dimP2) · ln

(1 +

8(1 + L)√

2K

1− η

)·[ln

2

1 + η

]−1.

Remark 7. We note that under the hypotheses of Theorem 18 the mapping Vpossesses the property (see Lemma 2.18 in [CL08a])

α(V B) ≤ ηα(B) for any B ⊂M,

where α(B) is the Kuratowski α-measure given by (1.60). This means thatV is α-contraction on M (in the terminology of [Hal88]). The note that thelatter property is not sufficient for finite-dimensionality of the set M . Indeed,let

X = l2 =

x = (x1;x2; . . .) :

∞∑i=1

x2i <∞

and

M =x = (x1;x2; . . .) ∈ l2 : |xi| ≤ i−2, i = 1, 2, . . .

We define a mapping V in X by the formula

1 Dissipative hyperbolic-like evolutions 39[V x]i

= fi(xi), i = 1, 2, . . .

where fi(s) = s for |s| ≤ i−2, fi(s) = i−2 for s ≥ i−2, and fi(s) = −i−2 fors ≤ −i−2. One can see that V is globally Lipschitz in X and V X = M = VM .Since M is a compact set, the mapping is α-contraction (with η = 0). On theother hand it is clear that dimf M =∞.

We also note that this example means that the statement of Theorem 2.8.1in [Hal88] is not true and thus it requires some additional hypotheses concern-ing the mapping.

In what follows we shall present a unified criterion which allows to proveboth finite-dimensionality and smoothness of attractors. This criterion in-volves a special class of systems that we call quasi-stable.

1.4.4 Quasi-stable systems

In this section following the presentation given in [CL10, Section 7.9] we intro-duce a class of dissipative dynamical systems which display rather special longtime behavior dynamic properties. This class will be referred to as quasi-stablesystems which are defined via quasi-stability inequality given in Definition 11.It turns out that this is quite large class of systems naturally occurring innonlinear PDE’s of hyperbolic type of second order in time possibly inter-acting with parabolic equation. The interest in this class of systems stemsfrom the fact that quasi-stability inequality almost automatically implies -inone shot- number of desirable properties such as asymptotic smoothness,finite dimensionality of attractors, regularity of attractors, expo-nential attraction etc. In what follows we shall provide brief introductionto the theory of quasi-stable systems.

We begin with the following assumption.

Assumption 19 Let X, Y , and Z be reflexive Banach spaces; X is compactlyembedded in Y . We endow the space H = X × Y × Z with the norm

|y|2H = |u0|2X + |u1|2Y + |θ|2Z , y = (u0;u1; θ0).

The trivial case Z = 0 is allowed. We assume that (H,St) is a dynamicalsystem on H = X × Y × Z with the evolution operator of the form

Sty = (u(t);ut(t); θ(t)), y = (u0;u1; θ0) ∈ H, (1.64)

where the functions u(t) and θ(t) possess the properties

u ∈ C(R+;X) ∩ C1(R+;Y ), θ ∈ C(R+;Z).

The structure of the phase space H and the evolution operator St in Assump-tion 19 is motivated by the study of the system generated by equation ofthe second order in time in X × Y possibly interacting with some first-orderevolution equation in space Z. This type of interaction arises in modeling ofthermoelastic plates and structural acoustic sytems.


Definition 11. A dynamical system of the form (1.64) is said to be stablemodulo compact terms (quasi-stable, for short) on a set B ⊂ H if there exista compact seminorm µX(·) on the space X and nonnegative scalar functionsa(t), b(t), and c(t) on R+ such that (i) a(t) and c(t) are locally bounded on[0,∞), (ii) b(t) ∈ L1(R+) possesses the property limt→∞ b(t) = 0, and (iii)for every y1, y2 ∈ B and t > 0 the following relations

|Sty1 − Sty2|2H ≤ a(t) · |y1 − y2|2H (1.65)

and

|Sty1 − Sty2|2H ≤ b(t) · |y1 − y2|2H + c(t) · sup0≤s≤t

[µX(u1(s)− u2(s))

]2(1.66)

hold. Here we denote Styi = (ui(t);uit(t); θi(t)), i = 1, 2.

Remark 8. The definition of quasi-stability is rather natural from the point ofview of long-time behavior. It pertains to decomposition of the flow into expo-nentially stable and compact part. This represents some sort of analogy withthe “splitting” method [BV92, Tem88], however, the decomposition refers tothe difference of two trajectories, rather than a single trajectory. In addition,the quadratic dependence with respect to the semi-norm in (1.66) is essen-tial in the definition. The relation (1.66) is called a stabilizability estimate(or quasi-stability inequality) and, in the context of long-time dynamics, wasoriginally introduced in [CL04a] (see also [CEL04, CL04b] and the discussionin [CL08a] and [CL10]). To obtain such an estimate proves fairly technical(in critical problems) and requires rather subtle PDE tools to prove it. Il-lustrations of the method are given in [CL10] for some abstract models andalso for a variety of von Karman models. We also refer to [BC08, BCL07,CEL04, CL04a, CL06a, CL07a, CL08a, CLT08, CLT09, Nab09] for similarconsiderations for other models.

The notion of quasi-stability introduced in Definition 11 requires a specialstructure of the semiflow and a special type of a (quasi-stability) inequality.However, the idea behind this notion can be applied in many other cases (see,e.g., [CL04b, CL08a, CL11] and also [CL10]. Systems with delay/memoryterms can be also included in this framework (see, e.g., [Fas07, Fas09, Pot09,Ryz05] and also [CL10]). The same idea was recently applied in [CK10] foranalysis of long-time dynamics in a parabolic-type model (see below in Sec-tion 1.6).

Remark 9. In order to write down a more explicit form of quasi-stability in-equality let us consider dynamical system (say of gradient form) generatedby some second order evolution equation in the space H = X × Y with anassociated energy functional -say Ey(t) and the damping integral denoted byDy(t), so that the energy identity reads

Ey(t) +

∫ t

s

Dy(τ)dτ = Ey(s), s < t.


In this case a sufficient condition for quasi-stability is the following [CEL04]stabilizability inequality: there exists a parameter T > 0, such that the differ-ence of two any trajectories z(t) ≡ y1(t)− y2(t) satisfies the relation

E0,z(T ) +

∫ T

0

E0,z(τ)dτ ≤ CT∫ T

0

Dz(τ) + CTLOTz, (1.67)

where E0,z(t) denotes a positive part of the energy, Dz is the dampingfunctional for the difference z and LOTz is a lower order term of the formLOTz = supτ∈(0,T ) ||z(τ)||2X1

, where x ⊂ X1 with a compact embedding. Ifone is already equipped with an existence of compact attractor, then stabiliz-ability inequality that is required is of milder form: there exist two parameters−∞ < s < T such that

E0,z(T ) +

∫ T

s

E0,z(τ)dτ ≤ Cs,T∫ T

s

Dz(τ)dτ + Cs,TLOTz. (1.68)

Here LOTz denotes lower order terms measured by

LOTz = supτ∈(s,T )

||z(τ)||2X1,

and parameters s, T may depend on the specific trajectories y1, y2 .The advantage of the above formulation is that these are inequalities that

are obtainable by multipliers when studying stabilization and controllability(see, e.g., the monographs [Las02, LT00] and the references therein). The sta-bilizability inequality in (1.67) provides a direct link between controllabilityand long time behavior. Indeed, when considering stabilization to zero equi-libria, the lower order terms LOTz are dismissed by applying compactness-uniqueness argument. Then the inequality applied to a single trajectory y1 = y

Ey(T ) ≤ C0,T

∫ T

0

Dy(τ) = C0,T [Ey(0)− Ey(T )] (1.69)

implies, by standard evolution method, exponential decay to zero equilibrium.Similarly, in the case of controllability the observability inequality required

[Las02] is precisely (1.69) with Dy(t) denoting the quantities observed (forinstance, velocity in the spatial domain or velocity on the subdomain). Ineither case, the crux of the matter is to establish such inequality.

In what follows our aim is to show that quasi-stable systems have far reach-ing consequences and enjoy many nice properties that include (i) existenceof global attractors that is both finite-dimensional and smooth, (ii)exponential attractors, and so on.

Asymptotic smoothness

Proposition 2. Let Assumption 19 be in force. Assume that the dynamicalsystem (H,St) is quasi-stable on every bounded forward invariant set B inH. Then, (H,St) is asymptotically smooth.


Proof. Let

X = Closurev ∈ X : |v|X ≡ µX(v) + |v|Y <∞

. (1.70)

One can see that X is compactly embedded in the Banach space X. We havethat the space

W 1∞,2(0, T ;X,Y ) = f ∈ L∞(0, T ;X) : f ′ ∈ L2(0, T ;Y )

is compactly embedded in C(0, T ; X). This implies that the pseudometric %tBin C(0, t;H) defined by the formula

%tB(Sτy1, Sτy2) = c(t) supτ∈[0,t]

µX(u1(t)− u2(t))

is precompact (with respect to H). Here we denote by Sτyi the element fromC(0, t;H) given by function yi(τ) = Sτyi ≡ (ui(t);uit(t); θ

i(t)). By (1.66) %tBsatisfies the hypotheses of Theorem 13. This implies the result.

Corollary 1. If the system (H,St) is dissipative and satisfies the hypothesesof Proposition 2, then it possesses a compact global attractor.

Proof. By Proposition 2 the system (H,St) is asymptotically smooth. Thusthe result follows from Theorem 16.

Finite dimension of global attractors

We start with the following general assertion.

Theorem 20. Let Assumption 19 be valid. Assume that the dynamical system(H,St) possesses a compact global attractor A and is quasi-stable on A (seeDefinition 11). Then the attractor A has a finite fractal dimension dimH

f A(this also implies the finiteness of the Hausdorff dimension).

Proof. The idea of the proof is based on the method of “short” trajectories(see, e.g., [MN96, MP02] and the references therein and also [CL08a]).

We apply Theorem 18 in the space HT = H×W1(0, T ) with an appropriateT . Here

W1(0, T ) =

z ∈ L2(0, T ;X) : |z|2W1(0,T ) ≡

∫ T

0

(|z(t)|2X + |zt(t)|2Y

)dt <∞

.

(1.71)The norm in HT is given by

‖U‖2HT = |y|2H + |z|2W1(0,T ), U = (y; z), y = (u0;u1; θ0). (1.72)


Let yi = (ui0;ui1; θi0), i = 1, 2, be two elements from the attractor A. We denote

Styi = (ui(t);uit(t); θi(t)), t ≥ 0, i = 1, 2,

and Z(t) = Sty1 − Sty2 ≡ (z(t); zt(t); ξ(t)), where

(z(t); zt(t); ξ(t)) ≡ (u1(t)− u2(t);u1t (t)− u2t (t); θ1(t)− θ2(t)).

Integrating (1.66) from T to 2T with respect to t, we obtain that∫ 2T

T

|Sty1 − Sty2|2Hdt ≤ bT |y1 − y2|2H + cT sup0≤s≤2T

[µX(z(s))]2, (1.73)

where

bT =

∫ 2T

T

b(t)dt and cT =

∫ 2T

T

c(t)dt.

It also follows from (1.66) that

|ST y1 − ST y2|2H ≤ b(T ) · |y1 − y2|2H + c(T ) · sup0≤s≤T

[µX(z(s))]2

and combining with (1.73) yields

|ST y1−ST y2|2H+

∫ 2T

T

|Sty1−Sty2|2Hdt ≤ bT |y1−y2|2H+cT sup0≤s≤2T

[µX(z(s))]2,

(1.74)where

bT = b(T ) +

∫ 2T

T

b(t)dt and cT = c(T ) +

∫ 2T

T

c(t)dt. (1.75)

Let A be the global attractor. Consider in the space HT the set

AT := U ≡ (u(0);ut(0); θ(0);u(t), t ∈ [0, T ]) : (u(0);ut(0); θ(0)) ∈ A ,

where u(t) is the first component of Sty(0) with y(0) = (u(0);ut(0); θ(0)), anddefine operator V : AT 7→ HT by the formula

V : (u(0);ut(0); θ(0);u(t)) 7→ (ST y(0);u(T + t)).

It is clear that V is Lipschitz on AT and V AT = AT .Because the space X given by (1.70) possesses the properties

X ⊂ X ⊂ Y and X ⊂ X is compact,

contradiction argument yields

[µX(u)]2 ≤ ε|u|2X + Cε|u|2Y for any ε > 0. (1.76)


Therefore it follows from (1.65) that

sup0≤s≤2T

[µX(z(s))]2 ≤ bT

cT|y1 − y2|2H + C(aT , bT , cT ) sup

0≤s≤2T|z(s)|2Y ,

where aT = sup0≤s≤2T a(s) and bT , cT given by (1.75). Consequently, from(1.74) we obtain

‖V U1 − V U2‖HT ≤ ηT ‖U1 −U2‖HT +KT · (nT (U1 −U2) + nT (V U1 − V U2)),

for any U1, U2 ∈ AT , where KT > 0 is a constant (depending on aT , bT , cT ,the embedding properties of X into Y , the seminorm µX), and

η2T = bT = b(T ) +

∫ 2T

T

b(t)dt. (1.77)

The seminorm nT has the form nT (U) := sup0≤s≤T |u(s)|Y . Because W1(0, T )is compactly embedded into C(0, T ;Y ), nT (U) is a compact seminorm on HT

and we can choose T > 1 such that ηT < 1. We also have from (1.65) that

‖V U1 − V U2‖HT ≤ LT ‖U1 − U2‖HTfor U1, U2 ∈ AT ,

where L2T = a(T ) +

∫ 2T

Ta(t)dt. Therefore we can apply Theorem 18 which

implies that AT is a compact set in HT of finite fractal dimension.Let P : HT 7→ H be the operator defined by the formula

P : (u0;u1; θ0; z(t)) 7→ (u0;u1; θ0).

A = PAT and P is Lipshitz continuous, thus dimHf A ≤ dimHT

f AT < ∞.Here dimW

f stands for the fractal dimension of a set in the space W . Thisconcludes the proof of Theorem 20.

Remark 10. By [CL08a] the dimension dimHf A of the attractor admits the

estimate

dimHf A ≤

[ln

2

1 + ηT

]−1· ln m0

(4KT (1 + L2

T )1/2

1− ηT

), (1.78)

Here m0(R) is the maximal number of pairs (xi; yi) in HT × HT possessingthe properties

‖xi‖2HT + ‖yi‖2HT ≤ R2, nT (xi − xj) + nT (yi − yj) > 1, i 6= j.

It is clear that m0(R) can be estimated by the maximal number of pairs (xi; yi)in W1(0, T )×W1(0, T ) possessing the properties ‖xi‖2W1(0,T ) + ‖yi‖2W1(0,T ) ≤R2 and nT (xi − xj) + nT (yi − yj) > 1 for all i 6= j. Thus the bound in (1.78)depends on the functions a, b, c and seminorm µX in Definition 11 and alsoon the embedding properties of X into Y .


Regularity of trajectories from the attractor

In this section we show how stabilizability estimates can be used in order toobtain additional regularity of trajectories lying on the global attractor. Thetheorem below provides regularity for time derivatives. The needed “space”regularity follows from the analysis of the respective PDE. It typically in-volves application of elliptic theory (see the corresponding results in [CL10,Chapter 9]).

Theorem 21. Let Assumption 19 be valid. Assume that the dynamical sys-tem (H,St) possesses a compact global attractor A and is quasi-stable on theattractor A. Moreover, we assume that (1.66) holds with the function c(t)possessing the property c∞ = supt∈R+

c(t) < ∞. Then any full trajectory(u(t);ut(t); θ(t)) : t ∈ R that belongs to the global attractor enjoys thefollowing regularity properties,

ut ∈ L∞(R;X) ∩ C(R;Y ), utt ∈ L∞(R;Y ), θt ∈ L∞(R;Z) (1.79)

Moreover, there exists R > 0 such that

|ut(t)|2X + |utt(t)|2Y + |θt(t)|2Z ≤ R2, t ∈ R, (1.80)

where R depends on the constant c∞, on the seminorm µX in Definition 11,and also on the embedding properties of X into Y .

Proof. It follows from (1.66) that for any two trajectories

γ = U(t) ≡ (u(t);ut(t); θ(t)) : t ∈ R,γ∗ = U∗(t) ≡ (u∗(t);u∗t (t); θ

∗(t)) : t ∈ R

from the global attractor we have that

|Z(t)|2H ≤ b(t− s)|Z(s)|2H + c(t− s) sups≤τ≤t

[µX(z(τ))]2

(1.81)

for all s ≤ t, s, t ∈ R, where Z(t) = U∗(t) − U(t) and z(t) = u∗(t) − u(t). Inthe limit s→ −∞ relation (1.81) gives us that

|Z(t)|2H ≤ c∞ sup−∞≤τ≤t

[µX(z(τ))]2

for every t ∈ R and for every couple of trajectories γ and γ∗. Using relation(1.76) we can conclude that

sup−∞≤τ≤t

|Z(τ)|2H ≤ C sup−∞≤τ≤t

|z(τ)|2Y , (1.82)

for every t ∈ R and for every couple of trajectories γ and γ∗ from the attractor.


Now we fix the trajectory γ and for 0 < |h| < 1 we consider the shiftedtrajectory γ∗ ≡ γh = y(t + h) : t ∈ R. Applying (1.82) for this pair oftrajectories and using the fact that all terms (1.82) are quadratic with respectto Z we obtain that

sup−∞≤τ≤t

|uh(τ)|2X + |uht (τ)|2Y + |θht (τ)|2Z

≤ C sup

−∞≤τ≤t|uh(τ)|2Y , (1.83)

where uh(t) = h−1 · [u(t+h)−u(t)] and θh(t) = h−1 · [θ(t+h)− θ(t)]. On theattractor we obviously have that

|uh(t)|Y ≤1

h·∫ h

0

|ut(τ + t)|Y dτ ≤ C, t ∈ R,

with uniformity in h. Therefore (1.83) implies that

|uh(t)|2X + |uht (t)|2Y + |θht (t)|2Z ≤ C, t ∈ R.

Passing with the limit on h then yields relations (1.79) and (1.80).

Fractal exponential attractors

For quasi-stable systems we can also establish some result pertaining to (gen-eralized) fractal exponential attractors.

We first recall the following definition.

Definition 12. A compact set Aexp ⊂ X is said to be inertial (or a fractalexponential attractor) of the dynamical system (X,St) iff A is a positivelyinvariant set of finite fractal dimension (in X) and for every bounded setD ⊂ X there exist positive constants tD, CD and γD such that

dXStD |Aexp ≡ supx∈D

distX(Stx, Aexp) ≤ CD · e−γD(t−tD), t ≥ tD.

If the dimension of A is finite in some extended space we call A generalizedfractal exponential attractor.

For more details concerning fractal exponential attractors we refer to [FEN94]and also to recent survey [MZ08]. We only note that the standard tech-nical tool in the construction of fractal exponential attractors is the so-called squeezing property which says [MZ08], roughly speaking, that eitherthe higher modes are dominated by the lower ones or that the semiflow iscontracted exponentially. Instead the approach we use is based on the quasi-stability property which says that the semiflow is asymptotically contractedup to a homogeneous compact additive term.

Theorem 22. Let Assumption 19 be valid. Assume that the dynamical system(H,St) is dissipative and quasi-stable on some bounded absorbing set B. We


also assume that there exists a space H ⊇ H such that t 7→ Sty is Holdercontinuous in H for every y ∈ B; that is, there exist 0 < γ ≤ 1 and CB,T > 0such that

|St1y − St2y|H ≤ CB,T |t1 − t2|γ , t1, t2 ∈ [0, T ], y ∈ B. (1.84)

Then the dynamical system (H,St) possesses a (generalized) fractal exponen-

tial attractor whose dimension is finite in the space H.

In contrast with Theorem 3.5 [MZ08] Theorem 22 does not assume Holdercontinuity in the phase space. At the expense of this we can guarantee finite-ness of the dimension in some extended space only. This is why we use thenotion of generalized exponential attractors.

Proof. We apply the same idea as in the proof of Theorem 20.We can assume that absorbing set B is closed and forward invariant (oth-

erwise, instead of B, we consider B′ = ∪t≥t0StB for t0 large enough, whichlies in B).

In the space HT = H ×W1(0, T ) equipped with the norm (1.72), whereW1(0, T ) is given by (1.71), we consider the set

BT := U ≡ (u(0);ut(0); θ(0);u(t), t ∈ [0, T ]) : (u(0);ut(0); θ(0)) ∈ B ,

where u(t) is the first component of Sty(0) with y(0) = (u(0);ut(0); θ(0)). Asabove we define the shift operator V : BT 7→ HT by the formula

V : (u(0);ut(0); θ(0);u(t)) 7→ (ST y(0);u(T + t)).

It is clear that BT is a closed bounded set in HT which is forward invariantwith respect to V .

It follows from (1.65) that

‖V U1 − V U2‖2HT ≤

(a(T ) +

∫ 2T

T

a(t)dt

)‖U1 − U2‖2HT , U1, U2 ∈ BT .

As in the proof of Theorem 20 we can obtain that

‖V U1 − V U2‖HT ≤ ηT ‖U1 − U2‖HT +KT · (nT (U1 − U2) + nT (V U1 − V U2))

for any U1, U2 ∈ BT and for some T > 0, where KT > 0 is a constant,nT (U) := sup0≤s≤T |u(s)|Y and ηT < 1 is given by (1.77). Therefore by Theo-rem 7.4.2[CL10] the mapping V possesses a fractal exponential attractor; thatis, there exists a compact set AT ⊂ BT and a number 0 < q < 1 such thatdimHT

f AT <∞, VAT ⊂ AT , and

sup

distHT (V kU,AT ) : U ∈ BT

≤ Cqk, k = 1, 2, . . . ,

for some constant C > 0. In particular, this relation implies that


sup distH(SkT y,A ) : u ∈ B ≤ Cqk, k = 1, 2, . . . , (1.85)

where A is the projection of AT of the first components:

A = (u(0);ut(0); θ(0)) ∈ B : (u(0);ut(0); θ(0);u(t), t ∈ [0, T ]) ∈ AT .

It is clear that A is a compact forward invariant set with respect to ST ; thatis, STA ⊂ A . Moreover dimH

f A ≤ dimHTf AT <∞. One can also see that

Aexp = ∪StA : t ∈ [0, T ]

is a compact forward invariant set with respect to St; that is, StAexp ⊂ Aexp.Moreover, it follows from (1.84) that

dimHf Aexp ≤ c

[1 + dimH

f A]<∞.

We also have from (1.85) and (1.65) that

sup distH(Sty,Aexp) : u ∈ B ≤ Ce−γt, t ≥ 0,

for some γ > 0. Thus Aexp is a (generalized) fractal exponential attractor.

Remark 11. Holder continuity (1.84) is needed in order to derive the finiteness

of the fractal dimension dimHf Aexp from the finiteness of dimH

f A . We donot know whether the same holds true without property (1.84) imposed insome vicinity of Aexp. This is because Aexp is a uncountable union of (finite-dimensional) sets StA . We also emphasize that fractal dimension depends onthe topology. Indeed, there is an example of a set with finite fractal dimensionin one space and infinite fractal dimension in another (smaller) space.

1.5 Long time behavior for canonical models describedin Section 1.2

The goal of this section is to show how to apply abstract methods presentedin the previous section in order to establish long time behavior propertiesof dynamical systems generated by PDE’s described in Section 1.2. An in-teresting feature is that though all the models considered are of hyperbolictype-without an inherent smoothing mechanism, the long time behavior canbe made “smooth” and characterized by finite dimensional structures. Ourplan is to focus on the following topics.

1. Control to finite dimensional and smooth attractors of nonlinear waveequation• with interior fully supported dissipation,• with boundary or partially localized dissipation.


2. Control to finite dimensional and smooth attractors of nonlinear platesdynamics• von Karman plates with a nonlinear fully supported interior feedback

control,• von Karman and Berger plates with boundary and partially localized

damping,• Kirchhoff-Boussinesq plates with a linear interior feedback control.

For each model considered we follow the plan: (i) state the assumptions andthe results, (ii) discuss possible generalizations, (iii) state open problems,(iv) provide a sketch of the proofs with precise references to the originalsources.

1.5.1 Wave dynamics

In this section we consider the existence of finite dimensional and smoothattractors associated with the dynamics of semilinear wave equation criticalexponents in both (feedback) sources and interior and boundary damping.

We start with interior damping and source model as described in (1.1).

Interior damping

In studying long time behavior we find convenient to collect all the assump-tions required for the source and damping:

Assumption 23 1. The source f ∈ C1(R) satisfies the dissipativity condi-tion in (1.25) and the growth condition

|f ′(s)| ≤M |s|2 for |s| ≥ 1.

2. The damping g(s) is an increasing function, g(0) = 0 and it satisfies thefollowing asymptotic growth conditions:

0 < mg|s|2 ≤ g(s)s ≤Mg|s|6 for |s| ≥ 1. (1.86)

3. The damping coefficient a satisfies a(x) ≥ a0 > 0, x ∈ Ω .

With reference to the model (1.1) we shall consider relatively simple, butrepresentative of the main challenges, polynomial structures for the sourceand the damping.

f(s) = −s3 + cs2 and g(s) = g1s+ |s|4s with g1 ≥ 0 and c ∈ R. (1.87)

Remark 12. 1.The power 3 associated with the source f and the power 5 asso-ciated with the damping g are often referred as ”double critical” parametersfor the wave equation with n = 3. The reason for this is that the mapsu → f(u) from H1(Ω) into L2(Ω) and u → g(u) from H1(Ω) into H−1(Ω)are bounded but not compact.2. More general forms of space dependent and localized damping coefficienta(x) are considered in [CLT08]


Under the above conditions the wellposedness results of Theorem 2 assertexistence of a continuous semiflow St which defines dynamical system (H , St)with H = H1

0 (Ω)×L2(Ω). We are ready to undertake the study of long-timebehavior. Our first main result is given below:

Theorem 24 (Compactness). With reference to the dynamics described by(1.1) and zero Dirichlet boundary conditions, we assume Assumption 23 forthe source f and the damping g. Then, there exists a global compact attractorA ⊂ H for the dynamical system (H , St). In addition (H , St) is gradientsystem.

To describe the structure of the attractor, we introduce the set of station-ary points of St denoted by N ,

N = V ∈H : StV = V for all t ≥ 0 . (1.88)

Every stationary point W ∈ N has the form W = (w; 0), where w = w(x)solves the problem

∆w + f(w) = 0 in Ω and w = 0 in Γ. (1.89)

Under the Assumption 23, standard elliptic theory implies that every sta-tionary solution satisfies w ∈ H2(Ω) and the set of all stationary solutionsis bounded in H1(Ω). In fact, more regularity of stationary solutions can beshown, but the above is sufficient for the analysis to follow.

The next result is a consequence of general properties of gradient systems(see, e.g., [BV92, Hal88] and also Section 1.4.3 above) and asserts that theattractor A coincides with this unstable manifold.

Theorem 25 (Structure). Under the assumptions of Theorem 24 we havethat

• A = Mu(N );• limt→+∞ distH (StW,N ) = 0 for any W ∈H ≡ H1

0 (Ω)× L2(Ω);• if (1.89) has a finite number of solutions3, then for any V ∈ H there

exists a stationary point Z = (z, 0) ∈ N such that StV → Z in H ast→ +∞.

Our second main result read refers to the dimensionality of attractor.

Theorem 26 (Finite dimensionality). Let f ∈ C2(R), g ∈ C1(R) andAssumption 23 be in force. In addition, we assume that for all s ∈ R:

1. |f ′′(s)| ≤ C|s|,2. 0 < m ≤ g′(s) ≤ C[1 + sg(s)]2/3.

Then the fractal dimension of the global attractor A of the dynamical system(H , St) is finite.

3 We note that the property that (1.89) has finitely many solutions is generic.


With reference to the model (1.1) we still can take f and g as in (1.87), butwith g1 > 0.

The proof of Theorem 26 follows from quasi-stability property satisfied forthe model (see Definition 11). The exactly same lemma allows to obtain “forfree” the following regularity result for elements from the attractor.

Theorem 27 (Regularity). Under the assumptions of Theorem 26 the at-tractor is a bounded set in

V =

(u, v) ∈ H10 (Ω)×H1

0 (Ω)∣∣−∆u+ g(v) ∈ L2(Ω),

,

i.e. there exist constants ci > 0 such that4

A ⊂

(u, v) ∈ H1(Ω)×H1(Ω)

∣∣∣∣ ‖u‖1,Ω + ‖v‖1,Ω ≤ c1;‖∆u− g(v)‖ ≤ c2,

.

A natural question that can be asked in this context is that of existence ofstrong (i.e., corresponding to topology of strong solutions) attractors. Thoughthis latter property is technically related to smoothness of attractors, howeverthe corresponding result does not follow from Theorem 27, unless the dampingis subcritical. Additional analysis is needed for that. A first question to addressin this direction is the existence of attractors for strong solutions. This is tosay that when restricting solutions to regular initial data, the correspondingtrajectories converge asymptotically to an attractor A1 which typically maybe smaller than A the latter corresponds to weak or generalized solutions. Afirst step toward this goal is to show dissipativity of strong solutions (whichof course does not follow at once from dissipativity of weak solutions, unlessa problem is linear). Fortunately, dsissipativity of strong solution is, again,direct consequence of quasi-stability. Thus, we have this property for “free”.

Theorem 28 (Dissipativity of strong solutions). Let the assumptions ofTheorem 26 be in force. Then there exists a number R0 > 0 such that for anyR > 0 we can find tR > 0 such that

‖wtt(t)‖2 + ‖wt(t)‖21,Ω ≤ R20 for all t ≥ tR (1.90)

for any strong solution w(t) to problem (1.1) with initial data (w0;w1) fromthe set

WR =

(w0, w1) ∈ H10 (Ω)×H1

0 (Ω) : ‖w0‖1,Ω + ‖w1‖1,Ω ≤ R, ‖w2‖ ≤ R,

(1.91)where w2 ≡ g(w1)−∆w0.

Theorem 28 refers to strong solutions. The existence of such, is guaranteedonce initial data are taken from WR. With additional calculations (using themultiplier ∆wt) one improves the statement and obtains the dissipativity for

4 See also Theorem 29 below which asserts an additional regularity of attractor.


‖w(t)‖22,Ω ≤ R20, t ≥ tR with initial data from H2(Ω). This, in particular,

shows that strong attractors (for strong solutions) are in H2(Ω) ∩ H10 (Ω) ×

H10 (Ω). An interesting question is whether the same regularity is enjoyed by

weak attractors. In other words, whether H2(Ω)∩H10 (Ω)×H1

0 (Ω) ⊃ A ? Theanswer to this question is given by

Theorem 29 (Regularity). Under the same assumptions as in Theorem 26the attractor is a bounded set in H2(Ω)× L2(Ω).

The proof of this theorem given in [Kha10] exploits the so called “backwardsmoothness on attractor”.

Decay rates to equilibrium: Under the conditions of Theorem 25 we know-ing that every trajectory converges to some equilibrium point one would liketo know “how fast”? The answer to this question is given by decay rates ofasymptotic convergence of solutions to point of equilibria. This is the topicwe present next.

We introduce concave, strictly increasing, continuous function k0 : R+ 7→R+ which captures the behavior of g(s) at the origin possessing the properties

k0(0) = 0 and s2 + g2(s) ≤ k0(sg(s)) for |s| ≤ 1. (1.92)

Such a function can always be constructed due to the monotonicity of g, see[LT93] or Appendix B in [CL10]. Moreover, on the strength of Assumption 23there exists a constant c > 0 such that k(s) ≡ k0(s) + cs is a concave, strictlyincreasing, continuous function k : R+ 7→ R+ such that

k(0) = 0 and |s|2 ≤ k(sg(s)) for all s ∈ R. (1.93)

Given function k we define

H0(s) =k

(s

c3

), G0(s) = c1(I +H0)−1(c2s), (1.94)

Q(s) =s− (I +G0)−1(s),

where ci are some positive constants. It is obvious that Q(s) is strictly mono-tone. Thus, the differential equation

dσ

dt+Q(σ) = 0, t > 0, σ(0) = σ0 ∈ R, (1.95)

admits global, unique solution σ(t) which, moreover, decays asymptotically tozero as t→∞. With these definitions we are ready to state our next result.

Theorem 30 (Rate of convergence to equilibria). In addition to Assu-mption 23, we assume that the set of stationary points N is finite, and everyequilibrium V = (v; 0) is hyperbolic in the sense that the problem

∆w + f ′(v) · w = 0 in Ω and w = 0 on Γ (1.96)


has no non-trivial solutions. Then, for any W0 = (w0;w1) ∈ H , there existsa stationary point V = (v; 0) such that

‖S(t)W0 − V ‖H ≤ C(σ([tT−1])

), t > 0, (1.97)

where C and T are positive constants depending on W0, [a] denotes the integerpart of a and σ(t) satisfies (1.95) with σ0 = C(W0, V ) where C(W0, V ) is aconstant depending on ||W0||H and ||V ||H and Q is defined in (1.94). Inparticular, if g′(0) > 0, then

‖S(t)W0 − V ‖H ≤ Ce−ωt

for some positive constants C and ω depending on W0.

Remark 13. Since Q(s) is strictly increasing and Q(0) = 0, the rates describedby the ODE in (1.95) (see, e.g., [LT93] or [CL10, Appendix B]) decay uniformlyto zero. The “speed” of decay depends on the behavior of g′(s) at the origin.If g′(s) decays to zero polynomially, then by solving the ODE in (1.95), oneobtains algebraic decay rates for the solutions to σ(t). If, instead δ = 0 andg′(0) > 0, then Q(σ) = aσ for some a > 0 and, consequently, the decay ratesderived from (1.95) are exponential (see [LT93] for details). In the latter caseusing the dissipitavity of strong solutions (see Theorem 28) one can prove byinterpolation that the exponential decay rate holds for strong solutions in astronger (than in H ) topology.

Boundary damping

In the case when boundary damping is the only active mode of dissipation(model in (1.14) and (1.15) with g = 0) one can still prove that the longtime behavior is essentially the same as in the case of internal damping. Thetask of achieving this goal is much more technical, due to the necessity ofpropagating the damping from the boundary into the interior. This is doneby using familiar by now ”flux” multipliers. However, the resulting analysisand PDE estimates are considerably more complicated and often resort tospecific unique continuation property as well as Carleman’s estimates (whendealing with critical cases). This topic has been considered in [CEL02, CEL04,CLT09]. The corresponding results is given below.

Theorem 31 (Boundary dissipation). With reference to the equations in(1.14)and (1.15), where we take g = 0, h(w) = −w, the source f satisfies dis-sipativity property imposed in (1.25), where λ1 corresponds to the first eigen-value of the Robin problem and such that |f ′′(s)| ≤ C(1 + |s|) for all s ∈ R.The boundary damping g0 is an increasing, differentiable function, g0(0) = 0and satisfies the asymptotic growth condition:

m ≤ g′0(s) ≤M, |s| > 1, m,M > 0, (1.98)

Then


1. Under the above assumption the statements of Theorem 24 and 25 hold.2. If, in addition, g′0(0) > 0 then the statements of Theorem 26 and also

Theorem 29, Theorem 30 (with obvious modifications) hold true.

One fundamental difference between boundary damping and interior damp-ing is that the restriction of linear bound at infinity for the boundary dampingis essential. One dimensional examples [Van00] disprove validity of uniformstability to a single equilibrium when the damping is superlinear at infinity.

Generalizations

With reference to interior damping

1. Wave equation with homogenous Neumann or Robin boundary conditionscan be considered in the same way.

2. The same wave model with additional boundary damping and source,as described in (1.14) and (1.15). The corresponding analysis is moreinvolved. Details are given in [CL07a].

3. Possibility of degenerate damping g(s), where the density function a(x)describes possibility of degeneracy of the damping. This situation is alsostudied in [CL07a].

4. Strong attractors. These are attractors corresponding to strong solutions.In the case when boundary source and damping are involved and thedamping may degenerate, the description of strong attractors is moreinvolved. These are bounded sets V × H1(Ω) where H2(Ω) ⊂ V . Thedetails are given in [CL07a].

5. Under additional assumptions imposed on regularity of the damping g(s)and the source f(s) one can prove that the attractors corresponding toTheorem 29 are infinitely smooth C∞. This can be done by the usualboot-strap argument.

6. Attractors with localized dissipation only. This corresponds to having a(x)localized in the layer near boundary. [CLT08]. As in the case of boundarydamping, requires linearly bounded damping.

With reference to boundary damping

1. Attractors with boundary damping which is further localized to a suitableportion of the boundary. Requires appropriate geometric conditions, see[CLT09] for details.

2. Relaxed regularity assumptions imposed on the damping function g0(s),see [CLT09] again.

Open problems

1. Higher regularity of attractors with minimal hypotheses on the dampingand source.


2. Under which conditions weak attractors coincide with strong attractorsin the case of boundary or partially localized damping?

3. Supercritical sources (e.g., f(s) = s5). Ball’s method will provide exis-tence of attractors with linear damping. However, linear damping is notsufficient for uniqueness of solutions. One needs to consider non-uniquesolutions.

4. Nonmonotone damping. Some particular results in this direction can befound in [CL08a].

5. Non-gradient structures. For instance models involving first order differ-entials as the potential sources.

Remarks

1. Theorem 24 -Compactness. Difficulty: double critical exponent -rules outprevious methods. Use Theorem 14 after an appropriate rewriting of thesource which then fits into the structure of the functional Ψ . This pointis explained below.

2. Theorems 26 and Theorem 29-finite dimensionality and smoothness. Relieson proving quasi-stability inequality. The main trick is to provide appro-priate decomposition of the source that takes advantage of dissipativityintegral which is L1(R). See [CL07a, CLT09].

3. Full statement of Theorem 29 - [Kha10]. Smoothness of the trajectorieson the attractor near −∞. The existence of strong attractors (for strongsolutions) is de facto consequence of quasi-stability inequality. However, inmany situation proving additional spatial regularity (H2(Ω), for instance)is problematic. It is for this task that special considerations related tobackward smoothness are important.

4. Decay rates - Theorem 30. The difficulty relates to the fact that con-vergence to equilibria is a very ”unstable” process. It is not possible tolocalize the analysis to the neighborhood of the equilibria. Decay ratesare derived by using convex analysis and reducing estimates to nonlinearODE’s. See [CL07a].

5. Boundary damping - Theorem 31. First results are given in [CEL02,CEL04] where finite-dimensionality was proved for subcritical sourcesonly. Complete proof of Theorem 31 (in fact in a more general version) isgiven in [CLT09], where flux multipliers are used along with Carleman’sestimates and also backward smoothness of trajectories.

Guide through the Proofs

We are not in a position to present complete proofs of the results. These aregiven in the cited references. Here, our aim instead is to orient the readerwhat are the main points one needs to go over in order to prove these results.Special emphasis is given to more subtle technical details where ”some” tricks-most of which recently introduced - need to be applied. It is our hope that


an attentive reader will be able to grasp the essence of the proof, the toolsrequired so that when guided to more specific reference he will be in a positionto reconstruct a complete proof.

Theorem 24 - Compactness

Step 1: The corresponding system is gradient system. This followsfrom the fact that Lyapunov function V (t) ≡ E (t) is bounded from above andbelow by the topology of the phase space. The latter results from dissipativitycondition imposed on f(s) in Assumption 23 (see (1.25)). Moreover Vt(t) = 0implies ((g(wt), wt)) = 0 hence wt(x, t) = 0 in Ω reduces the dynamics to astationary set defined by (1.88) and (1.89). The set N of stationary points isbounded due, again, to dissipativity condition imposed on f . This part of theargument is standard.

Step 2: Asymptotic smoothness. The main difficulty in carrying the proofof asymptotic smoothness is double criticality of both the source and thedamping. This is the reason why previous approaches (based on splitting orsqueezing) did require additional assumptions-such as subcriticality of thesource or a requirement that the damping parameter be sufficiently large. Toovercome this difficulty we use Theorem 14. In preparation for this theoremwe are using the equation

ztt −∆z + a(x)[g(wt)− g(ut)] = f(w)− f(u) in Ω × R+ (1.99)

with z = 0 on Γ , written for the difference of two solutions z = w − u, whereboth w and u are solutions to the original equations with trajectories residingin a bounded set B ∈ H, and also

• energy identity (multiply equation (1.99) by zt);• equipartition of energy (multiply equation (1.99) by z).

In addition, we recall dissipation relations for each solution u and w:∫ t

0

[((g(ut), ut))) + ((g(wt), wt))] ≤ CB, ∀ t > 0, (1.100)

and also

‖ut(t)‖2 + ‖u(t)‖21,Ω + ‖wt(t)‖2 + ‖w(t)‖21,Ω ≤ CB, ∀ t > 0.

Critical step consists of obtaining “recovery” estimate for the energy of z interms of the damping and the source. The important part is that the “recov-ery” estimate does not involve any initial data. This step is achieved by usingequipartition of the energy. The result of which is the following inequality[CL07a]:

1

2TEz(T )+

∫ T

0

Ez(t) ≤∫ T

0

[||zt||2 +D(zt) + |G(zt, z)|

]dt+Ψ(z, T ), (1.101)


for T ≥ 4, where Ez(t) ≡ ||zt(t)||2 + ||∇z(t)||2,

D(zt) ≡ ((g(wt)− g(ut), zt)), G(zt, z) ≡ ((g(wt)− g(ut), z)), (1.102)

and

Ψ(z, T ) ≡ −∫ T

0

((f(w)− f(u), zt))

+

∫ T

0

ds

∫ T

s

dτ((f(w)− f(u), zt)) +

∫ T

0

|((f(w)− f(u), z))|dt

Our goal is to bound the right hand side of the above relation by terms thatare (i) either bounded uniformly in T , or (ii) compact or (iii) small multiplesof energy integral.

From the energy relation for z we have that∫ T

0

D(zt)dτ ≤ CB +

∫ T

0

((f(w)− f(u), zt))dτ. (1.103)

One can also see from Proposition B.1.2 in [CL10] that

‖zt‖2 ≤ η + CηD(zt), ∀ η > 0.

Thus the terms ‖zt‖2 and D(zt) produce in (1.101) an admissible contribution.The last integral in the definition of Ψ is compact. Thus the noncompact

terms are the second one involving G and the first two integrals in definitionof Ψ . Since these two terms are similar, it suffices to analyze the main non-compact terms in (1.101) which are:∫ T

0

|G(zt, z)|dt and

∫ T

0

ds

∫ T

s

((f(w)− f(u), zt))dτ.

These terms reflect double criticality of the damping-source exponents. Weanalyze these next.

For the damping term G one uses split Ω into Ω1 = |ut(x, t)| > 1 andthe complement Ω2 = Ω \Ω1. With the use of critical growth condition for gand the fact that ‖z‖L6(Ω) ≤ C‖z‖1,Ω ≤ CB we obtain∫

Ω1

|g(ut)||z|dx ≤[∫

Ω1

|g(ut)|6/5dx]5/6‖z‖L6(Ω)

≤C[∫

Ω1

g(ut)utdx

]5/6‖z‖1,Ω ≤ CB

∫Ω

g(ut)utdx.

On Ω2 the damping is bounded -contributing to lower order terms. Thus weobtain for G(zt, z)

58 Igor Chueshov and Irena Lasiecka∫ T

0

G(zt, z)dt ≤CB

∫ T

0

[(g(ut), ut) + (g(wt), wt)] dt+ CBT supt∈[0,T ]

||z(t)||

≤CB + CBT supt∈[0,T ]

||z(t)||,

which gives the inequality in terms of the universal constant (independent ontime T ) and a lower order term ||z(t)||.

For the source, the key point is the following decomposition that allowsto prove sequential convergence of the functional Ψ in Theorem 14:∫ T

0

((f(w)− f(u), zt))

=

∫Ω

[F (w(T )) + F (u(T ))− F (w(0))− F (u(0))]

−∫ T

0

[((f(u), wt)) + ((f(w), ut))] dt,

where F (s) denotes the antiderivative of f . In line with Theorem 14 we con-sider these terms on sequences u = wn, w = wm converging weakly in H1(Ω)to a given element. We want to conclude that the corresponding functionalconverges sequentially to zero. Convergence on the first four terms is strongdue to compactness while convergence on the last term is sequential withwn → w weakly in H1(Ω) and wmt → wt weakly in L2(Ω). Applying (1.101)gives

Ez(T ) ≤ ε

2+CBT

+Cε [Ψ(z, T ) + LOT (z, T )] ≤ ε+Cε [Ψ(z, T ) + LOT (z, T )] ,

for T = Tε large enough, where LOT (z, T ) comprise of all compact terms.Now we use the fact that

Ψ(wn, wm, T ) ≡−∫ T

0

((f(wn)− f(wm), wnt − wmt))

+

∫ T

0

ds

∫ T

s

dτ((f(wn)− f(wm), wnt − wmt))

is weakly sequentially compact - see the details in [CL07a, CL08a].

Step 3: completion. We apply Theorem 17 to deduce the results stated inTheorem 24 and Theorem 25.

Theorem 25 -Structure -follows from asymptotic smoothness and gradientstructure via Theorem 17.

Theorems 26 and 29 - Finite dimensionality and regularity. The keytool is the quasi-stability inequality formulated in Definition 11. This inequal-ity, as in the case of asymptotic smoothness, should hold for the difference of


two solutions z = w − u taken from a bounded set-say an absorbing ball B.As before, we apply two multipliers z and zt.

For the damping term G(zt, z) given in (1.102) we have the following esti-mate (see Proposition 5.3 in [CL07a] for details):

Lemma 5 (Damping). For every ε > 0 there exists Cε such that∫ T

0

|G(zt, z)|dt ≤ε sup[0,T ]

‖z‖21,Ω

(T +

∫ T

0

[((g(wt), wt)) + ((g(ut), ut))

]dτ

)

+ Cε

∫ T

0

((g(ut)− g(wt), zt))dt.

The following decomposition of the source is also essential (see Proposition5.4 in [CL07a]).

Lemma 6 (Source). The following estimate holds true:∣∣∣∣∣∫ T

t

((f(u)− f(w), zt))

∣∣∣∣∣ ≤CB,T ||z||2 + ε

∫ T

0

‖z‖21,Ω

+ Cε,B

∫ T

0

‖z(t)‖21,ΩK(t)dt,

where K(t) = ||wt(t)||2 + ||ut(t)||2 which under the condition g′(s) ≥ m > 0,∀ s, by (1.100) satisfies the following dissipativity property∫ ∞

0

K(s)ds ≡∫ ∞0

[||wt(s)||2 + ||ut(s)||2

]ds < CB. (1.104)

By using dissipation properties in (1.100) and in (1.104) and combining withthe results of Lemma 5 and Lemma 6 one obtains (see [CL07a] for details)the inequality

Ez(t) ≤C1

[e−ωtEz(0) + ||z||2L∞(0,t;H1

0 (Ω)

]exp

C2

∫ t

0

K(s)ds

≤C

[e−ωtEz(0) + ||z||2L∞(0,t;H1

0 (Ω)

]where

K(t) = K(t) + ((g(ut(t)), ut(t))) + ((g(wt(t)), wt(t))).

This leads to the desired quasi-stability inequality.

Theorem 29 -Strong attractors - proof explores smoothness of backwardtrajectories on the attractor, see [Kha10].

Theorem 30-Decay rates. The details given in [CL07a, Section 8].

Theorem 31 -Boundary dissipation. The proof is more technical here.Let us point out the main differences.


Compactness:

Gradient structure: In the boundary case the question of gradient struc-ture has one additional aspect with respect to the interior damping. It in-volves a familiar in control theory question of unique continuation throughthe boundary. This property is known to hold due to various extensions ofHolmgren-type theorems. This aspect of the problem is central in inverseproblems when one is to reconstruct the source from the boundary measure-ments. Under suitable geometric conditions unique continuation results holdfor the wave equation with a potential and H1 solutions [Isa06, Rui92, EIN02].

Reconstruction of the energy: Since the damping is localized near theboundary, reconstruction of the energy involves propagation of the damp-ing from a boundary into the interior. While in the case of interior dampingequipartition of energy was sufficient for recovery of potential energy, this isnot enough in the present case. Additional multiplier is needed which is re-ferred as “flow multiplier” given by h∇w, where h is a suitable vector field.This multiplier -of critical order for the wave equationis successful in recon-structing the energy of a single solution -provided time T is sufficiently largeand the support of the damping on the boundary sufficiently large. However,when dealing with the differences of two trajectories (functions z), the methodis no longer successful. There is a competition of two critical terms. A majordetour is needed when one estimates the difference of two solutions. In theboundary case we have two critical multipliers h∇z and zt both of energylevel:

((f(u)− f(w), h∇z)) and ((f(u)− f(w), zt)).

While this is not a problem with subcritical sources, it becomes a majorissue in the case of critical sources. The remedy for the first term to thisissue is by invoking Carleman’s estimates with large parameter which allowsfor balancing the terms. This step is very technical-details can be found in[CLT09].

Dissipativity integrals: Note that in this case (1.103) requires additionalintegration over the boundary in the right hand side.This corresponds to theboundary source h(s) in (1.15). However, flux multiplier - again - with Car-leman’s estimates allows to get the estimate on this term. At the end of theday, with the use of restrictive growth conditions assumed on g (linear boundat infinity) one obtain reconstruction of the form as in (1.101 without ||zt||2term. The estimate of G(z, zt) is reasonably straightforward due to the linearbounds assumed on the damping functions g(s).

Functional Ψ : The estimate for the compactness part of Ψ is the sameas before. It depends on the source only. This allows for the applicability ofthe same Theorem 14 to conclude compactness of attractors with boundarydissipation.


Quasi-stability and the consequences: Finite dimensionality andsmoothness of attractors. Here the presence of the boundary dampingpresents another major challenge. This is at the level of the inequality inLemma 6. The result stated there depends critically on finiteness of K(t).This is no longer valid in the pure boundary case, where dissipativity inte-gral involves boundary integrals only. In that case one resorts to the analysisof “backward” smoothness on the attractor. This procedure consists of thefollowing four steps.

Step 1: Prove quasi-stability inequality for solutions on the attractor nearstationary points. This leads to consideration of negative time t→ −∞. Suchestimate is possible due to the fact that the system is gradient system andthe velocities wt(t) and ut(t) in (1.104) are “small” near equilibria.

Step 2: Quasi-stability inequality provides additional smoothness of the tra-jectories at some negative time Tf < 0. This smoothness is propagated for-ward on the strength of regularity theorem. The consequence of this is thatthe elements in the attractor A are contained in H2(Ω)×H1(Ω).

Step 3: The ultimate smoothness of attractors is achieved by claiming thatthe attractor is bounded in the topology of H2(Ω)×H1(Ω). The above con-clusion is obtained by exploiting compactness of attractors and finite net cov-erage.

Step 4: Having obtained smoothness of attractor, one proceed to prove finite-dimensionality. This becomes a straightforward consequence of quasi-stabilityproperty on the attractor resulting from the aforementioned smoothness.

It should be noticed that the method described above is fundamentallybased on gradient property of the dynamics. Complete details are given in[CLT09]. We also note that it is possible to avoid “backward” smoothnessmethod in we proof of the quasi-stability property on the attractor (seeLemma 9 below, which is devoted to the case of the boundary damping.

Remark 14. In the case of boundary damping (linearly bounded) there is an-other method that leads to compactness of attractors also in the presence ofcritical sources. This method relies on a suitable split of the dynamics [CEL02].In such case there is no need for Carleman’s estimates. However, propertiessuch as regularity and finite dimensionality of attractors intrinsically dependon quasi-stability estimate. The latter requires consideration of the differencesof two solutions. It is this step that forces Carleman’s estimates to operate.In the subcritical case this is not necessary and more direct estimates providefull spectrum of results also in the boundary case, see [CEL04].

1.5.2 Von Karman plate dynamics

Internal damping

We consider system (1.4) with clamped boundary conditions (1.5) and discusslong time behavior under the following hypothesis.


Assumption 32 We assume that Assumption 3 holds and, in addition, gi ∈C1 and g′i(s) ≥ m for |s| ≥ s0 in the case α > 0 and in the case α = 0: g ∈ C1

and g′(s) ≥ m for |s| ≥ s0.

We also want to be more specific about the nature of the source P (w). Forsake of simplicity, we do not consider non-conservative force cases which leadto systems that are no longer of gradient structure. Instead we limit ourselvesto conservatively forced models. We suppose that Assumption 4 holds withP1(w) ≡ 0, i.e., P (w) is a Frechet derivative of the functional −Π0(w) andthe following property holds: there exist a ∈ R and ε > 0 such that

Π0(w) + a‖w‖22−ε,Ω ≥ 0, ((P0(w), w)) + a‖w‖22−ε,Ω ≥ 0, ∀w ∈ H20 (Ω).(1.105)

Remark 15. In the specific example given in Remark 2 the functional Π0 takesthe form

Π0(u) = −1

2(([F0, u], u))− ((p, u)), F0 ∈ H3+δ(Ω) ∩H1

0 (Ω), p ∈ L2(Ω).

Thus, the assumption (1.105) is satisfied (in the case α > 0 we can take lessregular F0).

Under these conditions concerning P the energy function can be written as

E (u, ut) =1

2

[‖ut‖2 + α‖∇ut‖2

]+

1

4||∆F (u)||2 +Π0(u).

Under the general Assumption 3 we have shown that von Karman systemgenerates a continuous semiflow in the space Hα ≡ H2

0 (Ω) ×Hα(Ω), whereHα(Ω) is given by (1.39), i.e., Hα(Ω) = H1

0 (Ω) in the case α > 0 andHα(Ω) = L2(Ω) in the case α = 0. This leads to a well-defined dynamicalsystem (Hα, St).

The existence of a global attractor under the assumptions specified formodel (1.4) can be derived now from Theorem 14. The main long-time resultreads as follows:

Theorem 33 (Compactness). Let Assumption 32 be in force. Assume thatP (w) = [F0, w] + p, where F0 ∈ H3+δ(Ω) ∩ H1

0 (Ω) and p ∈ L2(Ω). Thenthe dynamical system (Hα, St) generated by equations (1.4) with the clampedboundary conditions (1.5) possesses a global compact attractor A in the spaceHα ≡ H2

0 (Ω)×Hα(Ω), where Hα is given by (1.39).

In the case α > 0 the proof of Theorem 33 follows from general abstractresults for second order equations with subcritical source (see [CL08a]). In thecase α = 0 the force is critical and the proof relies on Theorem 14 in the samemanner as in the case of wave equations. We refer to [CL10] for details.

Remark 16. One can easily obtain the same result for a more general class ofnonlinear sources P (w) which can be axiomatized by (1.105) with an addi-tional requirement that the map is compact.


It is easy to verify (internal damping) that the energy E (u, ut) is a strictLyapunov function for (Hα, St). Hence (Hα, St) is a gradient system (see Def-inition 9 in Section 1.4.3), and the application of Theorem 17 yields the fol-lowing result.

Theorem 34 (Structure). Let Assumption 32 be in force. Assume that Pin (1.4) satisfies (1.105). Let (Hα, St) be the dynamical system generated byequations (1.4) with clamped boundary conditions Assume that (Hα, St) pos-sesses a compact global attractor A. Let N∗ be a set of stationary solutions to(1.4) (see Proposition 1). Then

• A = M u(N ), where N = (w; 0) : w ∈ N∗ is the set of stationary pointof (Hα, St) and M u(N ) is the the unstable manifold M u(N ) emanatingfrom N which is defined as a set of all U ∈ Hα such that there exists afull trajectory γ = U(t) = (u(t);ut(t)) : t ∈ R with the properties

U(0) = U and limt→−∞

distHα(U(t),N ) = 0.

• The global attractor A consists of full trajectories γ = (u(t);ut(t)) : t ∈R such that

limt→±∞

‖ut(t)‖2Hα

+ infw∈N∗

‖u(t)− w‖22

= 0.

• Any generalized solution u(t) to problem (1.4) stabilizes to the set of sta-tionary points; that is,

limt→+∞

‖ut(t)‖2Hα

+ infw∈N∗

‖u(t)− w‖22

= 0. (1.106)

This theorem along with generic-type results on the finiteness of the num-ber of solutions to the stationary problem allow us to obtain the followingresult.

Corollary 2. Under the hypotheses of Theorem 34 there exists an open denseset R0 in L2(Ω) such that for every load function p ∈ R0 the set N ofstationary points for (Hα, St) is finite. In this case A = ∪Ni=1M

u(zi), wherezi = (wi; 0) and wi is a solution to the stationary problem Moreover,

• The global attractor A consists of full trajectories γ = (u(t);ut(t)) : t ∈R connecting pairs of stationary points; that is, any W ∈ A belongs tosome full trajectory γ and for any γ ⊂ A there exists a pair w−, w+ ⊂N ∗ such that

limt→−∞

‖ut(t)‖2Hα

+ ‖u(t)− w−‖22

= 0

andlim

t→+∞

‖ut(t)‖2Hα

+ ‖u(t)− w+‖22

= 0.


• For any (u0;u1) ∈ Hα there exists a stationary solution w ∈ N ∗ such that

limt→+∞

‖ut(t)‖2Hα

+ ‖u(t)− w‖22

= 0, (1.107)

where u(t) is a generalized solution to problem (1.4) with initial data(u0;u1) and with the clamped boundary conditions.

Remark 17. We note that the generic class R0 of loads in the statement ofCorollary 2 can be rather thin. The standard example (see, e.g., [Chu99, Sec-tion 2.5]) is the set R0 in the interval [0, 1] of the form

R0 =x ∈ (0, 1) : ∃ rk ∈ Q, |x− rk| ≤ ε2−k−1

,

where Q is the sequence of all rational numbers. This set is open and dense in[0, 1], but its Lebesgue measure less than ε. However using analyticity of thevon Karman force terms and the Lojasiewicz-Simon inequality we can obtaina result (see [Chu12b]) on convergence for all loads p under some additionalnon-degeneracy at the origin type conditions imposed on the damping terms.By assuming this additional non-degeneracy condition (always satisfied wheng′(0) > 0 or in the hyperbolic case when g(s)s ≥ ms2, |s| ≤ 1) the result ofCorollary 2 holds for all loads p ∈ L2(Ω). For general discussion of Lojasiewicz-Simon method we refer to [HJ99, HJ09] and to the references therein.

It follows from Assumption 32 (see [CL10], for instance) that there existsa strictly increasing continuous concave function H0(s) such that

s21 + s22 ≤ H0(s1gi(s1) + s2gi(s2)) for all s1, s2 ∈ R.

in the case α > 0 and s2 ≤ H0(sg(s)) for s ∈ R in the case α = 0.The above facts are used in formulation of the result on the rate of

convergence to an equilibrium.

Theorem 35 (Rates of convergence to equilibria). Assume that P sat-isfies (1.105) in (1.4) and the hypotheses above that guarantee the existenceof a compact global attractor A for the dynamical system (Hα, St) generatedby problem (1.4) with the clamped boundary conditions hold.

If the set of stationary solutions consists of finitely many isolated equilib-rium points that are hyperbolic,5 then for any initial condition y ∈ Hα thereexists an equilibrium point e = (w; 0), w ∈ H2

0 (Ω), such that

‖Sty − e‖2Hα ≤ C · σ([tT−1

]), t > 0, (1.108)

where C and T are positive constants depending on y and e, [a] denotes theinteger part of a and σ(t) satisfies the following ODE,

5 In the sense that the linearization of the stationary problem with the correspond-ing boundary conditions around every stationary solution has a trivial solutiononly.


dσ

dt+Q(σ) = 0, t > 0, σ(0) = C(y, e). (1.109)

Here C(y, e) is a constant depending on y and e,

Q(s) = s− (I +G0)−1

(s) with G0(s) = c1 (I +H0)−1

(c2s), (1.110)

where positive numbers c1 and c2 depend on y and e. If in Assumption 32s0 = 0, then the rate of convergence in (1.108) is exponential.

Remark 18. It is also possible to state a result on the convergence rate with-out assuming hyperbolicity of equilibrium points (see [Chu12b]). However asin the case described in Remark 17 this requires additional non-degeneracytype properties of the damping terms. Moreover, the lack of hyperbolicity iscompromised by slowing down the decay rates.

The following result describes finite dimensionality of attractors.

Theorem 36 (Finiteness of dimension). Assume the hypotheses whichguarantee the existence of a compact global attractor A for the system (Hα, St)generated by problem (1.4) with the boundary conditions (1.5) hold (see The-orem 33) and α ≥ 0. In addition, we assume that

(i) In the case α = 0 there exist m,M0 > 0 such that

0 < m ≤ g′(s) ≤M0[1 + sg(s)], s ∈ R, (1.111)

(ii) When α > 0 we assume that (i) g(s) is either of polynomial growth atinfinity or else satisfies (1.111), and (ii) there exists 0 ≤ γ < 1 such thatthe functions gi satisfy the inequality

0 < m ≤ g′i(s) ≤M [1 + sgi(s)]γ , s ∈ R, i = 1, 2, (1.112)

Then the fractal dimension of the attractor A is finite.

Remark 19. One can see that in the case α > 0 the condition imposed on gi in(1.112) are valid if in addition to Assumption 32 we assume that that thereexists m,M1,M2 > 0 such that

0 ≤ g′i(s) ≤M1[1 + |s|p−1], sgi(s) ≥ m|s|(p−1)/γ −M2, s ∈ R,

for some p ≥ 1 (see Remark 9.2.7 [CL10]).

The following assertion on regularity of elements in the attractor is astraightforward consequence of Theorem 21.

Theorem 37 (Regularity). Let the hypotheses of Theorem 36 be valid. Thenany full trajectory γ = (u(t);ut(t)) : t ∈ R from the attractor A of the dy-namical system (Hα, St) generated by problem (1.18) with the boundary con-ditions (1.5) possesses the properties

u(t) ∈ Cr(R;W ), (ut(t), utt(t)) ∈ Cr(R;Hα), (1.113)

where Cr means right-continuous functions and


• in the case α > 0: Hα = H20 (Ω)×H1

0 (Ω) and W = H3(Ω) ∩H20 (Ω);

• in the case α = 0: Hα = H20 (Ω)× L2(Ω) and W = H4(Ω) ∩H2

0 (Ω).

Moreover,

• in the case α > 0: A ⊂ H3(Ω)×H2(Ω) and

supt∈R

(‖u(t)‖23 + ‖ut(t)‖22 + ‖utt(t)‖21

)≤ CA <∞;

• in the case α = 0: A ⊂ H4(Ω)×H2(Ω) and

supt∈R

(‖u(t)‖24 + ‖ut(t)‖22 + ‖utt(t)‖21

)≤ CA <∞.

The validity of quasi-stability property for the system allows us to deduce,without an additional effort, another important property of dynamical sys-tem. This is existence of strong attractors that is, attractors in a strongtopology determined by the generators of the (linearized) dynamical system.The corresponding result is formulated below.

Theorem 38 (Strong attractors). We assume that the hypotheses of The-orem 37 are in force.

Case α > 0: Let the rotational damping be linear; that is, gi(s) = gi · sfor i = 1, 2. Then the global attractor A is also strong: for any bounded set Bfrom Hst = (H3(Ω) ∩H2

0 (Ω))×H20 (Ω) we have that

limt→∞

supy∈B

distHst(Sty,A) = 0.

Case: α = 0: Let Hst = (H4(Ω)∩H20 (Ω))×H2

0 (Ω). The global attractor Aof the system (H0, St) generated by the generalized solutions to problem (1.4)with clamped b.c. (1.5) is also strong, i.e. A is a global attractor for the system(Hst, St).

Moreover, in both cases α = 0 and α > 0 the global attractor A has a finitedimension as a compact set in Hst.

Proof. We follow the approach presented for the proof of Theorem 8.8.4[CL10].The hypotheses of Theorem 37 guarantees that the system (Hα, St) is quasi-stable.

An important property of the dynamical system is an existence of ex-ponential attractors, which attract trajectories at the exponential rate. Thequasi-stability property allows us to deduce an existence of exponentialattractors. The corresponding result is formulated below.

Theorem 39 (Exponential attractors). Let (Hα, St) be the dynamical sys-tem generated by problem (1.4) and (1.5).

Case α > 0: Under the condition (1.112) the dynamical system (Hα, St)has a (generalized) fractal exponential attractor A (see Definition 12) whose


dimension is finite in the space H1(Ω) ×W ′, where W ′ is a completion ofH1

0 (Ω) with respect to the norm ‖ · ‖W ′ = ‖(1− α∆) · ‖−2.Case α = 0: Assume the validity of the hypotheses above which guarantee

the existence of a global finite-dimensional attractor. Assume in addition that

|g(s)| ≤ C (1 + sg(s))γ, s ∈ R, (1.114)

for some 0 ≤ γ < 1. Then the system (H0, St) has a (generalized) fractal ex-ponential attractor A whose dimension is finite in the space L2(Ω)×H−2(Ω).

Main ingredients for the proofs. The rotational case α > 0 correspondsto the situation where von Karman nonlinearity is compact, hence subcritical.This alleviates number of technical issues when proving validity of quasi-stability inequality which, in turn, leads to the existence of smooth and finite-dimensional attractors. Indeed, build in compactness turns critical integralsinto lower order terms.

We thus focus here on the more subtle non-rotational case, α = 0. Here, themain challenge is to establish quasi-stability inequality. Once this is proved,finite-dimensionality of attractors - Theorem 36, smoothness of attractor -Theorem 37 and existence of strong and exponential attractors -Theorem 38and also Corollary 39 follow from the abstract tools presented in Section 1.4.4.

Thus, the core of the difficulty relates to the quasi-stability estimate ob-tained for a difference of two solutions z = u−w defined onQ = Ω×(0, T ), Σ =Γ × (0, T ). The solutions under consideration are confined to a bounded in-variant set (attractor, for instance), so we can assume

‖ut(t)‖2 + ‖u(t)‖22,Ω ≤ R2, ‖wt(t)‖2 + ‖w(t)‖22,Ω ≤ R2.

To present the main idea we assume that P (u) ≡ 0 in (1.4). In this case theequation for z can be written as:

ztt +∆2z = f in Ω, z = 0,∂

∂nz = 0 on Γ, (1.115)

where f = −a(g(ut) − g(wt)) + [v(u), u] − [v(w), w]. We have the followingenergy equality (satisfied for strong solutions)

Ez(T ) +DTt (z) = Ez(t) +

∫ T

t

(R(z), zt)Ωdt, (1.116)

where

Ez(t) =1

2

∫Ω

[|zt|2 + |∆z|2]dx, R(z) = [v(u), u]− [v(w), w],

and

DTt (z) =

∫ T

t

∫Ω

a[g(ut)− g(wt)]ztdxdτ.


Our goal is to prove the following recovery inequality:

Ez(T ) +

∫ T

0

Ez(t)dt ≤ CRDT0 (z) + LOT (z), (1.117)

whereLOT (z) ≤ CR sup

t∈[0,T ]

‖z(t)‖22−ε,Ω .

Inequality (1.117), via reiteration on the intervals [mT, (m+1)T ], leads to thequasi-stability inequality -see Definition 11.

To prove (1.117) one applies standard multipliers zt and z. After somecalculations (as for the wave equation) one obtains

Ez(T ) +

∫ T

0

Ez(t)dt ≤ CRDT0 (z) + CRLOT (z) + CT

∫ T

0

(R(z), zt)dt.

(1.118)

The difficulty, however, is in handling the critical term∫ T

0

(R(z), zt)Ωdt. (1.119)

In order to obtain quasi-stability inequality, this critical term should be rep-resented in terms of the damping DT

0 (z), small multiples of the integrals of

the energy, i.e., ε∫ T0Ez(t)dt and lower order terms (subcritical quantities) in

a quadratic form. The key to this argument is ”compensated compactnessstructure” of the von Karman bracket which leads to the following estimate:

|∫ T

0

(R(z), zt)dτ | ≤ CR maxt∈[0,T ]

‖z(t)‖22−ε + CR

∫ T

0

(‖ut‖+ ‖wt‖)‖z‖22,Ωdt

(1.120)The critical role is played by the presence in (1.120) of the velocities ‖ut‖,‖wt‖ which represent the damping and obey the estimate∫ +∞

0

(‖ut‖2 + ‖wt‖2)dt ≤ CR (1.121)

Indeed, once (1.120) is proved and inserted into (1.118), then (1.121) alongwith Cronwall’s inequality leads to the desired quasi-stability estimate. Thus,the key is to be able to prove (1.120). To this end the following “compensatedcompactness” decomposition of the von Karman bracket is used:

(R(z), zt) =1

4

d

dtQ(z) +

1

2P (t)

where


Q(z) = (v(u) + v(w), [z, z])− |∆v(u+ w, z)|2, (1.122)

and

P (z) =− (ut, [u, v(z, z)])− (wt, [w, v(z, z)])

− (ut + wt, [z, v(u+ w, z)]), (1.123)

where v(u,w) = ∆−2([u,w]

). The above yields∫ T

0

(R(z), zt)dt ≤ C(R)[Q(T )−Q(0)] +1

2

∫ T

0

P (z)dt. (1.124)

By using sharp Airy’s regularity

|v(u, v)|W 2,∞ ≤ C‖u‖2,Ω‖v‖2,Ω

given by Lemma 2, one obtains the estimates

|Q(z)| ≤ C‖z‖22−ε,Ω and |P (z)| ≤ CR(‖ut‖+ ‖wt‖)‖z‖22,Ω ,

which yields∫ T

0

(R(z), zt)dt ≤ CR∫ T

0

(‖ut‖+ ‖wt‖)‖z‖22,Ω + CR maxt∈[0,T ]

‖z(t)‖22−ε

as desired for (1.120), hence for quasi-stability inequality.

Boundary damping

We consider next the model in (1.18) with the boundary conditions in (1.19).This type of problems is often referred to as control problem with reducednumber of controls. This is to say that a dissipation affects only one of thetwo boundary conditions.

We impose the following hypotheses.

Assumption 40 • Assumption 3(1,2) is in force, a(x) ∈ L∞(Ω) is non-negative almost everywhere and P (w) ≡ p ∈ L2(Ω).

• The boundary damping g0 ∈ C1(R) is increasing, with the property g0(0) =0 and there exist positive constants m, M , and s0 such that

0 < m ≤ g′0(s) ≤M for |s| ≥ s0. (1.125)

We start with

Rotational Case α > 0. The state space H ≡[H2(Ω)∩H1

0 (Ω)]×H1

0 (Ω).Using energy equality one can easily prove the following assertion.

Theorem 41. Let Assumption 40 be in force and α > 0. Then


• There exists R∗ > 0 such that the set

WR = y = (u0;u1) ∈ H : E (u0, u1) ≤ R (1.126)

is a nonempty invariant set with respect to semiflow St generated by equa-tions (1.18) with hinged b.c. (1.19) for every R ≥ R∗. Moreover, the setWR is bounded for every R ≥ R∗ and any bounded set is contained in WR

for some R.• If in addition a(x) > 0 almost everywhere in Ω, and g1 > 0 in (1.8) the

system (H,St) is gradient.• The set N of all stationary points of the semiflow St is bounded in H.

Thus there exists R∗∗ > 0 such that N ⊂ WR for all R ≥ R∗∗.

We recall that

N = V ∈ H : StV = V for all t ≥ 0 .

and every stationary point W has the form W = (u; 0), where u = u(x) ∈H2(Ω) is a weak (variational) solution to the problem

∆2u = [v(u), u] + p in Ω; u = 0, ∆u = 0 on Γ, (1.127)

with function v(u) satisfying (1.6) with w = u, i.e., v(u) = ∆−2([u, u]

).

Theorem 42 (Compact attractors). Let Assumption 40 be in force andα > 0. Then

• The restriction (WR, St) of the dynamical system (H,St) on WR given by(1.126) has a compact global attractor AR ⊂ WR for every R ≥ R∗, whereR∗ is the same as in Theorem 41.

• If, in addition, a(x) > 0 almost everywhere in Ω, g1 > 0, then there existsa compact global attractor A for the system (H,St). Moreover, we haveA = M u(N ), where M u(N ) is the unstable manifold (see the definitionin Section 1.4.3) emanating from the set N of equilibria for the semiflowSt.

• The two attractors AR and A have finite fractal dimension provided rela-tion (1.125) holds for all s ∈ R. Moreover, in this latter case the attrac-tors are bounded sets in the space H3(Ω)×H2(Ω) and for any trajectory(u(t);ut(t)) we have the relation

‖utt(t)‖1,Ω + ‖ut(t)‖2,Ω + ‖u(t)‖3,Ω ≤ C, t ∈ R.

The important ingredient of the argument is the following ObservabilityEstimate.

Proposition 3 (Observability estimate). Assume that Assumption 40 isin force and α > 0. Let U(t) = (u(t);ut(t)) = Sty1 and W (t) = (w(t);wt(t)) =Sty2 be two solutions corresponding to initial conditions y1 and y2 from the


set WR given by (1.126). Then there exist T0 > 0 and constants C1(T ) andC2(R, T ) such that

TEz(T ) +

∫ T

0

Ez(t)dt ≤ C1(T )

∫Σ1

∣∣∣∣ ∂∂nzt

∣∣∣∣2 dΣ +C2(R, T ) ·LOT (z) (1.128)

for any T ≥ T0, where z ≡ u− w,

Ez(t) =1

2

∫Ω

[|zt|2 + α|∇zt|2 + |∆z|2

]dx,

and the lower-order terms have the form

LOT (z) = sup0≤τ≤T

‖z(τ)‖20,Ω +

∫ T

0

‖zt(τ)‖20,Ωdτ. (1.129)

We can obtain results on convergence rates of individual trajectoriesto equilibrium under the condition that the set N of stationary points isdiscrete.

As in the previous section we introduce a concave, strictly increasing,continuous function H0 : R+ 7→ R+ which captures the behavior of g0(s) atthe origin possessing the properties

H0(0) = 0 and s2 ≤ H0(sg0(s)) for |s| ≤ 1. (1.130)

We define a function Q(s) by relations (1.110) and consider the differentialequation

dσ

dt+Q(σ) = 0, t > 0, σ(0) = σ0 ∈ R+, (1.131)

which admits the global unique solution σ(t) decaying asymptotically to zeroas t→∞.

With these preparations we are ready to state our result.

Theorem 43 (Rate of stabilization). Let the hypotheses of Theorem 42be valid with a(x) > 0 a.e. Assume that there exist γ > 0 and s0 > 0 such thatthe interior damping g(s) satisfies the relation sg(s) ≥ γs2 for |s| ≤ s0. Inaddition assume that problem (1.127) has a finite number of solutions. Thenfor any V ∈ H there exists a stationary point E = (e; 0) such that StV → Eas t→ +∞. Moreover, if the equilibrium E is hyperbolic in the sense that thelinearization of (1.127) around each of its solutions has the trivial solutiononly, then there exist C, T > 0 depending on V,E such that the followingrates of stabilization

‖StV − E‖H ≤ Cσ([tT−1]), t > 0,

hold, where [a] denotes the integer part of a and σ(t) satisfies (1.131) with σ0depending on V,E ∈ H (the constants ci in the definition of Q also dependon V and E). In particular, if g′0(0) > 0, then


‖StV − E‖H ≤ Ce−ωt

for some positive constants C and ω depending on V,E ∈ H.

Remark 20. The additional assumption that g(s)s ≥ γs2, γ > 0 is a technicalassumption required to prove that generalized solutions are also weak solu-tions. This property is essential in proving convergence to equilibria states.

Now we consider

Irrotational case: α = 0 and H ≡[H2(Ω) ∩H1

0 (Ω)]× L2(Ω):

We consider a model with nonlinear boundary dissipation acting via hingedboundary conditions which does not account a for regularizing effects of rota-tional inertia. Thus, the corresponding solutions are less regular than in thecase of rotational models.

The following analog of Theorem 41 is valid.

Theorem 44. Let Assumption 40 hold and α = 0. Then

• There exists R∗ > 0 such that the set

WR = y = (u0;u1) ∈ H : E (u0, u1) ≤ R (1.132)

is a nonempty bounded set in H for all R ≥ R∗. Moreover, any boundedset B ⊂ H is contained in WR for some R and the set WR is invariantwith respect to the semiflow St.

• If in addition a(x) > 0 then the system (H,St) is gradient.

Our main goal is to prove the global attractiveness property for the dy-namical system (H,St). This property requires additional hypotheses imposedon the data of the problem.

Assumption 45 The interior damping function g(s) is globally Lipschitz,i.e., |g(s1)− g(s2)| ≤M |s1 − s2| for all s1, s2 ∈ R.

This global Lipschitz requirement is due to the fact that only one boundarycondition is used as a source of dissipation.

Our main result is the following.

Theorem 46 (Compact attractors). We suppose that Assumptions 40 and45 are in force. Then

• For any R ≥ R∗ there exists a global compact attractor AR for the restric-tion (WR, St) of the dynamical system (H,St) on WR, where WR is givenby (1.132).

• If we assume additionally that a(x) > 0 a.e. in Ω and g(s)s > 0 for alls 6= 0, then there is R0 > 0 such that AR does not depend on R for allR ≥ R0. In this case A ≡ AR0 is a global attractor for (H,St) and Acoincides with the unstable manifold M u(N ) emanating from the set Nof stationary points for St. Moreover, limt→+∞ distH(StW,N ) = 0 forany W ∈ H.


• The global attractors AR and A are bounded sets in H3(Ω)×H2(Ω) andhave a finite fractal dimension provided the relation in (1.125) holds forall s ∈ R.

From Theorem 46 we obtain the following corollary.

Corollary 3. Let the hypotheses of Theorem 46 be in force. Assume thata(x) > 0 a.e. in Ω and g(s)s > 0 for all s 6= 0. Then the global attractorA consists of full trajectories γ = W (t) : t ∈ R such that

limt→−∞

distH(W (t),N ) = 0 and limt→+∞

distH(W (t),N ) = 0.

In particular, if we assume that equation (1.127) has a finite number of so-lutions, then the global attractor A consists of full trajectories γ = W (t) :t ∈ R connecting pairs of stationary points: any W ∈ A belongs to some fulltrajectory γ and for any γ ⊂ A there exists a pair Z,Z∗ ⊂ N such thatW (t) → Z as t → −∞ and W (t) → Z∗ as t → +∞. In the latter case forany V ∈ H there exists a stationary point Z such that StV → Z as t→ +∞.

In analogy with the previous cases one can also provide statements forthe rate of stabilizations to equilibria. The statement (and the proofs) do notdepend on the specific boundary conditions. However, in this case we do notneed to assume additional coercivity estimates for the interior damping. Thepoint is that our basic Assumptions 40 and 45 are sufficient to conclude thatgeneralized solutions are weak (variational).

Main ingredients for the proofs: Similar to the interior damping case,Theorem 14 provides the main tool for establishing asymptotic smoothness.For sake of avoiding repetitions, we shall mainly emphasize the parts of theproof that are more specific to the boundary damping. The proof is dividedinto several steps which are presented below. As before, we use notation Q ≡Ω×(0, T ), Σ ≡ Γ ×(0, T ) and, for sake of simplicity of the exposition, assumethat P (u) ≡ 0 in (1.18).

We can assume that u and w are strong (smooth) solutions. Then thedifference z ≡ u− w solves the following problem

ztt +∆2z = f in Ω, z = 0, ∆z = ψ on Γ, (1.133)

where f = −a(x)(g(ut) − g(wt)) + [v(u), u] − [v(w), w] and the boundaryconditions are given by

ψ = −[g0

(∂

∂nut

)− g0

(∂

∂nwt

)].

We have the following energy equality (which is satisfied for strong solutions)

Ez(T ) +DTt (z) = Ez(t) +

∫ T

t

((R(z), zt))dt, (1.134)


where

Ez(t) =1

2

∫Ω

[|zt|2 + |∆z|2]dx, R(z) = [v(u), u]− [v(w), w],

and

DTt (z) =

∫ T

t

∫Ω

a[g(ut)− g(wt)]ztdxdτ + DTt (z),

with

DTt (z) =

∫ T

t

∫Γ

[g0

(∂

∂nut

)− g0

(∂

∂nwt

)]∂

∂nztdΓdτ.

We also appeal to the following technical lemma obtained by using “flux”multipliers -see [CL07b].

Lemma 7. Let T > 0 be given. Let φ ∈ C2(R) be a given function with supportin [δ, T − δ] , where δ ≤ T/4, such that 0 ≤ φ ≤ 1 and φ ≡ 1 on [2δ, T − 2δ].Then any strong solution z to problem (1.133) satisfies the following inequality∫ T

0

Ez(t)φ(t)dt ≤ C1

∫ T

0

Ez(t)|φ′(t)|dt+ C2

∫Σ

[|ψ|2 +

∣∣∣∣ ∂∂nz

∣∣∣∣2]dΣ

+

∫ T

0

∫Ω

fh∇zφdxdt+

∫ T

0

∫Ω

R(z)ztdxdt+ C3 ·BT (z), (1.135)

where the constants Ci do not depend on T and the boundary terms6 BT (z)is given by

BT (z) ≡∫Σ

∣∣∣∣∣(∂

∂n

)2

z

∣∣∣∣∣2

+

∣∣∣∣ ∂∂n

∂

∂τz

∣∣∣∣2φdΣ +

∫ T

0

∥∥∥∥ ∂∂n∆z

∥∥∥∥2−1,Γ

φdt.

(1.136)

On the next step we eliminate the second- and third-order boundary traceson the boundary in the expression (1.136) for BT (z). These second-order“supercritical” boundary traces are due to the fact that the dissipation isallowed to affect the system via only one boundary condition. Traces of thesecond and third order (see (1.136)) are above the energy level, so these cannotenter the estimates. It is the microlocal analysis argument, again, that allowsfor the elimination of these terms. Handling of the boundary terms requiresdelicate trace estimates used already in the case α > 0. However the treatmentof critical source (giving rise to the term R(z)) is more troublesome now. Thedissipativity integral is supported on the boundary and not in the interior.

6 They are not defined on the energy space. These are higher-order boundary tracesof solutions.


Here are few details. For the trace result we use a more general trace estimateproved in [JL99] (see Proposition 1 and Lemma 4) valid for the linear Kirchoffproblem (note that the estimates in [JL99] are independent of the parameterrepresenting rotational forces). Thus, these estimates are applicable to bothKirchoff and Euler–Bernoulli (1.133) models. As a consequence we have thefollowing

Lemma 8. Let z be a solution to linear problem (1.133) with given f and ψand BT (z) be given by (1.136). Then there exist constants CT > 0 such thatfor any 0 < η < 1

2 the following estimate holds

BT (z) ≤ C1,T

∫Σ

(|ψ|2 +

∣∣∣∣ ∂∂nzt

∣∣∣∣2)dΣ

+ C2,T

[‖z‖2C([0,T ];H2−η(Ω)) + ‖zt‖2L2([0,T ];H−η(Ω)) +

∫ T

0

‖f‖2−η,Ωdt

].

The above calculations along with the Compactness Theorem 14 imply ex-istence of compact attractor -first statement in Theorem 46. As for smooth-ness and finite dimensionality, one needs to prove quasi-stability estimate-which amounts to the proof of the estimate in Proposition 3 -but with α = 0.This means that von Karman bracket is no longer subcritical and the term(R(z), zt) needs to be estimated by appropriate damping on the boundary. Toachieve this we proceed as in the internal case- by decomposing the bracketinto Q(z) and P (z). However, in the boundary case the additional difficulty isdue to the fact that dissipativity estimate (1.121) no longer holds. What onehas, instead, is similar quantity on the boundary. In order to take advantageof this relaxed dissipation, one proceeds in the following fashion. In the casewhen the system is gradient (when a(x) > 0, g1 > 0) similar strategy as in thecase of wave equation with boundary dissipation can be applied. This relies onexploiting the structure of the attractor as unstable manifold along with theconvergence of solutions to stationary points. In doing this one is considering(i) smoothing effect of backward trajectories on the attractor, (ii) propagatingit forward and (iii) using the compactness of the attractor. This is technicalpart of the argument with full details given in [CL07b, Section 4.2].

However, when the system does not have gradient property, the relianceon convergence to stationary points is no longer a tool. In that case, quasi-stability estimate can be obtained differently by exploiting already obtainedcompactness of a local attractor AR for R > R∗ along with density of H2(Ω)in L2(Ω).

To explain this idea with more details we show how we can get the esti-mates for the noncompact term involving R(z) -see (1.119) on the attractorcan be handled (this is the most critical part of the argument leading to aquasi-stability estimate, see (1.120) in the case of internal damping). We dothis in the following assertion.


Lemma 9. Let

γu = (u(t), ut(t)) : t ∈ R and γw = (w(t), wt(t)) : t ∈ R

be two trajectories from the attractor AR and z(t) ≡ u(t)−w(t). Then for anyε > 0 there exists C(ε, R) such that∣∣∣ ∫ t

s

(R(z), zt)∣∣∣ ≤ C(ε, T,R) sup

τ∈[s,t]||z(τ)||22−η + ε

∫ t

s

||z(τ)||22 (1.137)

for all s, t ∈ R with 0 ≤ t− s ≤ T and for some η > 0.

Proof. The starting point leading to (1.137) is the formula in (1.124). Straight-forward estimates performed on compact term Q(t) lead to∣∣∣ ∫ t

s

(R(z), zt)∣∣∣ ≤ C(R) sup

τ∈[s,t]||z(τ)||22−η +

1

2

∫ t

s

|P (z)|, (1.138)

where P (z) constitutes critical term in the decomposition of von Karmanbracket and is given by (1.123), i.e.,

P (z) = − (ut, [u, v(z, z)])− (wt, [w, v(z, z)])− (ut + wt, [z, v(u+ w, z)]) .

Our main goal is to handle the second term on the right hand side of (1.138)which is of critical regularity. We emphasize that in contrast with argumentgiven in [CL10, Section 10.5.3] do not use backward decaying of the velocitieson the attractor, i.e., the fact that ||wt(t)||+ ||ut(t)|| → 0 as t→ −∞. Instead,we use the already established compactness of the attractor. We recall theattractor is a compact set in (H2 ∩H1

0 )(Ω)× L2(Ω).Since for every τ ∈ R, the element ut(τ) belongs to a compact set in L2(Ω),

by density of H2(Ω) in L2(Ω) we can assume, without a loss of generality, thatfor every ε > 0 there exists a finite set φj ⊂ (H2∩H1

0 )(Ω) , j = 1, 2, ..., n(ε),such that for each τ ∈ R we can find indices j1(τ) and j2(τ) so that

||ut(τ)− φj1(τ)||+ ||wt(τ)− φj2(τ)|| ≤ ε, for all τ ∈ R. (1.139)

Let

Pj1,j2(z) ≡ − (φj1 , [u, v(z, z)])− (φj2 , [w, v(z, z)])− (φj1 + φj2 , [z, v(u+ w, z)])

where z(t) = w(t)− u(t). It can be easily shown that

||P (z(τ))− Pj1(τ),j2(τ)(z(τ))|| ≤ εC(R)||z(τ)||22 (1.140)

uniformly in τ ∈ R.Now we estimate the terms Pj1,j2(z). Starting with the estimate

||[u,w]||−2 ≤ C||u||2−β ||w||1+β , ∀β ∈ [0, 1)


(see equation(1.4.17) in [CL10, p.41]) and exploiting elliptic regularity oneobtains

||[u, v(z, w)]||−2 ≤C||u||2−β ||∆−2[z, w]||β+1 ≤ C||u||2−β ||∆−2[z, w]||2≤C||u||2−β ||[z, w]||−2 ≤ C||u||2−β ||z||2−β1

||w||1+β1, (1.141)

where above inequality holds for every β, β1 ∈ [0, 1). Recalling the additionalsmoothness of φj ∈ (H2 ∩H1

0 )(Ω), along with the estimate in (1.141) appliedwith appropriate β, β1, and accounting the structure of Pj1,j2(z) terms oneobtains:

||Pj1,j2(z)|| ≤ C(R)(||φj1 ||2 + ||φj2 ||2

)||z(τ)||22−η

for some η > 0. So we have

supj1,j2

||Pj1,j2(z)|| ≤ C(ε, R)||z(τ)||22−η (1.142)

for some η > 0 and all ε > 0, where C(ε, R) → ∞ when ε → 0. Taking intoaccount (1.140) and (1.142) in (1.138) we obtain (1.137).

Final form of quasistability estimate is obtained now by inserting inequal-ities obtained in Lemma 7, Lemma 8, Lemma 9 into the previous program ofderivation of Quasi-stability estimate.

Generalizations

1. The case of nonconservative forces can also be considered. This is to saythat the force P (u) is not potential. In that case the system has no longergradient structure and one needs first to establish existence of an ab-sorbing set. This can be done under some additional restrictions on thedamping, see [CL10], where the case “potential operator + (ψ,∇)w” isconsidered.

2. More general sources P (w) can be considered, including nonlinear sources.However, in general, one needs to make appropriate compactness hypothe-ses, see the Remark 16.

3. Dissipation in other (e.g., free) boundary conditions, see [CL10, CL04b,CL07b].

Open questions

1. General theory of non-conservative forces which destroy gradient struc-ture.

2. Localized dissipation, i.e., dissipation localized in a suitable layer near theboundary.


3. Boundary damping problems without the light damping. This has to dowith the existence of strict Lyapunov function and related unique con-tinuation across the boundary. While various unique continuation resultsfrom the boundary are available also for plates [Alb00], the non-local na-ture of the von Karman bracket prevents Carleman’s estimates [Alb00]from applicability.

1.5.3 Kirchhoff -Boussinesq plate

Interior damping. Case α > 0:

In this case our main results presented in the following theorem (for the proofswe refer to [CL08a, Chapter 7]).

Theorem 47. Let α > 0 and P (w) = ∆[w2] − %|w|l−1w for some % ≥ 0and l ≥ 1. Assume that the damping functions g and G have the structure7

described in (1.8). Then problem (1.12) with α > 0 and the clamped boundaryconditions (1.5) generate a continuous semiflow St in the space H ≡ H2

0 (Ω)×H1

0 (Ω).Assume in addition that either

m ≤ 3, % > 0, l ≤ m, infx∈Ωa(x) > 0

or else

m < 3, % = 0, infx∈Ωa(x) > 0 is sufficiently large

.

Then the semiflow St possesses a global compact attractor A. Moreover, if theconstant G1 in (1.8) is positive, then

• the fractal dimension of the attractor A is finite;• the system possesses a fractal exponential attractor (see Definition 12)

whose dimension is finite in the space H10 (Ω)×W , where W is a completion

of H10 (Ω) with respect to the norm ‖ · ‖W = ‖(1− α∆) · ‖−2.

Remark 21. (1) As we see the additional potential term ρ|u|l−1u in the forceP (w) allows us to dispense with a necessity of assuming large values for thedamping parameter.

(2) The requirements concerning the damping terms in Theorem 47 are notoptimal. We choose them for the sake some transparency. The same resultsunder much more general hypotheses concerning damping functions can befound in [CL08a, Chapter 7].

7 see Remark 21(2) below.


Interior damping. Case α = 0 with linear damping function:

We assume that g(s) = s and consider the source term of the form

P (w) = σ∆[w2]− %|w|l−1w with σ, % ≥ 0. (1.143)

As a phase space we take H ≡ H20 (Ω)× L2(Ω).

Theorem 48 (Compact attractor). Let

σ2 <1

4kmin1, k with k ≡ inf

Ωa(x) > 0. (1.144)

Then the dynamical system (H,St) generated by equations (1.12) with α = 0possesses a compact global attractor A.

If σ = 0, then the system (H,St) is gradient and thus A = M u(N ) isunstable manifold in H emerging from the set N of equilibria.

We note that the system under consideration is not a gradient system whenσ 6= 0. As a consequence, the proof of existence of global attractor can not bejust reduced to the proof of asymptotic smoothness. One needs to establishfirst existence of an absorbing ball. For this we use Lyapunov type functionof the form

Vε(t) = E (t) + ε

∫Ω

w(t)wt(t)dx, (1.145)

with ε = 14 min1, k, see [CL06b, CL11] for details in the case when a(x) ≡ k

is a constant. Here the energy functional has the form

E (t) ≡ 1

2||wt(t)||2 +

1

2||∆w(t)||2 +

1

4

∫Ω

|∇w(x, t)|4dx

+ σ

∫Ω

w(x, t)|∇w(x, t)|2dx+%

l + 1

∫Ω

|w(x, t)|l+1dx.

In order to complete the proof of the existence of global attractor, henceof Theorem 48, we use Ball’s method (see Theorem 15).

Let us take Vε(t) with ε = k/2 and denote Ψ(t) = Vk/2(t). Using the factthat energy identity holds for all weak solutions one can easily see that thefollowing equality is satisfied for this Lyapunov’s function Ψ(t):

∂tΨ(t) + kΨ(t) +

∫Ω

(a(x)− k)|wt|2dx = K(w(t)),

where

K(w(t)) =

∫Ω

(σ|∇w|2 − k

2((a(x)− k)w

)wtdx

− kσ∫Ω

w|∇w|2dx− k

4

∫Ω

|∇w|4dx− k%(l − 1)

2l + 2

∫Ω

|w(x, t)|l+1dx.


It is clear that the term K(w(t)) is subcritical with respect to strong energytopology, therefore using the representation

Ψ(T ) +

∫ T

0

e−k(T−t)L(t)dt = e−kTΨ(0) +

∫ T

0

e−k(T−t)K(w(t))dt,

with

L(t) =

∫Ω

(a(x)− k)|wt(t)|2dx,

we can apply Theorem 15 to prove the existence of global attractor.To establish the statement in the case σ = 0 we note that in this case

the full energy E (w,w′) is a strict Lyapunov function and thus the system isgradient.

Now we consider long-time dynamics of strong solutions.Let Hst =

[H4(Ω) ∩ H2

0 (Ω)]× H2

0 (Ω). The argument given in the well-posedness section shows that the restriction Sstt of the semiflow St on Hst isa continuous (nonlinear) semigroup of continuous mappings in Hst.

Theorem 49 (Compact attractor for strong solutions). Let σ = 0 anda(x) ≡ k > 0 be a constant (for simplicity). Then semiflow Sstt has a compactglobal attractor Ast in Hst. This attractor Ast possesses the properties:

• Ast ⊆ A;• Ast = M u

st(N ) is unstable manifold in Hst emerging from the set N ofequilibria;

• Ast has a finite fractal dimension as a compact set in Hst.

Moreover, if we assume in addition l ≡ 2k − 1 is an odd integer, then anytrajectory (w(t);wt(t)) from the attractor Ast possesses the property

supt∈R‖w(n)(t)‖m,Ω ≤ Cn,m <∞, n,m = 0, 1, 2, . . . , (1.146)

where w(n)(t) = ∂nt w(t). In particular, Ast is a bounded set in Cm(Ω)×Cm(Ω)for each m = 0, 1, 2, . . ., and thus Ast ⊂ C∞(Ω)× C∞(Ω).

The proof of Theorem 49 consists of several steps. The situation corre-sponding to gradient flows (we deal with the case σ = 0).

Step 1 (dissipativity in Hst) by the “barrier” method: Due to dissi-pativity in H and energy relation it is sufficient to consider dissipativity ofstrong solutions w(t) possessing properties

‖wt(t)‖2 + ‖w(t)‖22 ≤ R, t ≥ 0,

∫ ∞0

‖wt(τ)‖2dτ ≤ CR <∞. (1.147)

Let w(t) be strong solution satisfying (1.147) and v(t) = wt(t). Consider thefunctional


G(v, v′) ≡ 1

2

(‖v′‖2 + ‖∆w‖2 + ‖v‖2

+

∫Ω

[|∇w|2|∇v(x)|2 + 2 |(∇w,∇v)R2 |2

]dx), (1.148)

and denoteH(t) ≡ H(v, vt) = a0 +G(v, vt) + ε(v, vt), (1.149)

where a0 and ε are positive parameters. Then using Bresis–Gallouet–Sedenkotype inequality (see Lemma 4 and Remark 4) and choosing ε small enough,we obtain that

dH(t)

dt+ γH(t) ≤ C1

R‖wt(t)‖2H(t) ln (1 +H(t)) + C2R

with positive γ (see [CL11] for details). The above inequality leads to thedissipativity property by barrier’s method.

Step 2 (attractor in Hst): To prove the existence of the attractor weneed to establish asymptotic compactness of the system in Hst. For this weuse again Ball’s method. Let ε = k/2 and a0 = 0 in (1.149). Then one can seethat

dH(t)

dt+ kH(t) = Kst(w(t), wt(t)),

where

Kst(w, v) = 3

∫Ω

(∇w,∇v)R2 |∇v|2dx

−∫Ω

[f ′(w)v − v]vtdx−k

2

∫Ω

[f ′(w)v − v]vdx,

where f(s) = %|s|l−1s. The termKst(w, v) is obviously subcritical with respectto the topology in Hst, therefore we can apply the same Ball’s argument toprove the existence of global attractor Ast. The inclusion Ast ⊆ A is evident.One can also see that Ast = M u

st(N ).

Step 3 (Finite dimension of the attractor in Hst): To prove finitedimensionality of the strong attractor in Hst we use smoothness properties oftrajectories and again the methods developed in [CL08a]. For this we need toestablish the quasi-stability estimate in Hst.

Proposition 4 (Quasi-stability inequality in Hst). Let the hypotheses ofTheorem 49 be in force. Let w(t) and w∗(t) be strong solutions to the problemin question satisfying the estimates

‖w(t)‖4 + ‖wt(t)‖2 + ‖wtt(t)‖+ ‖w∗(t)‖4 + ‖w∗t (t)‖2 + ‖w∗tt(t)‖ ≤ R

for all t ∈ R and for some constant R > 0. Let w(t) = w(t)− w∗(t). Then


‖w(t)‖24 + ‖wt(t)‖22 + ‖wtt(t)‖2 ≤ C1e−γt (‖w(0)‖24 + ‖wt(0)‖22

)+ C2

∫ t

0

e−γ(t−τ)(‖w(τ)‖22 + ‖wt(τ)‖2

)dτ

for all t > 0, where γ > 0 and C1, C2 > 0 may depend on R.

The proof (see [CL11]) follows the strategy applied to von Karman evolutionequations and presented in [CL10, Sect.9.5.3].

By Theorem 20 (see also Theorem 4.3 in [CL08a]) the relation in Propo-sition 4 is sufficient to prove that the fractal dimension of Ast as a compactset in Hst is finite.

Boundary damping

In the case of α > 0 we can use the same methods (see Section 10.3 in [CL10]for the details) as for von Karman model (1.18) with boundary damping(1.19). The case of the boundary damping with α = 0 is still open.

1.6 Other models covered by the methods presented

In this section we shortly describe several models whose long-time dynam-ics can be studied within the framework presented above. These include: (i)nonlocal Kirchhoff wave model with strong and structural damping; (ii) finitedimensional and smooth attractors for thermoelastic plates; (iii) control tofinite dimensional attractors in thermal-structure, flow-structure and fluid-plate interactions. We also consider dissipative wave models which arise inplasma physics.

1.6.1 Kirchhoff wave models

In a bounded smooth domain Ω ⊂ Rd we consider the following Kirchhoffwave model with a strong nonlinear damping:

utt − σ(‖∇u‖2)∆ut − φ(‖∇u‖2)∆u+ f(u) = h(x), x ∈ Ω, t > 0,

u|∂Ω = 0, u(0) = u0, ut(0) = u1.(1.150)

Here ∆ is the Laplace operator, σ and φ are scalar functions specified later,f(u) is a given source term, h is a given function in L2(Ω).

This kind of wave models goes back to G. Kirchhoff (d = 1, φ(s) = φ0+φ1s,σ(s) ≡ 0, f(u) ≡ 0) and has been studied by many authors under different setsof hypotheses (see, e.g., [Lio78] and also [Chu12a] and the references therein).The model in (1.150) is characterized by the presence of three nonlinearities:the source, the damping and the stiffness.


We assume that the source nonlinearity f(u) is a C1 function possessingthe following properties: f(0) = 0 (without loss of generality),

µf := lim inf|s|→∞

s−1f(s)

> −∞, (1.151)

and also (a) if d = 1, then f is arbitrary; (b) if d = 2 then

|f ′(u)| ≤ C(1 + |u|p−1

)for some p ≥ 1;

(c) if d ≥ 3 then either

|f ′(u)| ≤ C(1 + |u|p−1

)with some 1 ≤ p ≤ p∗ ≡

d+ 2

d− 2, (1.152)

or else

c0|u|p−1 − c1 ≤ f ′(u) ≤ c2(1 + |u|p−1

)with some p∗ < p < p∗∗, (1.153)

where p∗∗ ≡ d+4(d−4)+ , ci > 0 are constants and s+ = (s+ |s|)/2.

We note that the conditions above covers subcritical, critical and super-critical cases.

Under rather mild hypotheses concerning C1 functions φ and σ we canprove that the problem is well-posed. This holds even without the requirementthat φ is non-negative (see the details in [Chu12a]). However in order tostudy long-time dynamics of the problem (1.150) we need to assume that thefunctions σ and φ from C1(R+) are positive and either

φ(s)s→ +∞ as s→ +∞ and µf = lim inf|s|→∞

s−1f(s)

> 0.

or elseµφ = lim inf

s→+∞φ(s) > 0 and µφλ1 + µf > 0,

where λ1 is the first eigenvalue of the minus Laplace operator in Ω withDirichlet boundary conditions.

As a phase space we consider H = [H10 (Ω) ∩ Lp+1(Ω)]× L2(Ω) endowed

with partially strong8 topology: a sequence (un0 ;un1 ) ⊂ H is said to bepartially strongly convergent to (u0;u1) ∈ H if un0 → u0 strongly in H1

0 (Ω),un0 → u0 weakly in Lp+1(Ω) and un1 → u1 strongly in L2(Ω) as n → ∞ (inthe case when d ≤ 2 we take 1 < p <∞ arbitrary).

Under the conditions stated above (see [Chu12a]) problem (1.150) gener-ates an evolution semigroup St in the space H . The action of the semigroupis given by the formula Sty = (u(t);ut(t)), where y = (u0;u1) ∈ H and u(t)is a weak solution to (1.150).

8 It is obvious that the partially strong convergence becomes strong below super-critical level (H1

0 (Ω) ⊂ Lp+1(Ω)).


To describe the dynamical properties of St it is convenient to introducethe following notion.

A bounded set A ⊂ H is said to be a global partially strong attractor forSt if (i) A is closed with respect to the partially strong topology, (ii) A isstrictly invariant (StA = A for all t > 0), and (iii) A uniformly attracts inthe partially strong topology all other bounded sets: for any (partially strong)vicinity O of A and for any bounded set B in H there exists t∗ = t∗(O, B)such that StB ⊂ O for all t ≥ t∗.

The main result in [Chu12a] reads as follows:

Theorem 50. Assume in addition that f ′(s) ≥ −c for all s ∈ R in the non-supercritical case (when (1.153) does not hold). Then the semigroup St givenby (1.150) possesses a global partially strong attractor A in the space H .Moreover, A ⊂H1 = [H2(Ω) ∩H1

0 (Ω)]×H10 (Ω) and

supt∈R

(‖∆u(t)‖2 + ‖∇ut(t)‖2 + ‖utt(t)‖2−1,Ω +

∫ t+1

t

‖utt(τ)‖2dτ)≤ CA

for any full trajectory γ = (u(t);ut(t)) : t ∈ R from the attractor A. Wealso have that

A = M+(N ), where N = (u; 0) ∈H : φ(‖A 1/2u‖2)A u+ f(u) = h.

The attractor A has a finite fractal dimension as a compact set in the space[H1+r(Ω) ∩H1

0 (Ω)]×Hr(Ω) for every r < 1.

To prove this result we first establish quasi-stability properties of St ondifferent topological scales (see [Chu12a] for details).

In similar way we can also study a model with structural damping with anon-supercritical force of the form

utt + φ(‖A 1/2u‖2)A u+ σ(‖A 1/2u‖2)A θut + F (u) = 0, t > 0,

u(0) = u0, ut(0) = u1,(1.154)

with θ ∈ [1/2, 1), where A is a linear positive self-adjoint operator withdomain D(A ) and with a compact resolvent in a separable infinite dimensionalHilbert space H. In this case we assume that the damping σ and the stiffnessφ factors are positive C1 functions and

Φ(s) ≡∫ s

0

φ(ξ)dξ → +∞ as s→ +∞.

The nonlinear operator F is locally Lipschitz in the following sense

‖A −θ[F (u1)− F (u2)]‖ ≤ L(%)‖A 1/2(u1 − u2)‖, ∀‖A 1/2ui‖ ≤ % ,

and potential, i.e., F (u) = Π ′(u), where Π(u) is a C1-functional on D(A 1/2),and ′ stands for the Frechet derivative. We assume that Π(u) is locallybounded on D(A 1/2) and there exist η < 1/2 and C ≥ 0 such that


ηΦ(‖A 1/2u‖2) +Π(u) + C ≥ 0 , u ∈ H1 = D(A 1/2) .

This hypotheses cover critical case of the source F , but not supercritical. Fordetails concerning the model in (1.154) we refer to [Chu10].

1.6.2 Plate models with structural damping

We consider a class of plate models with the strong nonlinear damping theabstract form of which is the following Cauchy problem in a separable Hilbertspace H:

utt +D(u, ut) + A u+ F (u) = 0, t > 0; u|t=0 = u0, ut|t=0 = u1. (1.155)

In the case of plate models with hinged boundary conditions, A = (−∆D)2,where ∆D is the Laplace operator in a bounded smooth domain Ω in R2 withthe Dirichlet boundary conditions. We have then that H = L2(Ω) and

D(A ) =u ∈ H4(Ω) : u = ∆u = 0 on ∂Ω

.

The damping operator D(u, ut) may have the form

D(u, ut) = ∆ [σ0(u)∆ut]− div [σ1(u,∇u)∇ut] + g(u, ut), (1.156)

where σ0(s1), σ1(s1, s2, s3) and g(s1, s2) are locally Lipschitz functions of si ∈R, i = 1, 2, 3, such that σ0(s1) > 0, σ1(s1, s2, s3) ≥ 0 and g(s1, s2)s2 ≥ 0. Alsothe functions σ1 and g satisfy some growth conditions. We note that every termin (1.156) represents a different type of damping mechanisms. The first one isthe so-called viscoelastic Kelvin–Voight damping, the second one representsthe structural damping and the term g(u, ut) is the dynamical friction (orviscous damping). We refer to [LT00, Chapter 3] and to the references thereinfor a discussion of stability properties caused by each type of the dampingterms in the case of linear systems. We should stress that the conditionsconcerning σi above allow to have a pure viscoelastic damping (i.e., we cantake σ1 ≡ 0 and g ≡ 0). However a similar results remains valid for thecase when σ0 ≡ 0 and σ1 is independent of ∇u. In this case the presence (orabsence) of the friction g(u, ut) has also no importance for long-time dynamics.

The nonlinear feedback (elastic) force F (u) may have one of the followingforms (which represent different plate models):

(a) Kirchhoff model: F (u) is the Nemytskii operator

u 7→ −κ1 · div |∇u|q∇u− µ1|∇u|r∇u+ κ2|u|lu− µ2|u|mu

− p(x),

where κi ≥ 0, q > r ≥ 0, l > m ≥ 0, µi ∈ R are parameters, p ∈ L2(Ω).(b) Von Karman model: F (u) = −[u,F (u) + F0] − p(x), where F0 ∈ H4(Ω)

and p ∈ L2(Ω) are given functions, the von Karman bracket [u, v] is givenby (1.7) and the Airy stress function F (u) solves (1.6).


(c) Berger Model: In this case the feedback force has the form

F (u) = −[κ

∫Ω

|∇u|2dx− Γ]∆u− p(x), (1.157)

where κ > 0 and Γ ∈ R are parameters, p ∈ L2(Ω); for some details andreferences see, e.g., [Chu99, Chapter 4] and [CL08a, Chapter 7].

One can show that the system generated by (1.155) is quasi-stable. Thusmethods presented in these notes apply. They allow to show the existenceof global attractors of finite fractal dimension which in addition posses somesmoothness properties. See [CK10, CK12] for more details.

1.6.3 Mindlin-Timoshenko plates and beams

Let Ω ⊂ R2 be a bounded domain with a sufficiently smooth boundary Γ .Lets v(x, t) = (v1(x, t), v2(x, t)) be a vector function and w(x, t) be a scalarfunction on Ω ×R+. The system of Mindlin-Timoshenko equations describesdynamics of a plate taking into account transverse shear effects (see, e.g.,[LL88, Chap.1] and the references therein). This system has the form

αvtt + k · g(vt)−Av + κ · (v +∇w) = −f0(v) +∇x [f1(w)] , (1.158)

wtt + k · g0(wt)− κ · div(v +∇w) = −f2(w). (1.159)

We supplement this problem with the Dirichlet boundary conditions

v1(x, t) = v2(x, t) = 0, w(x, t) = 0 on Γ × R+. (1.160)

Here the functions v1(x, t) and v2(x, t) are the angles of deflection of a filament(they are measures of transverse shear effects) and w(x, t) is the bending com-ponent (transverse displacement). The vector f0(v) = (f01(v1, v2); f02(v1, v2))and scalar f1 and f2 functions represents (nonlinear) feedback forces, whereasg(v1, v2) = (g1(v1), g2(v2)) and g0 are monotone damping functions describingresistance forces (with the intensity k > 0). The parameter α > 0 describesrotational inertia of filaments. The factor κ > 0 is the so-called shear modu-lus (from mechanical point of view the limiting situation κ→ +0 correspondsto plane strain and the case κ → +∞ corresponds to absence of transverseshear). The operator A has the form

A =

∂2x1

+1− ν

2∂2x2

1 + ν

2∂2x1x2

1 + ν

2∂2x1x2

1− ν2

∂2x1+ ∂2x2

,where 0 < ν < 1 is the Poisson ratio. In 1D case (dimΩ=1) the correspondingproblem models dynamics of beams under the Mindlin-Timoshenko hypothe-ses. For details concerning the Mindlin-Timoshenko hypotheses and governing


equations see, e.g. [Lag89] and [LL88]. We also note that in the limit κ→ +∞the system in (1.158) and (1.159) becomes Kirchhoff-Boussinesq type equa-tion, see [Lag89, CL06a].

We refer to [CL08a] and [CL06a] for an analysis (based on quasi-stabilitytechnology) of long time behaviour of the Mindlin-Timoshenko plate underdifferent sets of assumptions concerning nonlinear feedback forces, dampingfunctions and parameters. By the same method long-time dynamics in thermo-elastic Mindlin-Timoshenko model was studied in [Fas07, Fas09].

1.6.4 Thermal-structure interactions

This problem has the formutt − αAθ +A2u = B(u), u|t=0 = u0, ut|t=0 = u1,θt + ηAθ + αAut = 0, θ|t=0 = θ0,

(1.161)

where A is a linear positive self-adjoint operator with a compact inverse ina separable infinite dimensional Hilbert space H. The nonlinear term B(u)can model von Karman (as in (1.4)) and Berger (see (1.157)) nonlinearities.In this case the variable θ represents the temperature of the plate.

For application the methods presented we refer to [CL08b] and [CL10,Chapter 11] in the von Karman case, see also [BC08] for the Berger nonlin-earity.

1.6.5 Structural acoustic interactions

The mathematical model under consideration consists of a semilinear waveequation defined on a bounded domain O, which is strongly coupled with theBerger or von Karman plate equation acting only on a part of the boundary ofthe domain O. This kind of models, known as structural acoustic interactions,arise in the context of modeling gas pressure in an acoustic chamber which issurrounded by a combination of hard (rigid) and flexible walls.

More precisely, let O ⊂ R3 be a bounded domain with a sufficiently smoothboundary ∂O. We assume that ∂O = Ω ∪ S, where Ω ∩ S = ∅,

Ω ⊂ x = (x1;x2; 0) : x′ ≡ (x1;x2) ∈ R2

with the smooth contour Γ = ∂Ω and S is a surface which lies in the subspaceR3− = x3 ≤ 0. The exterior normal on ∂O is denoted by n. The set Ω

is referred to as the elastic wall, whose dynamics is described some plateequation. The acoustic medium in the chamber O is described by a semilinearwave equation. Thus, we consider the following (coupled) PDE system

ztt + g(zt)−∆z + f(z) = 0 in O × (0, T ),

∂z

∂n= 0 on S × (0, T ),

∂z

∂n= α vt on Ω × (0, T ),

(1.162)


and vtt + b(vt) +∆2v +B(v) + βzt|Ω = 0 in Ω × (0, T ),

v = ∆v = 0 on ∂Ω × (0, T ),(1.163)

endowed with initial data

z(0, ·) = z0 , zt(0, ·) = z1 in Ω, v(0, ·) = v0 , vt(0, ·) = v1 in Ω .

Here above, g(s) and b(s) are non-decreasing functions describing the dissi-pation effects in the model, while the term f(z) represents a nonlinear forceacting on the wave component and B(v) is (nonlinear) von Karman or Bergerforce; α and β are positive constants; The part S of the boundary describesa rigid (hard) wall, while Ω is a flexible wall where the coupling with theplate equation takes place. The boundary term βzt|Ω describes back pressureexercised by the acoustic medium on the wall.

Well-posedness issues of the abstract second order system, which is a par-ticular case of the one studied in [Las02, Sect. 2.6] (see also [CL10, Chapter6]).

Long-time dynamics from point of view of quasi-stable systems were in-vestigated in [CL10, Chapter 12] in the von Karman case and in [BCL07]for the Berger nonlinearity. We also note that flow structure model (1.162)and (1.163) in combination with thermoelastic system (1.161) was studied in[BC08] and [CL10, Chapter 12].

1.6.6 Fluid-structure interactions

Our mathematical model is formulated as follows (for details, see [CR13a]).Let O ⊂ R3, Ω ⊂ R2 and the surface S be the same as in the

case of the model in (1.162) and (1.163). We consider the following lin-ear Navier–Stokes equations in O for the fluid velocity field v = v(x, t) =(v1(x, t); v2(x, t); v3(x, t)) and for the pressure p(x, t):

vt − ν∆v +∇p = Gf and div v = 0 in O × (0,+∞), (1.164)

where ν > 0 is the dynamical viscosity and Gf is a volume force. We supple-ment (1.164) with the (non-slip) boundary conditions imposed on the velocityfield v = v(x, t):

v = 0 on S; v ≡ (v1; v2; v3) = (0; 0;ut) on Ω. (1.165)

Here u = u(x, t) is the transversal displacement of the plate occupying Ω andsatisfying the following equation:

utt +∆2u+ F (u) = p|Ω in Ω × (0,∞). (1.166)

The nonlinear feedback (elastic) force F (u) as above may have one of theforms (Kirchhoff, Karman or Berger) which are present for plates models


with structural damping (see (1.155)). We also impose clamped boundaryconditions on the plate

u|∂Ω =∂u

∂n

∣∣∣∣∂Ω

= 0 (1.167)

and supply (1.164)–(1.167) with initial data of the form

v(0) = v0, u(0) = u0, ut(0) = u1, (1.168)

We note that (1.164) and (1.165) imply the following compatibility condition∫Ω

u(x′, t)dx′ = const for all t ≥ 0, (1.169)

which can be interpreted as preservation of the volume of the fluid (we canchoose this constant to be zero).

It was shown in [CR13a] that equations (1.164)–(1.169) generates an evo-lution operator St in the space

H =

(v0;u0;u1) ∈ X × H20 (Ω)× L2(Ω) : (v0, n) ≡ v30 = u1 on Ω

,

where X =v = (v1; v2; v3) ∈ [L2(O)]3 : div v = 0; (v, n) = 0 on S

, and

L2(Ω) (resp. H20 (Ω)) denotes the subspace in L2(Ω) (resp. in H2

0 (Ω)) con-sisting of functions with zero averages.

This evolution operator St is quasi-stable and thus possesses a compactglobal attractor, see [CR13a]. We emphasize that we do not assume any kindof mechanical damping in the plate component. Thus this results means thatdissipation of the energy in the fluid due to viscosity is sufficient to stabilizethe system.

In a similar way (see [Chu11]) the model which deals only with longitudi-nal deformations of the plate neglecting transversal deformations can be alsoconsidered (in contrast with the model (1.164)–(1.168) which takes into ac-count the transversal deformations only). This means that instead of (1.165)the following boundary conditions are imposed on the velocity fluid field:

v = 0 on S; v ≡ (v1; v2; v3) = (u1t ;u2t ; 0) on Ω,

where u = (u1(x, t);u2(x, t)) is the in-plane displacement vector of the platewhich solves the wave equation of the form

utt −∆u−∇ [div u] + ν(v1x3; v2x3

)|x3=0 + f(u) = 0 in Ω; ui = 0 on Γ.

This kind of models arises in the study of blood flows in large arteries (seethe references in [Gro08]).

One can also analyze the corresponding model based on the full Kar-man shell model with rotational inertia (see [CR13b]). In this case to obtainwell-posedness we need to apply Sedenko’s method. The study of long-timedynamics is based on J.Ball’s method (see Theorem 15).


1.6.7 Quantum Zakharov system

In a bounded domain Ω ⊂ Rd, d ≤ 3, we consider the following systemntt −∆

(n+ |E|2

)+ h2∆2n+ αnt = f(x), x ∈ Ω, t > 0,

iEt +∆E − h2∆2E + iγE − nE = g(x), x ∈ Ω, t > 0.(1.170)

Here E(x, t) is a complex function and n(x, t) is a real one, h > 0, α ≥ 0 andγ ≥ 0 are parameters and f(x), g(x) are given (real and complex) functions.

This system in dimension d = 1 was derived in [GHG05], by use of aquantum fluid approach, to model the nonlinear interaction between quantumLangmuir waves and quantum ion-acoustic waves in an electron-ion densequantum plasma. Later a vector 3D version of equations (1.170) was suggestedin [HS09]. In dimension d = 2, 3 the system in (1.170) is also known (see, e.g.,[SSS09] and the references therein) as a simplified ”scalar model” which is ina good agreement with the vector model derived in [HS09] (see a discussionin [SSS09]).

The Dirichlet initial boundary value problem for (1.170) is well-posed andgenerates a dynamical system in an appropriate phase space (see [Chu12c]).Using quasi-stability methods one can prove the existence of a finite-dimen-sional global attractor, for details we refer to [Chu12c].

We also note that in the case h = 0 we arrive to the classical Zakharovsystem (see [Zak72]). Global attractors in this case were studied in [Fla91,GM98] in the 1D case and in [CS05] in the 2D case. In the latter case toobtain uniqueness the Sedenko method was applied and Ball’s method wasused to prove the existence of a global attractor.

1.6.8 Schrodinger–Boussinesq equations

The methods similar to described above can be also applied in study of quali-tative behavior of the system consisting of Boussinesq and Schrodinger equa-tions coupled in a smooth (2D) bounded domainΩ ⊂ R2. The resulting systemtakes the form:

wtt + γ1wt +∆2w −∆(f(w) + |E|2

)= g1(x), (1.171a)

iEt +∆E − wE + iγ2E = g2(x), x ∈ Ω, t > 0, (1.171b)

where E(x, t) and w(x, t) are unknown functions, E(x, t) is complex andw(x, t) is real. Here above γ1 and γ2 are nonnegative parameters and g1(x) andg2(x) are given (real and complex) L2-functions. We equip equations (1.171)with the boundary conditions

w|∂Ω = ∆w|∂Ω = 0, E|∂Ω = 0, (1.172)

and with the initial data

wt(x, 0) = w1(x), w(x, 0) = w0(x), E(x, 0) = E0(x). (1.173)


Long-time dynamics in this system was studied in [CS12a] by the methodsdescribed above under the following hypotheses concerning the (nonlinear)function f : f ∈ C1 (R), f(0) = 0 and

∃c1, c2 ≥ 0 : F (r) =

∫ r

0

f(ξ)dξ ≥ −c1r2 − c2, ∀|r| ≥ r0, (1.174)

∃M ≥ 0, p ≥ 1 : |f ′(s)| ≤M(1 + |s|p−1), s ∈ R. (1.175)

We also note that the ideas which were used in the study of the Kirchfoff-Boussinesq system (see (1.12) with α = 0 can be also applied to the followingSchrodinger-Boussinesq-Kirchfoff model:

wtt −∆wt +∆2w − div|∇u|3∇u

−∆

(w2 + |E|2

)= g1,

iEt +∆E − wE + iγ2E = g2, x ∈ Ω, t > 0,

equipped with the boundary conditions (1.172) and initial data (1.173). Fordetails we refer to [CS12b].

We note that the above nonlinear Schrodinger–Boussinesq models are oftenin the use as models of interactions between short and intermediate longwaves, which arises in describing the dynamics of Langmuir soliton formationand interaction in a plasma and diatomic lattice system (see the references in[CS05, CS12a, CS12b]).

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2

Stability of finite difference schemes forhyperbolic initial boundary value problems

Jean-Francois Coulombel

CNRS and Universite de Nantes, Laboratoire de mathematiques Jean Leray (UMRCNRS 6629), 2 rue de la Houssiniere, BP 92208, 44322 Nantes Cedex 3, France.Research of the author was supported by the French Agence Nationale de laRecherche, contract [email protected]

Summary. The aim of these notes is to present some results on the stability of finitedifference approximations of hyperbolic initial boundary value problems. We first re-call some basic notions of stability for the discretized Cauchy problem in one spacedimension. Special attention is paid to situations where stability of the finite differ-ence scheme is characterized by the so-called von Neumann condition. This leads usto the important class of geometrically regular operators. After discussing the dis-cretized Cauchy problem, we turn to the case of initial boundary value problems. Weintroduce the notion of strongly stable schemes for zero initial data. The first mainresult characterizes strong stability in terms of a solvability property and an energyestimate for the resolvent equation. This first result shows that the so-called UniformKreiss-Lopatinskii Condition is a necessary condition for strong stability. The mainresult of these notes shows that the Uniform Kreiss-Lopatinskii Condition is alsoa sufficient condition for strong stability in the framework of geometrically regularoperators. We illustrate our results on the Lax-Friedrichs and leap-frog schemes andcheck strong stability for various types of boundary conditions. We also extend astability result by Goldberg and Tadmor for Dirichlet boundary conditions. In thelast section of these notes, we show how to incorporate nonzero initial data andprove semigroup estimates for the discretized initial boundary value problems. Weconclude with some remarks on possible improvements and open problems.

These notes have been prepared for a course taught by the author in Trieste dur-ing a trimester devoted to “Nonlinear Hyperbolic PDEs, Dispersive and TransportEquations” (SISSA, May-July 2011). The material in the notes covers three articles,one of which is a collaboration with A. Gloria (INRIA Lille, France). These notes arealso the opportunity to include some simplified proofs of known results and to givesome detailed examples, which may help in clarifying/demystifying the theory. Theauthor warmly thanks the organizers as well as the participants of the trimester forinviting him to deliver these lectures and for the very kind and stimulating atmo-sphere in SISSA. Special thanks are addressed to Fabio Ancona, Stefano Bianchini,Gianluca Crippa and Andrea Marson for all the nice moments spent during theauthor’s stay in Trieste.

98 Jean-Francois Coulombel

2.1 Introduction

2.1.1 What is and what is not inside these notes?

These notes review the results derived in [Cou09, Cou11a, Cou11b] on the stabilityof finite difference approximations for hyperbolic initial boundary value problems. Inorder to keep the length of the notes reasonable, the analogous results for hyperbolicpartial differential equations, which have sometimes been proved quite some timeago, will be referred to without proof. This is mainly done to save space and to avoidintroducing further notation. One crucial point in the analysis below is to understandwhy the techniques developed for partial differential equations are unfortunately notsufficient to handle finite difference schemes. Special attention is therefore paid tothe main new phenomena that appear when considering discretized equations. Someexamples are scattered throughout the text in order to explain how the generaltheory, which may look sometimes rather complicated, is often simplified when onefaces a specific example. In particular, the Lax-Friedrichs and leap-frog schemes,which are some of the most simple discretizations of a hyperbolic equation, serve asa guideline throughout Sections 2.2, 2.4 and 2.5.

The notes are essentially self-contained. All results but one are completelyproved. Of course, some familiarity with hyperbolic equations can do no harm, butthe only basic requirements to follow the proofs are a good knowledge of matrices,some tools from real and complex analysis and a little bit of functional analysis.

As far as hyperbolic boundary value problems are concerned, the reader mightfirst want to get familiar with the theory for partial differential equations before read-ing the discrete counterpart that is detailed here. In this case, the books [Cha82,chapter 7] or [BG07, chapters 3-5] are convenient references. However, the theoryfor finite difference schemes can also be seen as a first step towards the theory forpartial differential equations since, as detailed below, some parts of the analysisare actually simpler in the discrete case. Even though the original results were notproved historically in this way, discrete problems can also be a constructive approx-imation method to obtain solutions of partial differential equations (this approachis widely detailed, for instance, in [Hil68, Kre04] and in many other textbooks). Forone-dimensional hyperbolic boundary value problems, the well-posedness theory forthe continuous equation is rather trivial thanks to integration along the characteris-tics. On the contrary, the stability theory for the discretized equation may becomereally involved so constructing solutions to the continuous one-dimensional problemthrough a finite difference approximation seems quite absurd. It seems more fairto say that the present stability theory for one-dimensional problems can be help-ful as an introduction to the well-posedness theory for multidimensional continuousproblems. At this stage, a complete stability theory for multidimensional discretizedproblems remains to be done (see however [Mic83] for, it seems, the only attemptso far).

As far as numerical approximations are concerned, a convenient reference for ourpurpose is [GKO95, chapters 5, 6, 11 and 13] where stability issues are analyzed,in particular for the discrete Cauchy problem. The techniques developed below arerestricted to linear schemes for linear equations. Consequently, no knowledge offlux limiters, ENO/WENO schemes nor any other nonlinear high order approxima-tion procedure is assumed. Extending some of the results below to such numerical

2 Stability of finite difference schemes for boundary value problems 99

schemes is definitely an open and challenging issue (which would be very interestingfrom the point of view of applications).

2.1.2 Some notation

Throughout these notes, the following notation is used:

U := ζ ∈ C, |ζ| > 1 , U := ζ ∈ C, |ζ| ≥ 1 ,

D := ζ ∈ C, |ζ| < 1 , S1 := ζ ∈ C, |ζ| = 1 .

We let Md,D(K) denote the set of d ×D matrices with entries in K = R or C, andwe use the notation MD(K) when d = D. The linear group of non-singular matricesof size D is denoted GlD(K). If M ∈ MD(C), sp(M) denotes the spectrum of M ,ρ(M) denotes the spectral radius of M , while M∗ denotes the conjugate transposeof M . The notation MT is also used for the transpose of a matrix M (here M isnot necessarily a square matrix). The matrix (M +M∗)/2 is called the real part ofM ∈ MD(C) and is denoted Re (M). The real vector space of Hermitian matricesof size D is denoted HD.

ForH1, H2 ∈HD, we writeH1 ≥ H2 if for all x ∈ CD we have x∗ (H1−H2)x ≥ 0.We let I denote the identity matrix, without mentioning the dimension. The norm ofa vector x ∈ CD is |x| := (x∗ x)1/2. The corresponding norm for matrices in MD(C)is also denoted | · |. We let `2 denote the set of square integrable sequences, and weusually do not mention the set of indeces of the sequences (sequences may be valuedin Cd for some integer d).

The notation diag (M1, . . . ,Mp) is used to denote the diagonal matrix whoseentries are (in this order) M1, . . . ,Mp. If the Mj ’s are matrices themselves, then thesame notation is used to denote the corresponding block diagonal matrix.

The notation x1 = x2 ≤ x3 = x4 means that x1 equals x2, x3 equals x4, and x2

is not larger than x3 (and consequently, of course, x1 is not larger than x4).The letter C denotes a constant that may vary from line to line or even within

the same line. The dependence of the constants on the various parameters is madeprecise throughout the text.

2.1.3 General presentation of the stability problem

In one space dimension, a hyperbolic initial boundary value problem reads∂tu+A∂xu = F (t, x) , (t, x) ∈ R+ × R+ ,

B u(t, 0) = g(t) , t ∈ R+ ,

u(0, x) = f(x) , x ∈ R+ ,

(2.1)

where A ∈ MN (R) is diagonalizable with real eigenvalues, the unknown u(t, x) isvalued in RN , and B is a matrix - not necessarily a square matrix, see below - thatencodes the boundary conditions. The functions F, g, f are given source terms, re-spectively, the interior source term, the boundary source term and the initial data.In one space dimension, it is rather easy to solve such a linear problem by diagonaliz-ing A and integrating along the characteristics. More precisely, let r1, . . . , rN denotea basis of eigenvectors of A associated with eigenvalues µ1, . . . , µN . Let us assume


for simplicity that 0 does not belong to sp(A), the so-called non-characteristic case.Up to reordering the eigenvalues, we can label them so that

µ1, . . . , µN+ > 0 , µN++1, . . . , µN < 0 .

The integer N+ denotes the number of incoming characteristics. We decompose thesource terms F, f and the unknown u as

F (t, x) =

N∑i=1

Fi(t, x) ri , f(x) =

N∑i=1

fi(x) ri , u(t, x) =

N∑i=1

ui(t, x) ri .

Assuming for simplicity that the solution u is smooth, at least C 1 with respect to(t, x), (2.1) gives

∀ i = 1, . . . , N ,d

dt

[ui(t, x+ µi t)

]= Fi(t, x+ µi t) .

We integrate these equalities with respect to t, keeping in mind that the ui’s andthe Fi’s are only defined on R+ × R+. For i ∈ N+ + 1, . . . , N, that is when µi isnegative, we obtain the formula

ui(t, x) = fi(x− µi t) +

∫ t

0

Fi(s, x− µi (t− s)) ds . (2.2)

The latter formula makes sense for all (t, x) in the quarter-space R+ × R+ becausein that case, all quantities x − µi t and x − µi (t − s) in (2.2) are nonnegative. Inparticular, the trace of ui on the boundary x = 0 of the space domain is entirelydetermined by the data:

ui(t, 0) = fi(|µi| t) +

∫ t

0

Fi(s, |µi| (t− s)) ds .

One should be careful when performing the integration in the case i ∈ 1, . . . , N+.According to the sign of x− µi t, we obtain

ui(t, x) =

fi(x− µi t) +

∫ t0Fi(s, x− µi (t− s)) ds , if x ≥ µi t ,

ui(t− x/µi, 0) +∫ tt−x/µi

Fi(s, x− µi (t− s)) ds , if x ≤ µi t .(2.3)

Analyzing the formulas (2.2) and (2.3), we observe that the solution u is entirelydetermined provided that we can express the traces of the incoming characteristicsui(t, 0) , 1 ≤ i ≤ N+ in terms of the data F, g, f . Since the traces of the outgoingcharacteristics ui(t, 0) , N+ + 1 ≤ i ≤ N are already determined by the formula(2.2), the boundary condition in (2.1) reads

N+∑i=1

ui(t, 0)B ri = g(t)−N∑

i=N++1

ui(t, 0)B ri ,

where the right-hand side can be expressed in terms of F, g, f . Therefore the initialboundary value problem (2.1) can be well-posed in any reasonable sense (meaningat least existence and uniqueness of a solution, even though we do not make thefunctional framework precise) if and only if the matrix B belongs to MN+,N (R),and satisfies


Rp = Span(B r1, . . . , B rN+

). (2.4)

In particular, (2.4) implies that B should have maximal rank, but this could havealready been seen from (2.1) for otherwise there would have been an algebraic ob-struction to solving the boundary condition in (2.1). If the matrix B satisfies (2.4),then we get an explicit expression for the components of the solution u along theeigenvectors ri. Energy estimates of u in terms of F, g, f as well as qualitative prop-erties of the solution (regularity, finite speed of propagation etc.) are readily seenfrom these expressions. If we try to summarize the above discussion, we obtain thefollowing conclusion: well-posedness of (2.1) requires first a precise number of bound-ary conditions that is compatible with the hyperbolic operator, and the verificationof the algebraic condition (2.4). Consequently, rather than checking energy estimatesfor each possible boundary conditions in (2.1), we are just reduced to verifying (2.4)which is by far easier.

A remarkable result by Kreiss [Kre70] states that for the analogue of (2.1) in sev-eral space dimensions, well-posedness - that is existence, uniqueness and continuousdependence of a solution in a suitable functional framework - can still be charac-terized by an algebraic condition. The latter is usually referred to as the UniformKreiss-Lopatinskii Condition (UKLC in what follows). There is however a modifi-cation between the one-dimensional case and the multi-dimensional case. Observingthat in one space dimension, the condition (2.4) for well-posedness equivalently reads

Ker B ∩ Span(r1, . . . , rN+

)= 0 ,

the UKLC in several space dimensions reads

∀ ζ ∈ Σ , Ker B ∩ E(ζ) = 0 ,

where Σ is some infinite set of parameters and the vector spaces E(ζ) all havedimension N+. Verifying the UKLC in several space dimensions is therefore morecomplicated since it requires computing a basis of a vector space that depends onparameters, and then checking that an appropriate determinant does not vanish.This can sometimes be done with explicit computations, see for instance [BG07,chapter 14] for the case of gas dynamics, or it can also be done in a numericalway (this numerical strategy was used in other contexts such as the computation ofEvans functions). One of the most difficult steps in the theory of [Kre70] is to givea precise definition of the vector spaces E(ζ) that enter the definition of the UKLC.Not so surprisingly, we shall also face this difficulty when dealing with numericalschemes. However, as shown on some specific examples, the general theory can befar more complicated than what one faces with one particular numerical scheme.One should therefore not be afraid to try checking the UKLC on some examples: itis the best way to manipulate the objects, to get used to them and to understandbetter the general theory. The reader is therefore strongly encouraged to test all theresults below on his/her favourite numerical scheme.

Our main goal in these notes is to characterize - that is, find necessary and suffi-cient conditions - stability for the numerical schemes occurring after discretizing theinitial boundary value problem (2.1). Existence and uniqueness for the discretizedversion of (2.1) will be completely trivial in these notes, and stability should be un-derstood as continuous dependence of the solution with respect to the data, meaningthe last requirement for “Hadamard well-posedness”. In view of the existing theory


for (2.1) and its analogue in several space dimensions, we wish to obtain a generalresult of the form: “the discretization of (2.1) is stable if and only if an algebraiccondition (to be determined) is satisfied”. This result will be meaningful if testingthe algebraic condition is easier than checking the validity of energy estimates forthe numerical schemes. As usual when one deals with problems in infinite dimen-sional spaces, the choice of the norm in the stability definition is crucial. Our longterm goal is to develop an analogous theory for discretized multi-dimensional prob-lems to the one detailed here in the one-dimensional case. The functional frameworkshould therefore be compatible with such an extension, and this basically restrictsus to working with L2-type spaces (hence the use of many Hilbertian methods). Asfar as convergence of numerical schemes is concerned, we focus here on the stabil-ity problem since consistency is supposed to be an easier problem. In some sense,consistency of a numerical scheme follows from some Taylor expansions on an exactsmooth solution of the continuous problem (2.1). If we can derive a powerful stabilitytheory, convergence should follow as a more or less direct consequence by combiningstability with consistency. Instead of giving precise results in this direction, we shallrefer the interested reader to [Gus75] where this strategy is used.

Let us now detail the plan of these notes. As a warm-up, we begin in Section 2.2with some considerations on the discretized Cauchy problem. This will be the op-portunity to introduce some objects that are crucial in the analysis of the discretizedinitial boundary value problem. We also introduce and analyze some examples suchas the Lax-Friedrichs and leap-frog schemes. Sections 2.3 and 2.4 are devoted to theanalysis of the discretized initial boundary value problem with zero initial data. Thisis, technically speaking, the most difficult part of these notes. In the case of zeroinitial data, stability can be analyzed by applying the Laplace transform and theso-called normal modes analysis. Our main result characterizes stability by means ofan algebraic condition of the same type as the UKLC. The main results in Section2.3 generalize - and sometimes simplify - the fundamental contribution by Gustafs-son, Kreiss and Sundstrom [GKS72]. To clarify the theory, we explain in Section2.4 the behaviors of all the objects (stable eigenvalues, stable subspace, UKLC...)for the Lax-Friedrichs and leap-frog schemes. Section 2.5 deals with the problemof incorporating nonzero initial data and adapting the notion of stability to thisnew framework. For one-dimensional problems, the incorporation of initial data wasperformed by Wu [Wu95]. We shall explain his method and propose an alternative -though closely related - approach. The main advantage of this new approach is thefact that it can be adapted in a straightforward way to multi-dimensional problems,while Wu’s method is restricted to one-dimensional problems for reasons that weshall detail. Eventually, we shall present some (of the numerous) open problems inSection 2.6.

2.2 Fully discretized hyperbolic equations

2.2.1 Finite difference operators and stability for the discreteCauchy problem

We consider the Cauchy problem

2 Stability of finite difference schemes for boundary value problems 103∂tu+A∂xu = 0 , (t, x) ∈ R+ × R ,u(0, x) = f(x) , x ∈ R ,

(2.5)

on the whole real line. As in Section 2.1, A ∈ MN (R) is diagonalizable with realeigenvalues µ1, . . . , µN . For initial data f ∈ L2(R), there exists a unique solutionu ∈ C (R+;L2(R)) solution to (2.5). This solution can be explicitly computed byintegrating along the characteristics. Decomposing along the eigenvectors ri of A,we obtain

u(t, x) =

N∑i=1

fi(x− µi t) ri , f(x) =

N∑i=1

fi(x) ri .

In particular, the following energy estimate is straightforward

supt≥0

∫R|u(t, x)|2 dx ≤ C

∫R|f(x)|2 dx , (2.6)

with a numerical constant C that only depends on A. Another possibility for com-puting the solution u to (2.5) is to use Fourier transform with respect to the spacevariable x. Letting ξ denote the associated frequency variable, u(t, ξ) satisfies thelinear ordinary differential equation

d

dtu(t, ξ) = −i ξ A u(t, ξ) , u(0, ξ) = f(ξ) ,

which we solve to obtain

u(t, ξ) = exp(−i t ξ A) f(ξ) . (2.7)

Let us now introduce the discretizations of (2.5) that we consider in these notes.Let ∆x,∆t > 0 denote a space and a time step where the ratio λ := ∆t/∆x is a fixedpositive constant. In all what follows, λ is called the CFL (for Courant-Friedrichs-Lewy) number and ∆t ∈ ]0, 1] plays the role of a small parameter, while ∆x = ∆t/λvaries accordingly. Some of the assumptions in the theory are restrictions on λ.Typically, the results will hold provided that λ is chosen in a suitable interval of R+.

The solution to (2.5) is approximated by a sequence (Unj ) defined for n ∈ N andj ∈ Z. More precisely, we always identify the sequence (Unj ) defined for n ∈ N andj ∈ Z with the step function

U(t, x) := Unj for (t, x) ∈ [n∆t, (n+ 1)∆t[×[j ∆x, (j + 1)∆x[ .

The goal is to build a numerical scheme that produces a step function U that isclose to u for the L∞(R+;L2(R)) topology. This is a natural requirement in view of(2.6). The choice of the topology may look rather arbitrary, especially in one spacedimension, but as detailed in the introduction, our goal is to develop some toolsthat may be extended to multi-dimensional problems. Let us observe that thoughthe solution u to (2.5) lies in the space C (R+;L2(R)), the approximation U lies, ingeneral, in the larger space L∞(R+;L2(R)). It is only in the limit process, by letting∆t tend to zero, that continuity with respect to time can be recovered.

Discretizing the initial condition of (2.5) is usually performed by choosing

∀ j ∈ Z , fj :=1

∆x

∫ (j+1)∆x

j ∆x

f(x) dx .


This is not the only possible choice, but it has the good property of being stablewith respect to the L2 topology, that is1

∑j∈Z

∆x |fj |2 ≤∫R|f(x)|2 dx .

From now on, we assume that the initial discretization has been chosen, producinga sequence (fj) ∈ `2 such that the associated grid function is “close” - in some sensethat we do not make precise - to the initial condition f of (2.5).

Starting from a given sequence (fj) ∈ `2, for instance the sequence defined justabove, many classical finite difference approximations of (2.5) take the form

Un+1j = QUnj , j ∈ Z , n ≥ 0 ,

U0j = fj , j ∈ Z ,

(2.8)

where Q is a finite difference operator whose expression is given by

Q :=

p∑`=−r

A` T` , (T` V )k := Vk+` . (2.9)

Let us give a few explanations on (2.9). The shift operator T is an invertible operatoron `2(Z) so taking powers T` is legitimate. The integers p, r in (2.9) are fixed, that is,they do not depend on the index j on the grid where the numerical scheme is applied,and neither do they depend on the small parameter∆t. In the same way, the matricesA−r, . . . , Ap ∈MN (R) should not depend on ∆t, nor on the initial data (fj). In most(linear) finite difference schemes, the matrices A` are polynomial functions of thematrix λA. In that case, all matrices A` can be diagonalized in the same basis. Werefer to the following paragraphs for some examples. Summarizing, the numericalscheme (2.8) is defined by two integers p, r and by the matrices A−r, . . . , Ap. Thenthe sequence Un+1 is computed from Un by applying the operator Q defined in(2.9), which acts boundedly on `2. In particular, for all initial condition (fj) ∈ `2,there exists a unique sequence (Unj ) that is a solution to (2.8), and moreover thissolution satisfies (Unj )j∈Z ∈ `2 for all n ∈ N.

Let us briefly recall that for nonlinear schemes such as ENO or WENO schemes,the matrices A` are not fixed but depend on the solution that is computed ; forinstance, to compute the sequence (U1

j ), the matrices A` at the first time step dependon (fj), and they are updated at each time step in order to take the oscillations ofthe sequence (Unj ) into account. The theory developed below relies crucially on thefact that the matrices A` are independent of the sequence (Unj ). It therefore doesnot extend directly to such nonlinear schemes.

The definition of stability for the numerical scheme (2.8) requires that the solu-tion to (2.8) satisfies the discrete analogue of (2.6). More precisely, we introduce

Definition 1 (Stability for the discrete Cauchy problem). The numericalscheme defined by (2.8), (2.9) is (`2-) stable if there exists a constant C0 > 0 suchthat for all ∆t ∈ ]0, 1], for all initial condition (fj)j∈Z ∈ `2 and for all n ∈ N, thereholds

1 This estimate is easily proved by applying Cauchy-Schwarz inequality on eachinterval [j ∆x, (j + 1)∆x[.

2 Stability of finite difference schemes for boundary value problems 105∑j∈Z

∆x |Unj |2 ≤ C0

∑j∈Z

∆x |fj |2 .

Of course, we could simplify the factor ∆x on both sides of the stability estimateand Definition 1 is clearly independent of the small parameter ∆t, but we prefer tokeep the ∆x factor in order to highlight the fact that discrete `2 norms are nothingbut L2 norms for step functions defined on the grid with uniform space step ∆x.The factor ∆x corresponds to the measure of the cell [j ∆x, (j+ 1)∆x[. This obser-vation is useful in order to understand the similarities between stability estimatesfor numerical schemes and energy estimates for partial differential equations.

Stability for the numerical scheme (2.8) is characterized by the following result.

Proposition 1 (Characterization of stability for the fully discrete Cauchyproblem). The scheme (2.8) is stable in the sense of Definition 1 if and only if thematrices A` in (2.9) satisfy

∀n ∈ N , ∀ η ∈ R ,

∣∣∣∣∣(

p∑`=−r

ei ` η A`

)n∣∣∣∣∣2

≤ C0 , (2.10)

with the same constant C0 as in Definition 1.

For future use, it is convenient to introduce the notation

∀κ ∈ C \ 0 , A (κ) :=

p∑j=−r

κÀ` , (2.11)

so that (2.10) reads

∀n ∈ N , ∀ η ∈ R ,∣∣∣A (ei η)n

∣∣∣2 ≤ C0 .

The matrix A (ei η) is called the amplification matrix (or symbol) of the scheme (2.8).

Proof (Proof of Proposition 1). • Let us assume that the bound (2.10) holds, or inother words that the family A (ei η), η ∈ R is uniformly power bounded with thebound

√C0. Let us consider the scheme (2.8). Then for all n ∈ N, the step function

Un defined byUn(x) := Unj , for x ∈ [j ∆x, (j + 1)∆x[ ,

satisfies

∀x ∈ R , Un+1(x) =

p∑`=−r

A` Un(x+ `∆x) .

We already know that Un belongs to L2(R) for all n, so we can apply Fouriertransform on both sides of the latter equality2. This operation yields the relation

∀ ξ ∈ R , Un+1(ξ) = A (ei∆x ξ) Un(ξ) ,

from which we deduce

∀ ξ ∈ R , Un(ξ) = A (ei∆x ξ)n U0(ξ) .

2 This is the precise point where it is crucial to deal with constant matrices A`.


Using Plancherel Theorem and the bound (2.10), we obtain∫R|Un(x)|2 dx =

1

2π

∫R

∣∣∣Un(ξ)∣∣∣2 dξ

≤ C0

2π

∫R

∣∣∣U0(ξ)∣∣∣2 dξ = C0

∫R|U0(x)|2 dx .

Consequently, the scheme (2.8) is stable with the same constant C0 as in (2.10).• We now assume that the scheme (2.8) is stable with the constant C0, and we

fix an integer n as well as a real number η. Let also X ∈ CN have norm 1. Then foran integer k ≥ n max(p, r), we consider the initial condition

fj :=

ei j ηX , if |j| ≤ k ,0 , otherwise.

The following computation is elementary (just recall the notation (2.11))

U1j = A (ei η) fj , if |j| ≤ k −max(p, r) .

By a straightforward induction, we obtain

Unj = A (ei η)n fj , if |j| ≤ k − n max(p, r) . (2.12)

Then we have∑|j|≤k−n max(p,r)

∆x |Unj |2 ≤∑j∈Z

∆x |Unj |2 ≤ C0

∑j∈Z

∆x |fj |2 = C0 ∆x (2 k + 1) .

The left hand side of the latter inequality is computed by using (2.12) and by usingthe definition of the vector fj . We obtain

(2 k + 1− 2n max(p, r))∆x∣∣∣A (ei η)nX

∣∣∣2 ≤ C0 ∆x (2 k + 1) .

Dividing by ∆x (2 k + 1), letting k tend to infinity and taking the supremum withrespect to X, we obtain the result of Proposition 1.

Remark 1. The easiest case of stability is when the matrices A` satisfy

∀ η ∈ R ,∣∣∣A (ei η)

∣∣∣ ≤ 1 .

Then the solution to (2.8) is such that the sequence of norms (‖Un‖`2) is non-increasing. This more restrictive notion is called strong `2-stability, and is furtherstudied in [Tad86].

The main idea in the proof of Proposition 1 is to test the stability estimateon oscillations ei j η. Of course, the sequence (ei j η)j∈Z does not belong to `2 so weneed to make a truncation. Fourier’s inversion Theorem shows that functions can bedecomposed as a superposition of oscillations so stability of the numerical schemeis encoded in a stability estimate for pure oscillations that should be uniform withrespect to the frequency.


Let us now make an important remark. The grid function Un is supposed tobe an approximation of the solution u at time n∆t. Hence Un should approximateu(n∆t, ·). Recalling the relation (2.7), the matrix A should satisfy

A (ei∆x ξ)n ≈ exp(−i n∆t ξ A) = exp(−i∆t ξ A)n .

We do not wish to make the meaning of the symbol ≈ precise. However, a natu-ral requirement should be to impose that in the limit ∆t → 0 with n = 1, bothexpressions coincide. This yields the restriction

A (1) =

p∑`=−r

A` = I . (2.13)

A numerical scheme of the form (2.8), (2.9) that satisfies (2.13) is said to be con-sistent. Higher order accuracy of the numerical scheme is encoded in the Taylorexpansion of A (ei∆x ξ) as ∆t tends to 0. However, this notion will not be muchused in what follows, except when discussing some examples.

We shall go back to the result of Proposition 1 in the following paragraph. Beforedoing so, let us discuss a possible extension of the theory. The reader who is familiarwith numerical discretizations of ordinary differential equations will probably wonderwhy we have restricted to numerical schemes with only one time step. As a matterof fact, there is no reason for doing so and in some situations one could prefer usinga two steps (or more) numerical procedure. A well-known example is the leap-frogscheme. Another example is discussed in one of the following paragraphs. Numericalschemes with several time steps take the following form: let us consider three integersp, r, s. Starting from some sequences (f0

j ), . . . , (fsj ) in `2, the sequence (Unj ) is definedby U

n+1j =

s∑σ=0

Qσ Un−σj , j ∈ Z , n ≥ s ,

Unj = fnj , j ∈ Z , n = 0, . . . , s ,

(2.14)

where the shift operators Qσ are given by

Qσ :=

p∑`=−r

A`,σ T` . (2.15)

Again, the matrices A`,σ in (2.15) should not depend on the sequence to be computedso that the same scheme applies to all initial data and at each time step. The notionof stability for (2.14) is entirely analogous to Definition 1.

Definition 2 (Stability for the discrete Cauchy problem). The numericalscheme defined by (2.14), (2.15) is (`2-) stable if there exists a constant C0 > 0 suchthat for all ∆t ∈ ]0, 1], for all initial condition (f0

j )j∈Z, . . . , (fsj )j∈Z in `2 and for all

n ∈ N, there holds

∑j∈Z

∆x |Unj |2 ≤ C0

∑j∈Z

∆x |f0j |2 + · · ·+

∑j∈Z

∆x |fsj |2 .


Similarly to Proposition 1, Proposition 2 below characterizes stability of thescheme (2.14) in terms of the uniform power boundedness of the correspondingamplification matrix. For future use, we therefore introduce the notation

∀κ ∈ C \ 0 , A (κ) :=

Q0(κ) . . . . . . Qs(κ)I 0 . . . 0

0. . .

. . ....

0 0 I 0

∈MN(s+1)(C) ,

Qσ(κ) :=

p∑`=−r

κÀ`,σ , (2.16)

which coincides with our former notation (2.11) in the case s = 0 (one step scheme).To avoid any possible confusion, we emphasize that in (2.16), the matrix A (κ) isdecomposed into blocks, each of which is a square N × N matrix with complexcoefficients. Such block decompositions of matrices will occur at numerous places inthese notes.

Proposition 2 (Characterization of stability for the fully discrete Cauchyproblem). The scheme (2.14) is stable in the sense of Definition 2 if and onlyif there exists a constant C1 > 0 such that the amplification matrix A in (2.16)satisfies

∀n ∈ N , ∀ η ∈ R ,∣∣∣A (ei η)n

∣∣∣2 ≤ C1 . (2.17)

Moreover, if the scheme is stable with a constant C0, then one can take C1 = (s +1)C0 in (2.17), and conversely if (2.17) holds with a constant C1, then one can takeC0 = C1 for the stability estimate of Definition 2.

Proof (Proof of Proposition 2). • Let us assume that the bound (2.17) holds withthe constant C1, and let us consider the scheme (2.14) with initial data in `2. Thenfor all n ∈ N, the step function Un defined by

Un(x) := Unj , for x ∈ [j ∆x, (j + 1)∆x[ ,

satisfies

∀n ≥ s , ∀x ∈ R , Un+1(x) =

s∑σ=0

p∑`=−r

A`,σ Un−σ(x+ `∆x) .

It is clear that Un belongs to L2(R) for all n (the operators Qσ act boundedly on`2), so we can again apply Fourier transform and obtain

∀n ≥ s , ∀ ξ ∈ R , Un+1(ξ) =

s∑σ=0

Qσ(ei∆x ξ) Un−σ(ξ) ,

from which we deduce

∀n ∈ N , ∀ ξ ∈ R ,

Un+s(ξ)

...

Un(ξ)

= A (ei∆x ξ)n

Us(ξ)

...

U0(ξ)

.


Stability follows from Plancherel Theorem as in the proof of Proposition 1, and weget

∀n ∈ N ,∑j∈Z

∆x |Unj |2 ≤ C1

∑j∈Z

∆x |f0j |2 + · · ·+

∑j∈Z

∆x |fsj |2 .

• Let us now assume that the scheme (2.14) is stable in the sense of Definition2 with a constant C0. Let n ∈ N, η ∈ R, and let k ≥ n max(p, r). We also considersome vectors X0, . . . , Xs ∈ CN satisfying

|X0|2 + · · ·+ |Xs|2 = 1 .

We consider the initial data

f0j :=

ei j ηX0 , if |j| ≤ k ,0 , otherwise,

. . . , fsj :=

ei j ηXs , if |j| ≤ k ,0 , otherwise.

For |j| ≤ k −max(p, r), the relation (2.14) gives

Us+1j =

s∑σ=0

Qσ(ei η)Us−σj .

In particular, there holds Us+1j+` = ei ` η Us+1

j for |j| ≤ k − 2 max(p, r) and |`| ≤max(p, r). Proceeding by induction, we get

Us+m+1j =

s∑σ=0

Qσ(ei η)Us+m−σj ,

for all m = 0, . . . , n− 1 and for all j satisfying |j| ≤ k− (m+ 1) max(p, r). It is nownot difficult to obtain the relationU

n+sj

...Unj

= A (ei η)n

fsj

...f0j

, if |j| ≤ k − n max(p, r) .

Then we have∑|j|≤k−n max(p,r)

∆x(|Unj |2 + · · ·+ |Un+s

j |2)≤∑j∈Z

∆x(|Unj |2 + · · ·+ |Un+s

j |2)

≤ (s+ 1)C0

(|f0j |2 + · · ·+ |fsj |2

)= (s+ 1)C0 ∆x (2 k + 1) .

Eventually, we obtain

(2 k + 1− 2n max(p, r))∆x∣∣∣A (ei η)nX

∣∣∣2 ≤ (s+ 1)C0 ∆x (2 k + 1) ,

with X :=

Xs

...X0

.

Dividing by ∆x (2 k + 1), letting k tend to infinity and taking the supremum withrespect to X, we obtain the result of Proposition 2.


The following paragraph discusses how the results of Propositions 1 and 2 areuseful in practice.

Remark 2. When one tries to verify that the amplification matrix of a numeri-cal scheme satisfies (2.10), resp. (2.17), the choice of the norm on MN (C), resp.MN(s+1)(C), is arbitrary because all norms are equivalent. It may be easier to workwith the norm maxi,j=1,...,N |Mi,j |, as we shall sometimes do below.

2.2.2 Possible behaviors for the eigenvalues of the amplificationmatrix

In this paragraph, we recall some facts about families of matrices with uniformlybounded powers. We also analyze how the characterization of Propositions 1 and 2can be simplified for a special class of numerical schemes.

The following result is elementary.

Lemma 1. Let M ∈Md(C) be power bounded. Then ρ(M) ≤ 1.

Proof (Proof of Lemma 1). Let µ ∈ sp(M), and let us choose an eigenvector X ∈ Cdwith norm 1 associated with the eigenvalue µ. For all integer n, we have

|µ|n = |µnX| = |MnX| ≤ C ,

where the constant C is an upper bound for the norms of all powers Mn. The latterinequality gives |µ| ≤ 1 and the result follows.

Lemma 1 immediately implies the following well-known necessary condition forstability.

Corollary 1 (von Neumann condition). Let us assume that the scheme (2.8),resp. (2.14), is stable in the sense of Definition 1, resp. 2. Then the amplificationmatrix A defined by (2.11), resp. (2.16), satisfies the so-called von Neumann condi-tion

∀ η ∈ R , ρ(A (ei η)) ≤ 1 . (2.18)

Let us observe that for one step schemes satisfying the consistency condition(2.13), A (1) is the identity matrix so the upper bound 1 for the spectral radiusallowed by the von Neumann condition is attained. In particular, when η is small,the eigenvalues of A (ei η) should be close to 1 but remain within the closed unitdisk. Usually, when one performs an expansion of the eigenvalues for small η, therequirement that the eigenvalues satisfy the von Neumann condition indicates somerestrictions on the possible values of the CFL number.

The von Neumann condition in Corollary 1 is only a necessary condition forstability. However there is one case, that is always met in examples, where it is alsoa sufficient condition.

Lemma 2. Let us assume that the matrices A−r, . . . , Ap in (2.9) can be simulta-neously diagonalized (for instance when they are all polynomial functions of λA).Then the scheme (2.8) is stable if and only if the von Neumann condition (2.18)holds.


Proof (Proof of Lemma 2). The proof is elementary. Choosing an invertible matrixT that diagonalizes A−r, . . . , Ap, the definition (2.11) shows that T also diagonalizesthe amplification matrix A , that is

∀κ ∈ C \ 0 , T−1 A (κ)T = diag (z1(κ), . . . , zN (κ)) .

If the von Neumann condition holds, the eigenvalues satisfy |zj(ei η)| ≤ 1 for allη ∈ R. This property implies

|A (ei η)n| = |T diag (z1(ei η)n, . . . , zN (ei η)n)T−1| ≤ |T | |T−1| .

Proposition 1 shows that the scheme (2.8) is stable.

The stability criterion of Lemma 2 will apply to all one step numerical schemesthat appear in these notes. However, this criterion does not apply to multi-stepschemes since the companion matrix A (ei η) in (2.16) can not be diagonalized ina fixed basis that does not depend on η. We therefore need to work a little more.The following Lemma gives a more precise description of the properties of powerbounded matrices.

Lemma 3. A matrix M ∈ Md(C) is power bounded if and only if ρ(M) ≤ 1 andfurthermore the eigenvalues of M whose modulus equals 1 are semi-simple (that is,their geometric multiplicity equals their algebraic multiplicity).

Proof (Proof of Lemma 3). The proof is classical and appears in many textbookson numerical analysis. Let M ∈ Md(C) and let us consider an invertible matrix Tthat reduces M to its Jordan form

T−1 M T = diag (M1, . . . ,Mp) ,

where each block Mj is either of the form µj I or a Jordan blockµj 1 0 0

0. . .

. . . 0...

. . .. . . 1

0 . . . 0 µj

,

whose size equals at least 2. (In this decomposition, the eigenvalues of the blocksMj are not necessarily pairwise disctinct.) It is straightforward to check that Mis power bounded if and only if each block is power bounded. We can now proveLemma 3.• Let us assume that M is power bounded. From Lemma 1, we already have

ρ(M) ≤ 1. If M is diagonalizable, then the proof is finished, so let us consider aJordan block Mj that appears in the reduction of M and whose size is denoted d.Writing Mj = µj I +Nj , we have

Mnj =

n∑k=0

Ckn µn−kj Nk

j ,

so the (1, 2)-coefficient of Mnj equals nµn−1

j for all n ≥ 1. Since all norms on thespace Md(C) are equivalent, there exists a constant C such that


∀n ≥ 1 , n |µj |n−1 ≤ C ,

and this implies |µj | < 1. In other words, eigenvalues of M that belong to the unitcircle S1 must be semi-simple.• Let us now assume that M satisfies ρ(M) ≤ 1 and all eigenvalues of M that

belong to S1 are semi-simple. In the Jordan reduction of M , the diagonal blocks arepower bounded, so to prove Lemma 3, it only remains to prove that a Jordan blockassociated with an eigenvalue in D is power bounded. We keep the same notationMj = µj I + Nj as above. If µj = 0, then Mj is clearly power bounded, so we nowassume 0 < |µj | < 1. We have

Mnj =

d−1∑k=0

Ckn µn−kj Nk

j ,

for all n ≥ d− 1 (here we have used Ndj = 0). It is therefore sufficient to prove that

for all fixed k = 0, . . . , d − 1, the sequence (Ckn µnj )n∈N is bounded. This sequence

tends geometrically to zero (use the so-called d’Alembert’s criterion) so it is boundedand the sequence (Mn

j )n∈N is also bounded. The proof of Lemma 3 is complete.

For numerical schemes, Lemma 3 shows that in addition to the von Neumanncondition, a necessary condition for stability is that if η ∈ R is such that the matrix

A (ei η) has an eigenvalue z ∈ S1, then z should be semi-simple.Lemma 3 is unfortunately not sufficient to characterize uniform power bounded-

ness for an infinite family of matrices3. Indeed, let us consider the following matrices

M1(x) :=

(1− x x

0 1− x

), M2(x) :=

(1− x2 x

0 1− x2

),

which both depend on a real parameter x ∈ [0, 1]. For all fixed x ∈ [0, 1], Lemma 3shows that the matrices M1(x) and M2(x) are power bounded. However, it is nota difficult exercise to show that the family M1(x) , x ∈ [0, 1] is uniformly powerbounded while the family M2(x) , x ∈ [0, 1] is not uniformly power bounded. Asa matter of fact, there exists only one result that fully characterizes families ofuniformly power bounded matrices. This famous Theorem is due to Kreiss and canbe stated as follows.

Theorem 1 (Kreiss matrix Theorem). Let d ∈ N and let F ⊂ Md(C). Thefollowing conditions are equivalent.

(i) There exists a constant C1 such that for all M ∈ F and for all n ∈ N, |Mn| ≤ C1.(ii) There exists a constant C2 such that for all M ∈ F , ρ(M) ≤ 1 and for all

z ∈ U , there holds

|(M − z I)−1| ≤ C2

|z| − 1.

3 The main reason is that the bound provided by Lemma 3 depends on the matrixT that reduces M to its Jordan form, and the construction of T is a highlyill-conditionned problem so that |T | |T−1| can be very large when M varies.


(iii)There exists a constant C3 such that for all M ∈ F , there exists an invertiblematrix T such that T−1 M T is upper triangular and

|T |+ |T−1| ≤ C3 ,

∀ 1 ≤ i < j ≤ d ,

|(T−1 M T )i,j | ≤ C3 min(1− |(T−1 M T )i,i|, 1− |(T−1 M T )j,j |) .

Rather than giving a complete proof of Theorem 1, which would take muchspace, we shall refer the interested reader to the nice review [Str97] where additionalcharacterizations and historical references can be found. Showing that (i) implies (ii)is easy and follows from a series expansion. An elegant proof that (ii) implies (i) canbe found in [Tad81]. Improvements of [Tad81] with optimal constants are reportedin [Str97].

The problem for showing uniform power boundedness for a parametrized familyof matrices is to handle how a Jordan block may approach a diagonal block associatedwith an eigenvalue in S1 as the parameter varies. For numerical schemes in one spacedimension, the pathology of the matrix M2(x) above is usually ruled out by the factthat as ei η approaches a point ei η for which the amplification matrix has a semi-simple eigenvalue z ∈ S1, the eigenvalues of A (ei η) close to z are also semi-simple.Furthermore, eigenvalues and eigenvectors can usually be determined as smoothfunctions of η. A model situation for such behavior would be(

1− x2m1 00 1− x2m2

), m1,m2 ∈ N , x ∈ [−1, 1] .

To make this framework precise, we introduce the following terminology.

Definition 3 (Geometrically regular operator). The finite difference operatorQ in (2.9), resp. the operators Qσ in (2.15), is said to be geometrically regular ifthe amplification matrix A defined by (2.11), resp. (2.16), satisfies the followingproperty: if κ ∈ S1 and z ∈ S1 ∩ sp(A (κ)) has algebraic multiplicity α, then thereexist some functions β1(κ), . . . , βα(κ) that are holomorphic in a neighborhood W ofκ in C and that satisfy

β1(κ) = · · · = βα(κ) = z , det(z I −A (κ)

)= ϑ(κ, z)

α∏j=1

(z − βj(κ)

),

with ϑ a holomorphic function of (κ, z) in some neighborhood of (κ, z) such thatϑ(κ, z) 6= 0, and if furthermore, there exist α vectors e1(κ), . . . , eα(κ) ∈ CN , resp.CN(s+1), that depend holomorphically on κ ∈ W , that are linearly independent forall κ ∈ W , and that satisfy

∀κ ∈ W , ∀ j = 1, . . . , α , A (κ) ej(κ) = βj(κ) ej(κ) .

For instance, if the matrices A−r, . . . , Ap satisfy the assumption of Lemma 2,it is clear that the finite difference operator Q in (2.9) is geometrically regular.Even better, in that case the eigenvalues and corresponding eigenvectors can beparametrized globally for κ 6= 0. The eigenvectors do not even depend on κ ! Theframework of Definition 3 is therefore meaningful mostly for multi-step schemes,e.g. the leap-frog scheme. We hope that it will also be useful for the study of finite


difference schemes in several space dimensions. We end this paragraph with the fol-lowing characterization of stability by the von Neumann condition for geometricallyregular operators.

Proposition 3 (Characterization of stability for geometrically regular op-erators). Let the finite difference operator Q in (2.9), resp. the operators Qσ in(2.15), be geometrically regular. Then the scheme (2.8), resp. (2.14), is stable if andonly if the von Neumann condition (2.18) holds.

The precise expression, either (2.11) or (2.16), of the amplification matrix Ais not relevant for the proof of Proposition 3. To unify both cases, we shall thusconsider that the size of A is N (s+ 1), which amounts to setting s = 0 for one-stepschemes.

Proof (Proof of Proposition 3). Using Corollary 1, it is sufficient to prove that thevon Neumann condition implies stability. The proof of Proposition 3 consists insplitting the set of parameters η ∈ R into a first part for which the amplificationmatrix has eigenvalues close to S1 and a second part for which the eigenvalues ofthe amplification matrix are in D, uniformly away from S1. In the first part, we shalluse the geometric regularity assumption to control the powers of the amplificationmatrix. The second part will be easier to control. We begin with an easy consequenceof Theorem 1 which will be useful later on.

Lemma 4. Let d ∈ N and let F ⊂ Md(C) be a family of matrices such that thereexists δ ∈ ]0, 1] for which

∀M ∈ F , ρ(M) ≤ 1− δ . (2.19)

Then F is uniformly power bounded if and only if F is bounded in Md(C).

Proof (Proof of Lemma 4). It is obvious that boundedness is a necessary conditionfor uniform power boundedness. Let now a family F ⊂ Md(C) be bounded andsatisfy (2.19) for some positive δ, and let M ∈ F . By Schur’s Lemma, there exists aunitary matrix T such that T−1 M T is upper triangular. Since F is bounded, whileT and T−1 belong to the unitary group (a bounded subset of Md(C)), there existsa constant C that is independent of M and such that

∀ 1 ≤ i < j ≤ d , |(T−1 M T )i,j | ≤ C .

From the assumption of Lemma 4, we also have

mini=1,...,d

(1− |(T−1 M T )i,i|) ≥ δ > 0 ,

so it is easily seen that F satisfies condition (iii) of Theorem 1. The conclusion ofLemma 4 follows.

The following observation is trivial and is stated without proof.

Lemma 5. Let K := κ ∈ S1 , sp(A (κ)) ∩ S1 6= ∅. Then K is a closed (hencecompact) subset of S1.


If K is empty (which never happens in practice, but let’s pretend), then the vonNeumann condition implies that for all κ ∈ S1, the spectrum of A (κ) is included inthe open unit disk D. Moreover, A (κ) depends holomorphically on κ ∈ S1 and S1

is a compact set, so there exists a constant δ > 0 such that ρ(A (κ)) ≤ 1− δ for allκ ∈ S1. (Here we use the continuity of the spectral radius.) Since A (κ) , κ ∈ S1is a bounded family, Lemma 4 shows that A (κ) is uniformly power bounded forκ ∈ S1 which completes the proof of Proposition 3.

Let us now assume that K is not empty. The following Lemma gives a descrip-tion of A (κ) in the neighborhood of any point of K .

Lemma 6. For all κ ∈ K , there exist an integer q, two positive constants C andδ, an open neighborhood W of κ in C and an invertible matrix T (κ) that dependsholomorphically on κ ∈ W and that satisfies

• for all κ ∈ W , |T (κ)|+ |T (κ)−1| ≤ C,• for all κ ∈ W , T (κ)−1 A (κ)T (κ) = diag (β1(κ), . . . , βq(κ),A](κ)), with β1(κ),

. . . , βq(κ) ∈ C, A](κ) ∈MN(s+1)−q(C), there holds |A](κ)| ≤ C and ρ(A](κ)) ≤1− δ.

Let us complete the proof of Proposition 3 using Lemma 6. We use a finitecovering of the compact set K by open sets W1, . . . ,WK ⊂ C given in Lemma 6(on each Wk, we have a change of basis Tk(κ) that satisfies suitable properties).Let now κ = ei η ∈ S1 ∩ Wk with 1 ≤ k ≤ K. The von Neumann condition showsthat the eigenvalues of A (κ) belong to D ∪ S1. Moreover, there exist some positiveconstants Ck and δk that do not depend on κ ∈ S1 ∩ Wk such that the diagonalblock A](κ) satisfies |A](κ)| ≤ Ck and ρ(A](κ)) ≤ 1 − δk. Applying Lemma 4,we find that the family A](κ) , κ ∈ S1 ∩ Wk is uniformly power bounded. Usingthe block diagonal decomposition of A (κ), it follows that the family of matricesA (κ) , κ ∈ S1 ∩ Wk is also uniformly power bounded. (Here we use the property|βj(κ)| ≤ 1 for κ ∈ S1 ∩ Wk which follows from the von Neumann condition.) Inother words, there exists a constant C1 > 0 such that

∀κ ∈ S1 ∩ (W1 ∪ · · · ∪WK) , ∀n ∈ N , |A (κ)n| ≤ C1 .

For κ in the closed (hence compact) subset S1 \ (W1 ∪ · · · ∪WK) of S1, we knowthat the spectrum of A (κ) lies inside D. Consequently, there exists a constant δ′ > 0such that ρ(A (κ)) ≤ 1− δ′ for κ ∈ S1 \ (W1 ∪ · · · ∪WK). Applying Lemma 4 again,there exists a constant C2 > 0 such that

∀κ ∈ S1 \ (W1 ∪ · · · ∪WK) , ∀n ∈ N , |A (κ)n| ≤ C2 .

Consequently the matrix A (κ) is uniformly power bounded for κ ∈ S1, and theproof of Proposition 3 is complete. It only remains to prove Lemma 6... Since itis the first occurence in these notes of arguments that will appear in several otherplaces, we give a detailed proof of Lemma 6. Similar arguments will be sometimesused as a “black box” later on.

Proof (Proof of Lemma 6). Let κ ∈ K . From the von Neumann condition, theeigenvalues of the amplification matrix split into two groups: eigenvalues on S1 andeigenvalues in D. We let z1, . . . , zm denote the pairwise distinct eigenvalues of A (κ)on S1. The corresponding algebraic multiplicities are denoted α1, . . . , αm. We alsointroduce the notation q := α1 + · · ·+ αm.


Let us consider an open neighborhood W of κ and a positive constant δ suchthat for all κ ∈ W , the eigenvalues of A (κ) belong to one of the following sets:

ζ ∈ C , |ζ − z1| ≤ δ , . . . , ζ ∈ C , |ζ − zm| ≤ δ , ζ ∈ C , |ζ| ≤ 1− 3 δ .

Up to shrinking δ and W , we can always assume that these disks do not intersect.Hence the disk with center z1 contains α1 eigenvalues of A (κ), the disk with centerzm contains αm eigenvalues, and the disk centered at the origin containesN (s+1)−qeigenvalues.

For κ ∈ W , the spectral projector Π(κ) of A (κ) on the generalized eigenspaceE(κ) associated with eigenvalues in ζ ∈ C , |ζ| ≤ 1− 3 δ is given by the Dunford-Taylor formula

Π(κ) =1

2 i π

∫|w|=1−2 δ

(w I −A (κ))−1 dw .

The projector Π(κ) depends holomorphically on κ ∈ W , and its image has rankN (s+1)− q. Choosing a basis eq+1, . . . , eN(s+1) of the generalized eigenspace E(κ),the vectors

Π(κ) eq+1 , . . . , Π(κ) eN(s+1) ,

are linearly independent for κ sufficiently close to κ, and moreover they belongto E(κ). We have thus constructed a basis (eq+1(κ), . . . , eN(s+1)(κ)) of E(κ) thatdepends holomorphically on κ for κ sufficiently close to κ (that is, for all κ ∈ W upto shrinking W ).

The geometric regularity condition shows that the α1 eigenvalues of A (κ) whichbelong to the disk ζ ∈ C , |ζ−z1| ≤ δ behave holomorphically on κ. Collecting theeigenvalues of A (κ) which do not contribute to E(κ), there exist some holomorphicfunctions β1, . . . , βq on the neighborhood W of κ such that

∀κ ∈ W , sp(A (κ)) ∩ ζ ∈ C , |ζ| > 1− 3 δ = β1(κ) , . . . , βq(κ) .

The geometric regularity condition also shows that the eigenvalues βi(κ) admitsome eigenvectors ei(κ) that are defined holomorphically on the neighborhoodW . To complete the proof of Lemma 6, it remains to observe that the vectorse1(κ), . . . , eN(s+1)(κ) are linearly independent, so this property remains true forall κ ∈ W , up to shrinking W once again. We have therefore constructed the columnvectors of the invertible matrix T (κ). Since T and T−1 are holomorphic, they arealso bounded up to shrinking W .

To conclude this paragraph, we show that geometric regularity can also arise as anecessary condition for stability of a finite difference scheme. In one space dimension,this notion seems to be central in the analysis of the discrete Cauchy problem andwe shall see in the next sections that it also plays a central role in the analysis ofdiscrete initial boundary value problems. Our result is the following.

Lemma 7. Let us consider the numerical scheme (2.8), resp. (2.14), in the scalarcase N = 1. If (2.8), resp. (2.14), is stable in the sense of Definition 1, resp. Defini-tion 2, then the finite difference operator Q, resp. the operators Qσ, is geometricallyregular.


Proof (Proof of Lemma 7). In the case of the one step scheme (2.8), the amplificationmatrix A (κ) in (2.11) is a complex number so geometric regularity is trivial. Theonly coefficient of A depends holomorphically on κ ∈ C \ 0 and the eigenvectoris independent of κ. We thus turn to the case of multistep schemes. The proof ofLemma 7 relies on a very simple observation which we state separately since it willbe useful later on.

Lemma 8. Let M ∈Mm(C) be a companion matrix, that is

M =

µ1 . . . . . . µm1 0 . . . 0

0. . .

. . ....

0 0 1 0

.

Then for all eigenvalue λ of M , the dimension of Ker (M − λ I) equals 1 and theeigensapce is spanned by the vector (λm−1, . . . , λ, 1)T .

The proof of Lemma 8 follows from a simple calculation and is omitted. Let uscomplete the proof of Lemma 7. If (2.14) is stable, we know that the amplificationmatrix A (κ) in (2.16) is uniformly power bounded for κ ∈ S1. Let us now considera point κ ∈ S1 for which A (κ) has an eigenvalue z ∈ S1. According to Lemma 3, weknow that z is a semi-simple eigenvalue. Since Lemma 8 shows that the geometricmultiplicity of z equals 1, we can conclude that z is a simple eigenvalue of A (κ). Inparticular, the Weierstrass preparation Theorem shows that for κ in a neighborhoodof κ, A (κ) has a unique simple eigenvalue β(κ) that depends holomorphically onκ such that β(κ) = z. Lemma 8 shows that the eigenspace Ker (A (κ) − β(κ) I) isspanned by the vector (β(κ)m−1, . . . , β(κ), 1)T which also depends holomorphicallyon κ. We have thus proved that the operators Qσ in (2.14) are geometrically regular.

We now show on a series of examples that either Lemma 2 or Proposition 3can be used to prove stability for various well-known numerical schemes. In theseexamples, we shall also be interested in giving a precise description of the eigenvaluesof the amplification matrix near a point where its spectrum meets S1.

2.2.3 The Lax-Friedrichs and leap-frog schemes

The Lax-Friedrichs approximation of (2.5) corresponds to the scheme (2.8) where

p = r = 1 , QLF :=I + λA

2T−1 +

I − λA2

T .

In other words, the corresponding numerical scheme readsUn+1j =

Unj−1 + Unj+1

2− λA

2(Unj+1 − Unj−1) , j ∈ Z , n ≥ 0 ,

U0j = fj , j ∈ Z .

(2.20)

We recall that the CFL number λ is a constant that is fixed from the beginning andthat stands for the ratio ∆t/∆x. Since A is diagonalizable with real eigenvalues, theresult of Lemma 2 applies and stability for (2.20) is encoded in the von Neumann


condition. Letting µ1, . . . , µN denote the eigenvalues of A and letting T denote aninvertible matrix that diagonalizes A, an easy computation gives

T−1 ALF (ei η)T = diag (z1(η), . . . , zN (η)) , zj(η) := cos η − i λ µj sin η .

In particular, we have

|zj(η)|2 = cos2 η + (λµj)2 sin2 η = 1 + [(λµj)

2 − 1] sin2 η . (2.21)

It is easy to deduce from (2.21) that if λ satisfies λ ρ(A) ≤ 1, then the von Neumanncondition (2.18) is satisfied and the scheme (2.20) is stable. Conversely, let us con-sider the case where λ satisfies λ ρ(A) > 1, with for instance λ |µ1| > 1. For small η,we compute

|z1(η)|2 = 1 + [(λµ1)2 − 1] η2 +O(η4) .

In particular, there holds |z1(η)| > 1 for all η 6= 0 sufficiently small. Corollary 1 thenshows that (2.20) is not stable. Summing up our computations, we have proved thatthe Lax-Friedrichs scheme (2.20) is stable if and only if λ ρ(A) ≤ 1.

Let us now fix a constant λ > 0 such that λ ρ(A) ≤ 1. We wish to study thebehavior of the eigenvalues zj(η) near points where these eigenvalues touch the unitcircle. A first possible case is when λ satisfies λ |µj | = 1 for some index j. Thenzj(η) ∈ S1 for all η ∈ R. Moreover, it is easy to verify the property z′j(η) 6= 0 in thiscase. Consequently, the parametrized curve zj(η) , η ∈ R coincides with S1 andcontains only regular points. The second possible case is when λ satisfies λ |µj | < 1.Then (2.21) shows that zj(η) ∈ S1 if and only if η ∈ Zπ. Furthermore, there holdsz′j(0) = −z′j(π) = −i λ µj . Assuming for simplicity that A is invertible, so that all theeigenvalues µj are nonzero, the parametrized curve zj(η) , η ∈ R is an ellipse thatis included in the unit disk, and that meets the unit circle at ±1 which correspondto regular points. (When 0 is an eigenvalue of A, the corresponding eigenvalue of theamplification matrix yields a parametrized curve that describes the segment [−1, 1],whose contact points ±1 with S1 are singular points.)

The leap-frog scheme is more or less the most simple approximation of (2.5) witha two-steps scheme. It corresponds to the scheme (2.14) where

s = p = r = 1 , Qlf,0 := −λA (T−T−1) , Qlf,1 := I .

In other words, the corresponding numerical scheme readsUn+1j = Un−1

j − λA (Unj+1 − Unj−1) , j ∈ Z , n ≥ 0 ,

U0j = f0

j , j ∈ Z ,U1j = f1

j , j ∈ Z .(2.22)

In this case, the amplification matrix is the block companion matrix

Alf (κ) :=

(−λ (κ− κ−1)A I

I 0

).

Diagonalizing A and permuting rows and columns, there exists a fixed invertiblematrix T such that

T−1 Alf (κ)T := diag

((−λµ1 (κ− κ−1) 1

1 0

), . . . ,

(−λµN (κ− κ−1) 1

1 0

)).


Our goal is first to determine the CFL parameters λ for which the von Neumanncondition holds. For a fixed index j and a real number η, we need to determine theeigenvalues of the matrix (

−2 i λ µj sin η 11 0

).

The eigenvalues are the roots to the polynomial equation

(ω + i λ µj sin η)2 + (λµj)2 sin2 η − 1 = 0 . (2.23)

Let us first consider the case λ |µj | > 1. Then choosing η = π/2, there exists onepurely imaginary root of (2.23) whose modulus equals λ |µj |+

√(λµj)2 − 1 and the

von Neumann condition is not satisfied. Let us now consider the case λ |µj | = 1.Choosing η = π/2 sgn(λµj), −i is a double eigenvalue and the corresponding 2× 2matrix is not diagonalizable. This shows that when λ ρ(A) equals 1, there exists anon-semi-simple eigenvalue z ∈ S1 of Alf (ei η). Using Lemma 3, the scheme can notbe stable.

Eventually, let us show that in the case λ ρ(A) < 1 the leap-frog scheme (2.22)is stable. We are going to apply Proposition 3. Since λ |µj | < 1, the roots to thepolynomial equation (2.23) are

ω1,j(η) :=√

1− (λµj)2 sin2 η − i λ µj sin η ,

ω2,j(η) := −√

1− (λµj)2 sin2 η − i λ µj sin η .

Both roots ω1,j , ω2,j are real analytic functions, and ω1,j − ω2,j does not vanish.Furthermore, it is straightforward to check that both eigenvalues ω1,j(η), ω2,j(η)belong to S1 for all η ∈ R. Let κ ∈ S1. We have already seen that the spectrum ofthe amplification matrix A (κ) is included in S1. The eigenvalues of each matrix(

−λµj (κ− κ−1) 11 0

),

are simple. For κ ∈ C in a sufficiently small neighborhood of κ, the two eigenvaluesand corresponding eigenvectors of(

−λµj (κ− κ−1) 11 0

),

depend holomorphically on κ. Collecting the eigenvalues and eigenvectors of eachdiagonal block in A (κ), we have proved that the operators in the leap-frog schemeare geometrically regular when λ ρ(A) < 1. Applying Proposition 3, we concludethat the leap-frog scheme is stable (and in this case it is also geometrically regular)if and only if λ ρ(A) < 1. In that case, the parametrized curve ω1,j(η) , η ∈ Rdescribes part of the unit circle S1, and it has exactly two singular points of order2 corresponding to the values η − π/2 ∈ Zπ. (The curve parametrized by ω2,j hasexactly the same behavior.)

Let us develop here an elementary calculation which shows stability for the leap-frog scheme (2.22) when λ ρ(A) < 1. We make the additional assumption that thematrix A is symmetric, and therefore |A| = ρ(A). We start from the relation (2.22),take the scalar product with the vector Un+1

j +Un−1j and sum with respect to j ∈ Z.

This yields

120 Jean-Francois Coulombel∑j∈Z

|Un+1j |2 −

∑j∈Z

|Un−1j |2

= −∑j∈Z

(Un+1j )∗

[λA (Unj+1 − Unj−1)

]−∑j∈Z

(Un−1j )∗


].

Using the symmetry of A and performing a “discrete integration by parts”, we obtain∑j∈Z

|Un+1j |2 −

∑j∈Z

|Un−1j |2

=∑j∈Z

[λA (Un+1

j+1 − Un+1j−1 )

]∗Unj −

∑j∈Z

(Un−1j )∗


].

We use the latter relation for both cases n odd and n even, and sum the correspond-ing two equalities. Using new indeces and summing with respect to n, we obtain∑

j∈Z

|U2nj |2 +

∑j∈Z

|U2n+1j |2 −

∑j∈Z

|f0j |2 −

∑j∈Z

|f1j |2

=∑j∈Z

[λA (U2n+1

j+1 − U2n+1j−1 )

]∗U2nj −

∑j∈Z

[λA (f1

j+1 − f1j−1)

]∗f0j .

Applying Cauchy-Schwarz inequality and collecting terms, we obtain

(1− λ |A|)∑j∈Z

|U2nj |2 + |U2n+1

j |2 ≤ (1 + λ |A|)∑j∈Z

|f0j |2 + |f1

j |2 .

Multiplying by ∆x, we have thus proved stability for (2.22) under the assumptionthat A is symmetric and satisfies λ ρ(A) < 1. Of course, this “energy method” basedon integration by parts does not predict instability in the case λ ρ(A) ≥ 1, neitherdoes it give information about the behavior of the eigenvalues of the amplificationmatrix.

For the Lax-Friedrichs and leap-frog schemes, we have focused on the descriptionof the parametrized curves zj(η), where zj(η) is an eigenvalue of the amplificationmatrix A (ei η). In these two examples, the eigenvalues can be parametrized globallyby smooth 2π-periodic functions of η ∈ R. Such curves are represented in Figure2.1.

At this stage, one can try to determine what are all the possible singular pointsfor the eigenvalues curves zj(η) when considering all stable geometrically regularoperators. For instance, the leap-frog scheme produces singular points of order 2.It does not seem so straightforward to find consistent and stable discretizations of(2.5) for which the amplification matrix has eigenvalues with an even more singularbehavior (singular points of order 3 or more). Some examples were constructed in[Cou09, Cou11a] but they are very limited from a practical point of view sincethey involve many grid points while achieving only a low order of approximation.Their interest is mainly theoretical and we postpone their detailed construction toAppendix A. Let us simply conclude by saying that it is important to understandwhich are the possible behaviors for the eigenvalues curves since these observationswill play an important role in Section 2.3 (see in particular the discussion on the“discrete block structure”).


Fig. 2.1. Left : parametrized curves of eigenvalues for the Lax-Friedrichs scheme(2.20) (the unit circle in black, the eigenvalue curve for λ |µj | = 0.8 in red, and theeigenvalue curve for λ |µj | = 0.5 in blue). Right : parametrized curves of eigenvaluesfor the leap-frog scheme (2.22) (the unit circle in black, the eigenvalues curves forλ |µj | = 0.8 in red and blue).

2.2.4 A few facts to remember in view of what follows, and a (notvery interesting) conjecture

We try to summarize here a few facts that should be kept in mind since they willplay an important role in the following Section. The von Neumann condition is anecessary condition for stability. However, in one space dimension, the geometricregularity condition for the amplification matrix is more or less always satisfied.This is rather clear for one step schemes (s = 0) since usually the matrices A` canbe simultaneously diagonalized. For multistep schemes as the leap-frog scheme, thisis a little less obvious but it can usually be checked by rather simple arguments. Themain advantage of the geometric regularity property is that it characterizes stabilityby means of the von Neumann condition, thus ruling out pathological Jordan blocks.

The main difference between the theory of hyperbolic partial differential equa-tions and their discrete counterparts lies in the behavior of eigenvalues of the am-plification matrix. For the continuous Cauchy problem, one passes from u(0, ξ) tou(∆t, ξ) through a multiplication by the matrix exp(−i∆t ξ A), see (2.7). In partic-ular, the eigenvalues of this exact amplification matrix belong to S1 for all frequencyξ. On the Fourier side, this property shows that modes associated with any fre-quency are not damped so that the L2 norm of the solution is conserved (at leastup to an appropriate change of unknown that diagonalizes A). At the discrete level,the eigenvalues of the amplification matrix are not necessarily located on S1 sincethey can also belong to D. Eigenvalues in D correspond to an exponential dampingas what happens more or less for parabolic equations. In order to have the lowestdissipation, the eigenvalues of the amplification matrix should remain as close aspossible to S1 (compare for instance the two pictures in Figure 2.6 and guess whichscheme dissipates less).

What makes the situation for discrete problems far more complicated is that forhigh frequencies, eigenvalues of the amplification matrix may approach S1 again. (Forconsistent schemes, there is always a group of eigenvalues located at 1 for η = 0.) This


may give rise to singular contact points of the eigenvalues curves that are analogousto glancing frequencies in the theory of partial differential equations. We refer to[Hig86, Tre82] for some more details on this issue. Here, such glancing frequenciesare also associated with some kind of dissipation phenomenon. The examples inAppendix A are given in order to show that many possible singular contact pointsmay appear. As a matter of fact, our conjecture is the following: if we consider thescalar transport equation

∂tu+ ∂xu = 0 , u(0, x) = f(x) ,

and if we consider two integers m1 ∈ N, m2 ∈ N such that m1 > 0 and 2m2 ≥ m1,then there exists a stable and geometrically regular numerical scheme such thatthere exists one eigenvalue curve defined in the neighborhood of some η ∈ R andsatisfying

z(η) = z +α (η− η)m1 + o((η− η)m1) , |z(η)| = 1− c (η− η)2m2 + o((η− η)2m2) ,

where z ∈ S1, α ∈ C \ 0 and c > 0. Of course, the numerical scheme should alsobe consistent with the transport equation in order to be meaningful. The examplesin Appendix A show that the conjecture is true at least for m1 = 2m2, m2 ∈ N, aswell as for m1 = 3 and m2 = 2. We do not think however that this conjecture isreally meaningful from a mathematical point of view. Our message is the following:geometrically regular numerical schemes are quite natural in one space dimensionsince all ”standard” schemes seem to fall into this class. One may therefore wish todevelop a stability theory for discretized initial boundary value problems that coversall geometrically regular discretizations of the hyperbolic operator. There is a priceto pay though, which is the appearance of infinitely many possible singular contactpoints with S1 for the eigenvalues of the amplification matrix. Dealing with thesesingular contact points makes the main gap between the seminal work [GKS72] and[Cou09, Cou11a]. Eventually, we note that such glancing/dissipative frequencies donot appear in the analogous theory for partial differential equations, see for instance[BG07, chapter 4].

2.3 Fully discrete initial boundary value problems:strong stability

2.3.1 Finite difference discretizations and strong stability

From now on, we consider the continuous problem (2.1) which we discretize bymeans of a finite difference scheme. Let us assume that we have already chosen onediscretization of the hyperbolic operator, as in Section 2.2, and that this schemeinvolves r points on the left and p points on the right, see (2.9) or (2.15). Herethe space grid is not indexed by Z anylonger since we consider a problem on ahalf-line. Up to using a translation on the indeces, we can always assume that thespace grid is indexed by j ∈ Z , j ≥ 1 − r. This means that the solution u to(2.1) is approximated by a sequence (Unj ) defined for j ≥ 1 − r and n ≥ 0. If theinitial condition (U0

j )j≥1−r is known, then we can not apply the discretization of thehyperbolic operator at points j = 1− r, . . . , 0 because this would require using some


values U0` with ` ≤ −r. Consequently, a discretization of (2.1) must involve (i) one

discretization of the hyperbolic operator to be used at the grid points j ≥ 1, and(ii) one way to discretize the boundary conditions to be used at the grid points j =1−r, . . . , 0. As we have already seen in Section 2.2, there are many possible choices fordiscretizing the hyperbolic operator and the reader will no doubt imagine that thereis also a wide choice of possibilities for discretizing the boundary conditions. We donot aim here at considering the most general schemes but we shall try neverthelessto encompass a wide class of discretizations, both in terms of the hyperbolic operatorand in terms of the boundary conditions. Some rather simple examples are givenin the following Section. More examples may be found in [GKS72] and [GKO95,chapters 11, 13] as well as in the references cited therein. In the examples that weshall detail in these notes, we shall see that discretizing the boundary conditions isnot especially difficult in one space dimension since one can then separate incomingfrom outgoing characteristics. Achieving high order approximation together withstability is however more delicate.

After this short introduction, let us now introduce the finite difference approxi-mation of (2.1). We let ∆x,∆t > 0 denote a space and a time step where λ = ∆t/∆xis a fixed positive constant, and we also let p, q, r, s denote some fixed integers.The solution to (2.1) is approximated by a sequence (Unj ) defined for n ∈ N, andj ∈ 1 − r + N. For j = 1 − r, . . . , 0, the vector Unj should be understood as anapproximation of the trace u(n∆t, 0) on the boundary x = 0, and possibly thetrace of normal derivatives. For instance, in the case r = 1, there is one grid pointin the discrete boundary, and Un0 is an approximate value of u(n∆t, 0). In the caser = 2, there are two grid points in the discrete boundary: Un0 is still an approximatevalue of u(n∆t, 0) and the scheme can be built in such a way that (Un0 − Un−1)/∆xis an approximation of ∂xu(n∆t, 0). In some sense, the integer r can give a measureof the order of approximation at the boundary. (It is rather clear that with onlyone grid point in the discrete boundary, one will hardly reach an approximation oforder 10...) The boundary meshes [j ∆x, (j + 1)∆x[, j = 1− r, . . . , 0, shrink to 0as ∆x tends to 0, so the formal continuous limit problem as ∆x tends to 0 is seton the half-line R+. In these notes, we consider finite difference approximations of(2.1) that read

Un+1j =

s∑σ=0

Qσ Un−σj +∆tFnj , j ≥ 1 , n ≥ s ,

Un+1j =

s∑σ=−1

Bj,σ Un−σ1 + gn+1

j , j = 1− r, . . . , 0 , n ≥ s ,

Unj = fnj , j ≥ 1− r , n = 0, . . . , s ,

(2.24)

where the operators Qσ and Bj,σ are given by:

Qσ :=

p∑`=−r

A`,σ T` , Bj,σ :=

q∑`=0

B`,j,σ T` . (2.25)

In (2.25), all matrices A`,σ, B`,j,σ belong to MN (R) and are independent of thesmall parameter ∆t, while T still denotes the shift operator as in Section 2.2. Letus emphasize that we deal here with explicit schemes for simplicity. If the solutionis known up to the time index n ≥ s, then the scheme first determines Un+1

j forj ≥ 1 by applying the discretization of the hyperbolic operator. Then the scheme


determines the values Un+11−r , . . . , U

n+10 by applying the operators Bj,σ. We believe

that most of the arguments below can be adapted to some implicit discretizationsas in [GKS72].

In Section 2.2, we have studied the stability of fully discrete hyperbolic equationson the whole real line. Stability for a numerical scheme had been defined in orderto reproduce the energy estimate (2.6) that was known to hold for the continuousproblem. The definition of stability for (2.24) follows the same approach, except thathere we wish to study the sensitivity of the solution with respect to three possiblesource terms: the interior source term (Fnj ), the boundary source term (gnj ) and theinitial data f0, . . . , fs. We shall in some sense cut the problems into two pieces anddeal first with the case of zero initial data. Nonzero initial data will be considered inSection 2.5. For zero initial data, an appropriate notion of stability was introducedin [GKS72]:

Definition 4 (Strong stability [GKS72]). The finite difference approximation(2.24) is said to be strongly stable if there exists a constant C0 such that for allγ > 0 and all ∆t ∈ ]0, 1], the solution (Unj ) of (2.24) with f0 = · · · = fs = 0 satisfiesthe estimate:

γ

γ∆t+ 1

∑n≥s+1

∑j≥1−r

∆t∆xe−2γn∆t|Unj |2 +∑n≥s+1

p∑j=1−r

∆te−2γn∆t|Unj |2

≤ C0

γ ∆t+ 1

γ

∑n≥s

∑j≥1

∆t∆x e−2 γ (n+1)∆t |Fnj |2

+∑n≥s+1

0∑j=1−r

∆t e−2 γ n∆t |gnj |2 . (2.26)

In Definition (4), the stability estimate (2.26) should be understood as follows:if the source terms (Fnj ), (gnj ) are such that the right hand-side of the inequality isfinite, then the solution (Unj ) should satisfy the latter inequality and the constantC0 is independent of γ > 0 and ∆t ∈ ]0, 1]. If the source terms are such that the righthand-side of the inequality is infinite, then (2.24) still uniquely defines a sequence(Unj ) but we do not require this solution to satisfy anything. The terminology “strongstability” is used to emphasize that the solution is estimated in the same norm asthe data. Here there are an interior source term and a boundary source term sothe natural requirement is to ask for a control of U in the interior domain and acontrol of the “trace” of U . To be completely honest, we should warn the readerthat Definition 4 above is not exactly the notion of strong stability introduced in[GKS72]. The difference is the following. In [GKS72], the authors considered in theleft-hand side of the inequality the term

∑n≥s+1

0∑j=1−r

∆t e−2 γ n∆t |Unj |2 ,

in order to estimate the trace of the solution (Unj ) while here we make the sumrun from 1 − r to p. This modification is motivated by the results of the followingparagraphs where we wish to characterize - as easily as possible - strong stability by


means of an estimate for the so-called resolvent equation. Such a characterization iseasily proved if we consider this slightly stronger notion of stability, while we havenot been able to fill the gap in [GKS72] with their weaker notion. But this does notso much matter since we show a better property on the solution that what appearedin [GKS72].

There are two ways to remember the stability estimate of Definition (4), and tounderstand why the various weights involving γ and ∆t are meaningful. Studyingfirst the limit ∆t → 0, we should recover formally an energy estimate for the con-tinuous problem (2.1). Indeed, if we let formally ∆t tend to 0, assuming that allquantities have a limit, we obtain

γ

∫ ∫R+×R+

e−2 γ t|u(t, x)|2 dt dx+

∫R+

e−2 γ t|u(t, 0)|2 dt

≤ C0

1

γ

∫ ∫R+×R+

e−2 γ t|F (t, x)|2 dt dx+

∫R+

e−2 γ t|g(t)|2 dt

.

The latter energy estimate is known to hold for solutions of (2.1) with zero initialdata as soon as the well-posedness condition (2.4) holds. This can be checked byusing the formulae (2.2), (2.3). Of course, the above limit is completely formal sincethere is already some problem with the size of the source terms on the boundary:in (2.24), the vectors gnj belong to RN while, for the continuous problem (2.1), g(t)belongs to Rp, and in general p is strictly smaller than N . However, the above formallimit shows the link between the energy estimate for (2.1) and the stability estimate(2.26) of Definition 4. We also note that in the first sum on the left-hand side of(2.26), the factor ∆t∆x is the measure of the mesh [n∆t, (n + 1)∆t[×[j ∆x, (j +1)∆x[ so the sum represents an L2 norm in the variables (t, x) of a piecewise constantfunction. All other sums in (2.26) represent L2 norms in t or in (t, x) as well.

Another interesting observation is to consider the limit γ → +∞ in (2.26). Ata formal level, the term exp(−2 γ m∆t) is negligible with respect to exp(−2 γ n∆t)for m > n. Multiplying (2.26) by exp(2 γ (s+1)∆t) and letting γ tend to +∞ (recallthat the initial data vanish), the scheme (2.24) should verify

1

∆t

∑j≥1−r

∆t∆x |Us+1j |2 +

p∑j=1−r

∆t |Us+1j |2

≤ C0

∆t∑j≥1

∆t∆x |F sj |2 +

0∑j=1−r

∆t |gs+1j |2

,

or equivalently

1

λ

∑j≥1−r

|Us+1j |2 +

p∑j=1−r

|Us+1j |2 ≤ C0

1

λ∆t2

∑j≥1

|F sj |2 +

0∑j=1−r

|gs+1j |2

.

Such an estimate can be easily deduced from (2.24) with U0 = · · · = Us = 0.

Remark 3. The estimate (2.26) can be made independent of ∆t by simply observingthat in (2.24), the small parameter ∆t appears only in the source term ∆tFnj . Oneeasily sees that strong stability for (2.24) is equivalent to requiring that the solution(Unj ) to


Un+1j =

s∑σ=0

Qσ Un−σj + Fnj , j ≥ 1 , n ≥ s ,

Un+1j =

s∑σ=−1

Bj,σ Un−σ1 + gn+1

j , j = 1− r, . . . , 0 , n ≥ s ,

Unj = 0 , j ≥ 1− r , n = 0, . . . , s ,

(2.27)

satisfies the estimate

γ

γ + 1

∑n≥s+1

∑j≥1−r

e−2 γ n |Unj |2 +∑n≥s+1

p∑j=1−r

e−2 γ n |Unj |2

≤ C

γ + 1

γ

∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 +∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 , (2.28)

for all γ > 0 and a constant C that is independent of γ. In other words, one canalways assume ∆t = 1 (and ∆x = 1/λ) when checking strong stability.

In the following paragraph, we shall explain how strong stability can be char-acterized by means of an estimate for the so-called resolvent equation. This charac-terization relies on the Laplace transform. The strategy is entirely analogous to theanalysis for the continuous problem, see [BG07, chapter 4].

2.3.2 The nomal modes analysis and the Godunov-Ryabenkiicondition

The resolvent equation is obtained by formally looking for solutions to (2.24) ofthe form Unj = znWj , z ∈ C \ 0. The source terms in (2.24) should have similarexpressions. Of course, this is a formal procedure since such sequences do not satisfyU0 = · · · = Us = 0, while we are restricting here to the case of zero initial data!Solutions to (2.24) should be thought of as superpositions of such elementary solu-tions that we call normal modes (this is the same strategy as in Section 2.2 whenwe performed some plane wave analysis by looking for solutions to (2.8) under theform of pure oscillations). Plugging the expression Unj = znWj in (2.24) yields asystem of the form

Wj −s∑

σ=0

z−σ−1 QσWj = Fj , j ≥ 1 ,

Wj −s∑

σ=−1

z−σ−1 Bj,σW1 = gj , j = 1− r, . . . , 0 ,(2.29)

where we do not wish to make the expression of the source terms precise since itwould be useless. Our goal here is to characterize strong stability for the scheme(2.24) in terms of an estimate for the solution to the resolvent equation (2.29).The main advantage for doing so is that studying (2.29) has reduced the dimensionsince time has been replaced by one complex parameter. For clarity, we shall dividesome of the arguments below into several intermediate results. The main results aresummarized at the end of this paragraph for future use. Our first main result is


Theorem 2 (Gustafsson, Kreiss, Sundstrom [GKS72]). Assume that thescheme (2.24) is strongly stable in the sense of Definition 4 with a constant C0 > 0such that (2.28) holds. Then for all z ∈ U , for all (Fj) ∈ `2 and for all vectorsg1−r, . . . , g0 ∈ CN , the resolvent equation (2.29) has a unique solution (Wj) ∈ `2

and this solution satisfies

|z| − 1

|z|∑j≥1−r

|Wj |2+

p∑j=1−r

|Wj |2 ≤ 4C0

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 . (2.30)

The proof of Theorem 2 relies on two preliminary results, which we prove now.

Lemma 9. For all x > 0, there holds

x

1 + x≤ ex − 1

ex≤ 2

x

1 + x,

or equivalently1

2

1 + x

x≤ ex

ex − 1≤ 1 + x

x.

Proof (Proof of Lemma 9). The inequality

x

1 + x≤ ex − 1

ex,

is easily seen to be equivalent to ex ≥ 1 + x, and this inequality follows from thepower series expansion of the exponential function.

On the other hand, the inequality

ex − 1

ex≤ 2

x

1 + x,

is equivalent to (x− 1) ex + x+ 1 ≥ 0. The latter function of x vanishes at 0 and isincreasing on R+ so the result follows.

Lemma 10. For all ν ≥ 1, we define the function ρν on R by

∀ θ ∈ R , ρν(θ) :=1√ν

ν−1∑k=0

e−i k θ .

Then the sequence (ρν) satisfies the following properties:

(i) For all ν ≥ 1, ρν is 2π-periodic and

1

2π

∫ π

−π|ρν(θ)|2 dθ = 1 .

(ii) For all α ∈ ]0, π/2], there holds

limν→+∞

∫ π

α

|ρν(θ)|2 dθ = 0 .


(iii)For all continuous function H on R verifying supθ∈R(1+θ2), |H(θ)| < +∞, thereholds

limν→+∞

1

2π

∫RH(θ) |ρν(θ)|2 dθ =

∑k∈Z

H(2 k π) .

Proof (Proof of Lemma 10). • The proof of property (i) follows from a straightfor-ward computation:∫ π

−π|ρν(θ)|2 dθ =

1

ν

ν−1∑k1,k2=0

∫ π

−πei (k1−k2) θ dθ = 2π .

• For α ∈ ]0, π/2] and θ ∈ [α, π], we have

|ρν(θ)| = 1√ν

∣∣∣∣1− e−i ν θ

1− e−i θ

∣∣∣∣ ≤ 2√ν√

(1− cos θ)2 + sin2 θ

≤

2/√ν if θ ∈ [π/2, π],

2/(√ν sinα) if θ ∈ [α, π/2].

Property (ii) follows by integrating the latter inequalities.• Let us first observe that both the integral on R and the sum over Z in property

(iii) are well-defined thanks to the assumption on H. Moreover, property (i) gives

Aν :=

∣∣∣∣∣ 1

2π

∫RH(θ) |ρν(θ)|2 dθ −

∑k∈Z

H(2 k π)

∣∣∣∣∣≤ 1

2π

∑k∈Z

∫ (2 k+1)π

(2 k−1)π

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ .

Our goal is to show that the sequence (Aν)ν≥1 converges towards 0. Let thereforeε > 0. We first note that there exists an integer Kε, that is independent of ν, suchthat

∀ ν ≥ 1 ,1

2π

∑|k|>Kε

∫ (2 k+1)π

(2 k−1)π

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ ≤ ε .

Indeed the assumption on H yields, for a suitable constant M that depends on Hbut not on ν,

1

2π

∑|k|>K

∫ (2 k+1)π

(2 k−1)π

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ

≤ 1

2π

∑|k|>K

∫ (2 k+1)π

(2 k−1)π

M

1 + k2|ρν(θ)|2 dθ = M

∑|k|>K

1

1 + k2.

The right-hand side of the latter inequality is small provided that K is large, inde-pendently of ν. We thus have

Aν ≤ ε+1

2π

∑|k|≤Kε

∫ (2 k+1)π

(2 k−1)π

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ .


The continuity of H at the points 2 k π, |k| ≤ Kε, gives the existence of someα ∈ ]0, π/2], that is independent of ν, such that

∀ k = −Kε, . . . ,Kε , ∀ θ ∈ [2 k π − α, 2 k π + α] ,

|H(θ)−H(2 k π)| ≤ ε

2Kε + 1.

For all ν ≥ 1, we thus have

Aν ≤ 2 ε+1

2π

∑|k|≤Kε

∫ 2 k π−α

(2 k−1)π

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ

+1

2π

∑|k|≤Kε

∫ (2 k+1)π

2 k π+α

|H(θ)−H(2 k π)| |ρν(θ)|2 dθ

≤ 2 ε+4 ‖H‖L∞(R)

2π(2Kε + 1)

∫ π

α

|ρν(θ)|2 dθ ,

where we have used property (i) for the integrals on [2 k π − α, 2 k π + α] and thefact that |ρν | is even. Using property (ii), we can complete the proof of property(iii) because we have obtained Aν ≤ 3 ε for ν sufficiently large.

Let us now prove Theorem 2.

Proof (Proof of Theorem 2). Before proving that the resolvent equation (2.29) has aunique solution for all data in `2, we shall prove an a priori estimate for any solutionto (2.29). In other words, we shall consider that we already have a solution to theresolvent equation and we wish to prove that this solution satisfies the estimate(2.30). We introduce the notation

∀ z ∈ U , L(z) : W ∈ `2 7−→ L(z)W ∈ `2

with (L(z)W )j :=

Wj −

∑sσ=0 z

−σ−1 QσWj , if j ≥ 1,

Wj −∑sσ=−1 z

−σ−1 Bj,σW1 , if 1− r ≤ j ≤ 0.(2.31)

Let now (Wj)j≥1−r ∈ `2, and let z0 ∈ U . For all integer ν ≥ 1, we define thesequence

∀ j ≥ 1− r , ∀n ≥ 0 , Unj (ν) :=

zn0 Wj/

√ν , if s+ 1 ≤ n ≤ s+ ν,

0 , otherwise,

as well as the source terms

∀ j ≥ 1 , ∀n ≥ s , Fnj (ν) := Un+1j (ν)−

s∑σ=0

Qσ Un−σj (ν) ,

∀ j = 1− r, . . . , 0 , ∀n ≥ s+ 1 , gnj (ν) := Unj (ν)−s∑

σ=−1

Bj,σ Un−1−σ1 (ν) .

In other words, the sequence (Unj (ν)) satisfies


Un+1j (ν) =

s∑σ=0

Qσ Un−σj (ν) + Fnj (ν) , j ≥ 1 , n ≥ s ,

Un+1j (ν) =

s∑σ=−1

Bj,σ Un−σ1 (ν) + gn+1

j (ν) , j = 1− r, . . . , 0 , n ≥ s ,

Unj (ν) = 0 , j ≥ 1− r , n = 0, . . . , s .

(2.32)

It is not difficult to check that for all fixed n, (Unj (ν)) and (Fnj (ν)) belong to `2.Moreover, Fnj (ν) = 0 for n ≥ 2 s+ ν + 1, and gnj (ν) = 0 for n ≥ 2 s+ ν + 2. We canapply the strong stability estimate (2.28) for γ = γ0 := ln |z0| > 0:

γ0

γ0 + 1

∑n≥s+1

∑j≥1−r

e−2 γ0 n |Unj (ν)|2 +∑n≥s+1

p∑j=1−r

e−2 γ0 n |Unj (ν)|2

≤ C0

γ0 + 1

γ0

∑n≥s

∑j≥1

e−2 γ0 (n+1) |Fnj (ν)|2

+∑n≥s+1

0∑j=1−r

e−2 γ0 n |gnj (ν)|2 . (2.33)

The right-hand side of (2.33) is finite because the sum over n involves finitely manyterms. The above definition of Unj (ν) gives

∀ j ≥ 1− r ,∑n≥s+1

e−2 γ0 n |Unj (ν)|2 =

s+ν∑n=s+1

e−2 γ0 n |z0|2n|Wj |2

ν= |Wj |2 ,

so (2.33) reduces to

γ0

γ0 + 1

∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2

≤ C0

γ0 + 1

γ0

∑n≥s

∑j≥1

e−2 γ0 (n+1) |Fnj (ν)|2

+∑n≥s+1

0∑j=1−r

e−2 γ0 n |gnj (ν)|2 .

Using Lemma 9, we have

|z0| − 1

|z0|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2

≤ 4C0

|z0||z0| − 1

∑n≥s

∑j≥1

e−2 γ0 n |z0|−2 |Fnj (ν)|2

+∑n≥s+1

0∑j=1−r

e−2 γ0 n |gnj (ν)|2 . (2.34)


The left-hand side of the inequality (2.34) does not depend on ν, and we are nowgoing to compute the limit of the right-hand side in (2.34) as ν tends to +∞.

Let us define the following functions on R+:

Uj(ν, t) :=

0 , if t ∈ [0, s+ 1[,

Unj (ν) , if t ∈ [n, n+ 1[, n ≥ s+ 1,

Fj(ν, t) :=

0 , if t ∈ [0, s[,

Fnj (ν) , if t ∈ [n, n+ 1[, n ≥ s,

gj(ν, t) :=

0 , if t ∈ [0, s+ 1[,

gnj (ν) , if t ∈ [n, n+ 1[, n ≥ s+ 1.

It is not difficult to check that the Laplace transform of each function Uj(ν, ·),Fj(ν, ·), gj(ν, ·) is well-defined and holomorphic on C, because each one of thesefunctions is bounded with compact support in R+. To avoid any possible confusion,we recall that the Laplace transform of a function f defined on R+ is

f(τ) :=

∫R+

e−τ t f(t) dt ,

for all complex number τ such that the above integral makes sense. The system(2.32) equivalently reads

Uj(ν, t+ 1) =

s∑σ=0

Qσ Uj(ν, t− σ) + Fj(ν, t) , j ≥ 1 , t ≥ s ,

Uj(ν, t) =

s∑σ=−1

Bj,σ U1(ν, t− 1− σ) + gj(ν, t) , j = 1− r, . . . , 0, t ≥ s+ 1 .

Multiplying the above equation by e−τ t and integrating over [s,+∞[ or [s+1,+∞[,we obtain

Uj(ν, τ)−s∑

σ=0

z−σ−1 Qσ Uj(ν, τ) = z−1 Fj(ν, τ) , j ≥ 1 ,

Uj(ν, τ)−s∑

σ=−1

z−σ−1 Bj,σ Uj(ν, τ) = gj(ν, τ) , j = 1− r, . . . , 0 ,(2.35)

where we use the short notation z := eτ .The Laplace transform Uj(ν, τ) can be explicitely computed from the definition

of Unj (ν). If we consider one τ0 ∈ C such that z0 := eτ0 , then we get

∀ θ ∈ R , Uj(ν, τ0 + i θ) =1− z−1

0 e−i θ

τ0 + i θe−i (s+1)θ ρν(θ)Wj , (2.36)

where ρν stands for the function defined in Lemma 10. Using the first relation in(2.35), we obtain

z−10 e−i θ Fj(ν, τ0 + i θ)

=1− z−1

0 e−i θ

τ0 + i θe−i (s+1)θ ρν(θ)

(Wj −

s∑σ=0

z−σ−10 e−i (σ+1)θ QσWj

).


Applying Plancherel’s Theorem and Fubini’s Theorem, we have∑n≥s

∑j≥1

e−2 γ0 n |z0|−2 |Fnj (ν)|2

=2 γ0

1− e−2 γ0

∑j≥1

|z0|−2

∫R+

e−2 γ0 t |Fj(ν, t)|2 dt

=2 γ0

2π (1− e−2 γ0)

∑j≥1

|z0|−2

∫R

∣∣Fj(ν, τ0 + i θ)∣∣2 dθ

=2 γ0

1− e−2 γ0

1

2π

∫RH(θ) |ρν(θ)|2 dθ ,

where we have used the notation

∀ θ ∈ R , H(θ) :=

∣∣∣∣1− z−10 e−i θ

τ0 + i θ

∣∣∣∣2 ∑j≥1

∣∣∣∣∣Wj −s∑

σ=0

z−σ−10 e−i (σ+1)θ QσWj

∣∣∣∣∣2

.

It is not so difficult to check that the function H satisfies the assumptions of property(iii) of Lemma 10. We thus have (recall the notation (2.31)):

limν→+∞

∑n≥s

∑j≥1

e−2 γ0 n |z0|−2 |Fnj (ν)|2

=2 γ0

1− e−2 γ0

∑k∈Z

|1− z−10 |2

|τ0 + 2 i k π|2∑j≥1

|(L(z0)W )j |2 . (2.37)

With completely similar arguments, we can also obtain

limν→+∞

∑n≥s+1

0∑j=1−r

e−2 γ0 n |gnj (ν)|2

=2 γ0

1− e−2 γ0

∑k∈Z

|1− z−10 |2

|τ0 + 2 i k π|20∑

j=1−r

|(L(z0)W )j |2 . (2.38)

Passing to the limit in (2.34) and using (2.37), (2.38), we get

|z0| − 1

|z0|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2

≤ 4C0

|z0||z0| − 1

∑j≥1

|(L(z0)W )j |2 +

0∑j=1−r

|(L(z0)W )j |2

× 2 γ0

1− e−2 γ0

∑k∈Z

|1− z−10 |2

|τ0 + 2 i k π|2 . (2.39)

Recalling the expression (2.36) for the Laplace transform of Uj(ν, ·), we have


|Wj |2 =∑n≥s+1

e−2 γ0 n |Unj (ν)|2

=2 γ0

1− e−2 γ0

∫R+

e−2 γ0 t |Uj(ν, t)|2 dt

=2 γ0

2π (1− e−2 γ0)

∫R

∣∣Uj(ν, τ0 + i θ)∣∣2 dθ

=2 γ0

2π (1− e−2 γ0)

∫R

∣∣∣∣1− z−10 e−i θ

τ0 + i θ

∣∣∣∣2 |Wj |2 |ρν(θ)|2 dθ

→ 2 γ0

1− e−2 γ0

∑k∈Z

|1− z−10 |2

|τ0 + 2 i k π|2 |Wj |2 .

We have thus derived the formula

2 γ0

1− e−2 γ0

∑k∈Z

|1− z−10 |2

|τ0 + 2 i k π|2 = 1 ,

so we can simplify in (2.39) and obtain

|z0| − 1

|z0|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2

≤ 4C0

|z0||z0| − 1

∑j≥1

|(L(z0)W )j |2 +

0∑j=1−r

|(L(z0)W )j |2 . (2.40)

The inequality (2.40) is only an a priori estimate for the operators L(z), z ∈U . We emphasize that the constant 4C0 is independent of z0 ∈ U and W ∈ `2.To complete the proof of Theorem 2, we only need to prove that each (bounded)operator L(z) is invertible. This property is shown by combining two arguments.

Lemma 11. There exists R0 ≥ 1 such that for all z ∈ C with |z| > R0, the operatorL(z) defined by (2.31) is an isomorphism on `2.

Proof (Proof of Lemma 11). Let L∞ be defined by

L∞ : W ∈ `2 7−→ L∞W ∈ `2

with (L∞W )j :=

Wj , if j ≥ 1,

Wj −Bj,−1 W1 , if 1− r ≤ j ≤ 0.

It is easy to check that L∞ is a bounded invertible operator on `2. Moreover, forz ∈ U and W ∈ `2, we have

z((L∞ − L(z))W

)j

=

∑sσ=0 z

−σ QσWj , if j ≥ 1,∑sσ=0 z

−σ Bj,σW1 , if 1− r ≤ j ≤ 0.

Consequently there exists a constant C such that

∀ z ∈ U , ‖L∞ − L(z)‖B(`2) ≤C

|z| ,

with B(`2) the set of bounded operators on `2. This property implies that for |z| >C ‖L−1

∞ ‖B(`2), L(z) is an isomorphism.


Lemma 12. Let E be a Banach space, and let T denote a nonempty connected set.Let L be a continuous function on T with values in the space B(E) of boundedoperators on E. Assume moreover that the two following conditions are satisfied:

• there exists a constant M > 0 such that for all t ∈ T and for all x ∈ E, we have‖x‖E ≤M ‖L (t)x‖E,

• there exists some t0 ∈ T such that L (t0) is an isomorphism.

Then L (t) is an isomorphism for all t ∈ T .

Proof (Proof of Lemma 12). We already know that B(E) is a Banach space andthat the set of isomorphisms Gl(E) is an open subset of B(E). This first propertyshows that the set t ∈ T /L (t) ∈ Gl(E) is open because L is continuous. It onlyremains to show that this set is closed and the claim will follow (this set is nonemptythanks to the assumption of Lemma 12). We thus consider a sequence (tn) in Tthat converges towards t ∈ T and such that for all n, L (tn) belongs to Gl(E). Weare going to show that L (t) also belongs to Gl(E). Using the Banach isomorphismTheorem, it is enough to prove that L (t) is a bijection.

Due to the uniform bound ‖x‖E ≤M ‖L (t)x‖E , it is clear that L (t) is injectiveand that for all n we have ‖L (tn)−1‖B(E) ≤ M . It remains to show that L (t) issurjective. Let y ∈ E. For all integers n and p, we have:∥∥L (tn+p)

−1 y −L (tn)−1 y∥∥E≤∥∥L (tn+p)

−1 −L (tn)−1∥∥

B(E)‖y‖E

≤∥∥L (tn+p)

−1 (L (tn)−L (tn+p)) L (tn)−1∥∥

B(E)‖y‖E

≤M2 ‖L (tn+p)−L (tn)‖B(E) ‖y‖E .

These inequalities show that (L (tn)−1 y) is a Cauchy sequence in E and thereforeconverges towards x ∈ E. Moreover we have L (tn) L (tn)−1 y = y for all n so,passing to the limit, we get L (t)x = y. Here we use again the continuity of L . Thisshows that L (t) is surjective, which completes the proof.

Lemma 12 shows that the resolvent equation can be uniquely solved for allz ∈ U . Indeed, for all integer ν sufficiently large, the mapping L restricted to theannulus z ∈ C , 1 + 2−ν ≤ |z| ≤ 2ν satisfies the assumptions of Lemma 12 (useLemma 11 for the existence of one point where L is an isomorphism and (2.40) for theuniform bound). We leave to the reader the verification that L(z) ∈ B(`2) dependscontinuously on z ∈ U . Eventually we can conclude that L(z) is an isomorphism forall z ∈ U and L(z)−1 satisfies the estimate (2.40) which is nothing else but (2.30).The proof of Theorem 2 is now complete.

Theorem 2 has an important consequence, which is the following well-knownnecessary condition for strong stability.

Corollary 2 (Godunov-Ryabenkii condition). Assume that the numericalscheme (2.24) is strongly stable in the sense of Definition 4. Then for all z ∈ U ,any W ∈ `2 satisfying

Wj −s∑

σ=0

z−σ−1 QσWj = 0 , j ≥ 1 ,

Wj −s∑

σ=−1

z−σ−1 Bj,σW1 = 0 , j = 1− r, . . . , 0 ,

must be zero.


The Godunov-Ryabenkii condition is a preliminary test in view of showing strongstability. It is analogous to the Lopatinskii condition for hyperbolic initial boundaryvalue problems. As we shall see later on, it is unfortunately not a sufficient conditionfor strong stability (see the following paragraphs for more comments).

In the remaining part of this paragraph, we are going to show the converse resultof Theorem 2.

Theorem 3 (Gustafsson, Kreiss, Sundstrom [GKS72]). Assume that thereexists a constant C1 > 0 such that for all z ∈ U , for all (Fj) ∈ `2 and for all vectorsg1−r, . . . , g0 ∈ CN , the resolvent equation (2.29) has a unique solution (Wj) ∈ `2


|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ C1

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Then the scheme (2.24) is strongly stable and satisfies (2.26) with the constantC1 max(1, λ)/min(1, λ).

The proof of Theorem 3 splits in several steps. In what follows, we shall saythat a sequence (Unj ) has compact support if the terms Unj vanish except for a finitenumber of indeces (j, n).

Proof (Proof of Theorem 3). •We consider some source terms (Fnj ), (gnj ) for (2.27)with compact support. We also let (Unj ) denote the solution to (2.27). It is easy toshow by induction on n that for all n, the sequence (Unj )j≥1−r belongs to `2. Forγ > 0, we introduce the quantities

IN (γ) :=

N∑n=0

∑j≥1

e−2 γ n |Unj |2 , BN (γ) :=

N∑n=0

0∑j=1−r

e−2 γ n |Unj |2 ,

SF (γ) :=∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 , Sg(γ) :=∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 .

Performing very crude estimates in (2.27), we immediately see that there exists aconstant C > 0 that is independent of F, g, U such that

∀ j ≥ 1 , ∀n ≥ s , |Un+1j |2 ≤ C

(|Fnj |2 +

s∑σ=0

p∑`=−r

|Un−σj+` |2

),

∀ j = 1− r, . . . , 0 , ∀n ≥ s , |Un+1j |2 ≤ C

(|gn+1j |2 +

s∑σ=−1

q∑`=0

|Un−σ1+` |2

).

Multiplying each inequality by exp(−2 γ (n+ 1)) and taking the sum, we obtain

∀N ≥ s+ 1 , IN (γ) ≤ C SF (γ) + C e−2 γs∑

σ=0

IN−1−σ(γ) + BN−1−σ(γ) ,

BN (γ) ≤ CIN (γ) + C Sg(γ) + C e−2 γs∑

σ=0

IN−1−σ(γ) ,


with a possibly larger constant C. Using the obvious inequalities

IN−1−σ(γ) ≤ IN (γ) , BN−1−σ(γ) ≤ BN (γ) ,

and combining the above estimates for IN (γ),BN (γ), we obtain that for some largeenough γ > 0, that is independent of F, g, U and N , there holds

∀ γ ≥ γ , ∀N ≥ s+ 1 , IN (γ) + BN (γ) ≤ C(SF (γ) + Sg(γ)

).

In other words, for γ ≥ γ, we have∑n≥s+1

∑j≥1−r

e−2 γ n |Unj |2 < +∞ . (2.41)

• As in the proof of Theorem 2, it is convenient to introduce the followingfunctions defined on R+:

Uj(t) :=

0 , if t ∈ [0, s+ 1[,

Unj , if t ∈ [n, n+ 1[, n ≥ s+ 1,

Fj(t) :=

0 , if t ∈ [0, s[,

Fnj , if t ∈ [n, n+ 1[, n ≥ s,

gj(t) :=

0 , if t ∈ [0, s+ 1[,

gnj , if t ∈ [n, n+ 1[, n ≥ s+ 1.

Then (2.41) reads

∀ γ ≥ γ ,∑j≥1−r

∫R+

e−2 γ t |Uj(t)|2 dt < +∞ . (2.42)

The Laplace transforms Fj , gj are well-defined and holomorphic on C, and Fj is iden-tically zero for j large enough. Moreover, (2.42) shows that the Laplace transforms

Uj , j ≥ 1− r, are well-defined and holomorphic on Re τ > γ, with γ independentof j. Applying Plancherel’s Theorem in (2.42), we find that for all γ > γ and for

almost every θ ∈ R, the sequence(Uj(γ + i θ)

)j≥1−r belongs to `2.

Applying the Laplace transform on (2.27) with Re τ > γ, we getUj(τ)−

s∑σ=0

z−σ−1 Qσ Uj(τ) = z−1 Fj(τ) , j ≥ 1 ,

Uj(τ)−s∑

σ=−1

z−σ−1 Bj,σ Uj(τ) = gj(τ) , j = 1− r, . . . , 0 ,(2.43)

where we still use the short notation z := eτ as in the proof of Theorem 2.Since Fj vanishes for j large, we have

∀ τ ∈ C ,∑j≥1

|z|−2∣∣Fj(τ)

∣∣2 < +∞ .

For all τ ∈ C with Re τ > 0, we can thus define (Wj(τ))j≥1−r as the unique solutionin `2 to the resolvent equation

2 Stability of finite difference schemes for boundary value problems 137Wj(τ)−

s∑σ=0

z−σ−1 QσWj(τ) = z−1 Fj(τ) , j ≥ 1 ,

Wj(τ)−s∑

σ=−1

z−σ−1 Bj,σWj(τ) = gj(τ) , j = 1− r, . . . , 0 .(2.44)

Moreover, (Wj(τ))j≥1−r satisfies

∀ τ ∈ C , Re τ > 0 ,|z| − 1

|z|∑j≥1−r

|Wj(τ)|2 +

p∑j=1−r

|Wj(τ)|2

≤ C1

|z||z| − 1

∑j≥1

|z|−2∣∣Fj(τ)

∣∣2 +

0∑j=1−r

∣∣gj(τ)∣∣2 . (2.45)

The difference between (2.43) and (2.44) is that (2.43) holds only for Re τ > γ while

(2.44) holds for Re τ > 0. Our goal is to identify (Wj) and (Uj) and to show that(2.43) holds for Re τ > 0. This is based on the following result, the proof of whichis left to the reader.

Lemma 13. The operator L(z) ∈ B(`2) in (2.31) depends holomorphically onz ∈ C \ 0. Consequently, under the assumptions of Theorem 3, L(eτ )−1 dependsholomorphically on τ for Re τ > 0.

Lemma 13 implies that for all j ≥ 1 − r, Wj is holomorphic on Re τ > 0because the source terms in (2.44) depend holomorphically on τ . Furthermore, we

know that Uj is holomorphic on Re τ > γ, and that for all γ > γ and almost

every θ ∈ R, (Uj(γ + i θ)) belongs to `2 and is a solution to (2.43). This implies

Uj(γ+ i θ) = Wj(γ+ i θ) for γ > γ and for almost every θ ∈ R. Since both functionsare holomorphic, we have obtained

∀ j ≥ 1− r , ∀ γ > γ , ∀ θ ∈ R , Uj(γ + i θ) = Wj(γ + i θ) .

Let now γ0 > 0. We integrate (2.45) with respect to θ ∈ R for τ = γ + i θand γ > γ0, and use Plancherel’s Theorem to compute the right-hand side of theinequality. We thus obtain

supγ>γ0

∑j≥1−r

∫R|Wj(γ + i θ)|2 dθ < +∞ .

Applying the Paley-Wiener Theorem for which we refer to [Rud87], this means thatfor all j ≥ 1− r, there exists a measurable function Vj on R+ such that∫

R+

e−2 γ0 t |Vj(t)|2 dt < +∞ ,

and Wj = Vj on Re τ > γ0. By injectivity of the Laplace transform, Vj must equalUj . In other words, we have just proved that for all γ0 > 0, exp(−γ0 t)Uj belongs

to L2(R+), so Uj is well-defined on Re τ > 0 and coincides with Wj . Hence (2.45)

holds with Uj instead of Wj . We now integrate (2.45) with respect to θ = Im τ anduse Plancherel’s Theorem, which yields


eγ − 1

eγ

∑n≥s+1

∑j≥1−r

e−2 γ n |Unj |2 +∑n≥s+1

p∑j=1−r

e−2 γ n |Unj |2

≤ C1

eγ

eγ − 1

∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 +∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 ,

for all γ > 0. Applying Lemma 9, we get

γ

γ + 1

∑n≥s+1

∑j≥1−r

e−2 γ n |Unj |2 +∑n≥s+1

p∑j=1−r

e−2 γ n |Unj |2

≤ C1

γ + 1

γ

∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 +∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 ,

• It is useful to recall that the latter estimate was derived under the assumptionthat the source terms had compact support. To complete the proof of Theorem 3,it is sufficient to prove Lemma 14 below.

Lemma 14. Assume that for all data (Fnj ) and (gnj ) with compact support, thesolution (Unj ) to (2.27) satisfies

γ

γ + 1

∑n≥s+1

∑j≥1−r

e−2 γ n |Unj |2 +∑n≥s+1

p∑j=1−r

e−2 γ n |Unj |2

≤ C1

γ + 1

γ

∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 +∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 ,

for all γ > 0. Then the scheme (2.24) satisfies (2.26) with the constantC1 max(1, λ)/min(1, λ).

Proof (Proof of Lemma 14). Let us consider some source terms (Fnj ), (gnj ) for (2.27),not necessarily with compact support. Let γ > 0 such that the right-hand side of(2.28) is finite. For ν ≥ 1, we define

Fnj (ν) :=

Fnj , if s ≤ n ≤ s+ ν − 1 and 1 ≤ j ≤ 1 + q + ν p,

0 , otherwise,

gnj (ν) :=

gnj , if s+ 1 ≤ n ≤ s+ ν and 1− r ≤ j ≤ 0,

0 , otherwise.

A direct induction argument shows that the corresponding solution (Unj (ν)) to (2.27)satisfies Unj (ν) = Unj for 0 ≤ n ≤ s+ ν and 1− r ≤ j ≤ ν. We thus have

γ

γ + 1

∑n≤s+ν

∑1−r≤j≤ν

e−2 γ n |Unj |2 +∑

n≤s+ν

p∑j=1−r

e−2 γ n |Unj |2

≤ C1

γ + 1

γ

∑n≥s

∑j≥1

e−2 γ (n+1) |Fnj |2 +∑n≥s+1

0∑j=1−r

e−2 γ n |gnj |2 ,


for all γ > 0 and all ν ≥ p+ 1. Passing to the limit ν → +∞, we have proved that(2.28) holds with the constant C1 and without any assumption of compact supporton the data. To prove that (2.26) holds, it is sufficient to apply (2.28) with the sourceterm ∆tFnj instead of Fnj , and with the parameter γ ∆t > 0 instead of γ. Recallingthe relation ∆t = λ∆x, we obtain the result. The details are left to the reader.

We summarize the main results of this paragraph into the following result.

Theorem 4 (Characterization of strong stability [GKS72]). The scheme(2.24) is strongly stable in the sense of Definition 4 if and only if there exists aconstant C > 0 such that for all z ∈ U , for all (Fj) ∈ `2 and for all vectorsg1−r, . . . , g0 ∈ CN , the resolvent equation (2.29) has a unique solution (Wj) ∈ `2


|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ C

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

In particular, if the scheme (2.24) is strongly stable, then the Godunov-Ryabenkiicondition of Corollary 2 holds.

It is also useful to keep in mind that showing the unique solvability of the resol-vent equation relies on a rather simple argument of functional analysis that reducesto the verification of an a priori estimate. Furthermore, the resolvent equation be-comes trivially solvable for |z| large. More precisely we state a slightly refined versionof Theorem 4 which will be useful in the following paragraph. Theorem 5 below showsthat it is sufficient to consider the resolvent equation for bounded parameters z.

Theorem 5 (Characterization of strong stability). The scheme (2.24) isstrongly stable in the sense of Definition 4 if and only if for all R ≥ 2, there existsa constant CR > 0 such that for all z ∈ U with |z| ≤ R, for all (Fj) ∈ `2 and forall vectors g1−r, . . . , g0 ∈ CN , the resolvent equation (2.29) has a unique solution(Wj) ∈ `2 and this solution satisfies

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Proof (Proof of Theorem 5). • Let us first assume that the scheme (2.24) is stronglystable in the sense of Definition 4. Then we apply Theorem 4: the resolvent equationcan be uniquely solved in `2 for all z ∈ U with the estimate

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ C

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

This shows that the conclusion of Theorem 5 holds with a constant CR := C thatis independent of R ≥ 2.• Let us now assume that for all R ≥ 2, the resolvent equation can be uniquely

solved in `2 for all z ∈ U , |z| ≤ R, with the estimate


|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

We first apply Lemma 11 and keep the notation introduced in the proof of thisLemma. There exists R0 ≥ 2 such that for all z ∈ C with |z| ≥ R0, the mappingL(z) ∈ B(`2) is an isomorphism and ‖L(z)−L∞‖B(`2) ≤ ‖L−1

∞ ‖−1B(`2)

/2. In particu-

lar, there exists a constant C > 0 such that for all z ∈ C with |z| ≥ R0, the uniquesolution (Wj) ∈ `2 to (2.29) satisfies

∑j≥1−r

|Wj |2 ≤ C

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

This estimate yields

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ 2∑j≥1−r

|Wj |2

≤ 2C

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2

≤ 2C

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

It remains to use the assumption for the radius R0 and consider the constantmax(2C,CR0). Theorem 4 then shows that the scheme (2.24) is strongly stable.

In the following paragraph, we shall write the resolvent equation into an equiva-lent but more convenient form. This will lead to the formulation of our main resultwhich characterizes strong stability in terms of an algebraic condition that is anal-ogous to the so-called uniform Kreiss-Lopatinskii condition.

2.3.3 An equivalent form of the resolvent equation

The equation

Wj −s∑

σ=0

z−σ−1 QσWj = Fj ,

defines an induction relation of order p+ r on the sequence (Wj). It is convenient torewrite this induction relation as an induction of order 1 for an augmented sequence.This is a classical procedure. For ` = −r, . . . , p, we define the matrices

∀ z ∈ C \ 0 , A`(z) := δ`0 I −s∑

σ=0

z−σ−1 A`,σ , (2.46)

where δ`1`2 denotes the Kronecker symbol. We also define the matrices

∀ ` = 0, . . . , q , ∀ j = 1− r, . . . , 0 , ∀ z ∈ C \ 0 ,

B`,j(z) :=

s∑σ=−1

z−σ−1 B`,j,σ . (2.47)


With these definitions, the reader will easily verify that (2.29) equivalently reads(use (2.25))

p∑`=−r

A`(z)Wj+` = Fj , j ≥ 1 ,

Wj −q∑`=0

B`,j(z)W1+` = gj , j = 1− r, . . . , 0 .(2.48)

To rewrite (2.48) as an induction relation of order 1, we make, as in [GKS72], thefollowing assumption.

Assumption 1 (Noncharacteristic discrete boundary). The matrices A−r(z) andAp(z) are invertible for all z ∈ U , or equivalently for all z ∈ C with |z| > 1− ε0

for some ε0 ∈ ]0, 1/2].

Let us first consider the case q < p. In that case, all the Wj ’s involved in theboundary conditions for the resolvent equation (2.48) are coordinates of the aug-mented vector4 W1 := (Wp, . . . ,W1−r) ∈ CN(p+r). Using Assumption 1, we candefine a block companion matrix M(z) that is holomorphic on some open neighbor-hood V := z ∈ C , |z| > 1− ε0 of U :

∀ z ∈ V , M(z) :=

−Ap(z)−1 Ap−1(z) . . . . . . −Ap(z)−1 A−r(z)

I 0 . . . 0

0. . .

. . ....

0 0 I 0

∈MN(p+r)(C) . (2.49)

We also define the matrix that encodes the boundary conditions for (2.48), namely

∀ z ∈ C \ 0 , B(z) :=

0 . . . 0 −Bq,0(z) . . . −B0,0(z) I 0...

......

.... . .

0 . . . 0 −Bq,1−r(z) . . . −B0,1−r(z) 0 I

∈MNr,N(p+r)(C) , (2.50)

with the B`,j ’s defined in (2.47). With such definitions, it is a simple exercise torewrite the resolvent equation (2.48) as an induction relation for the augmentedvector Wj := (Wj+p−1, . . . ,Wj−r) ∈ CN (p+r), j ≥ 1. This induction relation takesthe form

Wj+1 = M(z) Wj + Fj , j ≥ 1 ,

B(z) W1 = G ,(2.51)

where the new source terms (Fj),G in (2.51) are given by:

Fj := (Ap(z)−1 Fj , 0, . . . , 0) , G := (g0, . . . , g1−r) .

Remark 4. It is easy to check that the matrix B(z) in (2.50) depends holomorphicallyon z ∈ C \ 0 and has maximal rank N r for all z (just consider the N r×N r sub-matrix formed by the last columns). Consequently, the kernel of B(z) has dimensionN p for all z ∈ C \ 0.4 Vectors are now written indifferently in rows or columns in order to simplify the

redaction.


Let us now characterize strong stability for (2.24) in terms of an estimate for(2.51). Of course we shall use Theorem 5 and the strong relationship between (2.29)and (2.51).

Proposition 4 (Characterization of strong stability). Let Assumption 1 besatisfied, and let us assume q < p. Then the scheme (2.24) is strongly stable in thesense of Definition 4 if and only if for all R ≥ 2, there exists a constant CR > 0such that for all z ∈ U with |z| ≤ R, for all (Fj) ∈ `2 and for all G ∈ CNr, theequation (2.51) has a unique solution (Wj) ∈ `2 and this solution satisfies

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 + |G |2 . (2.52)

The main point to keep in mind is that in Proposition 4, the source terms Fj

may be arbitrary in CN(p+r), while when we rewrote (2.29) under the form (2.51),only the first coordinate of Fj was nonzero.

Proof (Proof of Proposition 4). • Let us first assume that the scheme (2.24) isstrongly stable so we can apply Theorem 5. Our goal is to show that (2.51) hasa unique solution for all source terms in `2 and that the estimate (2.52) holds fora suitable constant CR. As a warm-up, let us first show that if a solution in `2 to(2.51) exists, then it is necessarily unique. By linearity, this amounts to proving thatif (Wj)j≥1 ∈ `2 satisfies

Wj+1 = M(z) Wj , j ≥ 1 ,

B(z) W1 = 0 ,

then (Wj)j≥1 is zero (this is a new formulation of the Godunov-Ryabenkii condition).We thus consider such a sequence (Wj)j≥1, and we introduce the decomposition

Wj = (W (1)j , . . . ,W (p+r)

j ), where each vector W (k)j belongs to CN . Using the block

decomposition of M(z) - recall the definition (2.49) - we obtain

∀ ` = −r, . . . , p− 1 , ∀ j ≥ 1 , W (p−`)j = W (p+r)

j+`+r .

Furthermore, the sequence defined by Wj := W (p+r)j+r , j ≥ 1− r, satisfies the homo-

geneous resolvent equation

p∑`=−r

A`(z)Wj+` = 0 , j ≥ 1 ,

Wj −q∑`=0

B`,j(z)W1+` = 0 , j = 1− r, . . . , 0 .

The Godunov-Ryabenkii condition (Corollary 2) gives (Wj)j≥1−r = 0, which yields(Wj)j≥1 = 0. If a solution in `2 to (2.51) exists, it is necessarily unique.

Let now R ≥ 2, let z ∈ U satisfy |z| ≤ R, and let us consider (Fj) ∈ `2, G ∈ CNr.We wish to construct a solution (Wj) ∈ `2 to (2.51). We use again the decomposition

Wj = (W (1)j , . . . ,W (p+r)

j ) as well as the notation Wj := W (p+r)j+r , j ≥ 1−r. The source

terms are also decomposed as Fj = (F (1)j , . . . ,F (p+r)

j ), G = (G (0), . . . ,G (1−r)).Inspecting the system (2.51) shows that we should necessarily have


∀ ` = −r, . . . , p− 1 , ∀ j ≥ 1 , W (p−`)j = Wj+` −

`−1∑k=−r

F (p−k)j+`−1−k . (2.53)

Moreover, the sequence (Wj)j≥1−r should be a solution to (2.48) with source terms(Fj), g1−r, . . . , g0 defined by

∀ j ≥ 1 , Fj :=

p∑`=−r

A`(z)`−1∑k=−r

F (p−k)j+`−1−k , (2.54)

∀ j = 1− r, . . . , 0 , gj := G (j) +

j−2∑k=−r

F (p−k)j−1−k −

q∑`=0

B`,j(z)`−1∑k=−r

F (p−k)`−k . (2.55)

An important remark in view of what follows is that the matrices A` and B`,j definedby (2.46), (2.47) are bounded on U . Consequently, it is rather easy to check that therelations (2.54), (2.55) define a sequence (Fj) ∈ `2 and vectors g1−r, . . . , g0 ∈ CNsuch that, for a given constant M that does not depend on z nor on R, there holds

∑j≥1

|Fj |2 ≤M∑j≥1

|Fj |2 ,0∑

j=1−r

|gj |2 ≤M

∑j≥1

|Fj |2 + |G |2 . (2.56)

Applying Theorem 5, we know that there exists a unique solution (Wj) ∈ `2 to(2.48) with the source terms defined in (2.54), (2.55), and that for some constantCR independent of z and of the source terms, there holds

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

The relation (2.53) defines a sequence (Wj)j≥1 in CN(p+r); it is not difficult tocheck that this sequence belongs to `2 and that it is a solution to (2.51). Moreover,combining (2.53), (2.56) and the latter estimate for (Wj), we obtain

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 + |G |2 .

As already shown above, such a solution (Wj) ∈ `2 to (2.51) is unique.• Let us now assume that (2.51) has a unique solution in `2 for all source terms

(Fj),G and that the estimate (2.52) holds. Let now R ≥ 2, let z ∈ U with |z| ≤ R,and let us consider some source terms (Fj) ∈ `2, g1−r, . . . , g0 ∈ CN for the resolventequation (2.29). We define the vectors

Fj := (Ap(z)−1 Fj , 0, . . . , 0) , G := (g0, . . . , g1−r) .

The assumption yields the existence of a sequence (Wj) ∈ `2 to (2.51), satisfying

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 + |G |2 ,


with a constant CR that only depends on R. The above definition of the sourceterms (Fj),G gives5

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ C′R

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Using the decomposition Wj = (W (1)j , . . . ,W (p+r)

j ) as well as the notation Wj :=

W (p+r)j+r , j ≥ 1−r, we can check that (Wj) ∈ `2 is a solution to the resolvent equation

(2.29) and that it satisfies

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ C′R

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Again, we can also verify that such a solution (Wj) in `2 is necessarily unique(because the solution to (2.51) is unique in `2). The details are left to the reader.Theorem 5 completes the argument.

Remark 5. The result of Proposition 4 explains why in Definition 4 we have consid-ered the trace estimate ∑

n≥s+1

p∑j=1−r

∆t e−2 γ n∆t |Unj |2

in the left-hand side of (2.26). The main purpose for doing so is to obtain the term|W1|2 in the left-hand side of the estimate (2.52) in Proposition 4. Obtaining suchan estimate is possible only if in the characterization of Theorem 4 or Theorem 5,the estimate for the resolvent equation involves |W1−r|2 + · · · + |Wp|2 in the left-hand side and not only |W1−r|2 + · · ·+ |W0|2. If we had kept the definition of strongstability in [GKS72], the left-hand side of (2.52) would have involved |Π W1|2 insteadof |W1|2, where Π would be the projection from CN(p+r) to CNr that retains thelast N r components.

Remark 6. The definition of M(z) in (2.49) only depends on the fulfillment of As-sumption 1 and not on the integer q. We could have defined M(z) in the same wayeven if q had not been smaller than p.

We now examine the case q ≥ p. There is a slight modification to make here.If we wish to rewrite the boundary conditions of (2.48) as a linear system for someaugmented vector W1, then the coordinates of W1 should involve W1−r, . . . ,Wq+1,and q+1 > p. However we can still write the resolvent equation under a form similarto (2.51) up to defining

5 Here we observe that it is crucial to consider a bounded parameter z, becauseotherwise we could not use a uniform bound for |Ap(z)−1|. This is the main reasonwhy we have proved Theorem 5, because Theorem 4 would not have been suffi-cient. It is also crucial that the norm |Ap(z)−1| remains bounded as z approachesS1, which amounts to assuming that Ap(z) is invertible not only on U but on U(same argument as Lemma 12).


∀ z ∈ V , M(z) :=

−Ap(z)−1 Ap−1(z) . . . −Ap(z)−1 A−r(z) 0 . . . 0

I 0 . . . . . . 00 0 00 0 . . . 0 I 0

∈MN(q+r+1)(C) . (2.57)

We also define the matrix that encodes the boundary conditions for (2.48) and thefirst q + 1− p steps in the induction, namely

∀ z ∈ C \ 0 , B(z) :=

−Bq,0(z) . . . −B0,0(z) I 0...

.... . .

−Bq,1−r(z) . . . −B0,1−r(z) 0 I0 Ap(z) . . . . . . A−r(z)

. . .. . .

Ap(z) . . . . . . A−r(z) 0

∈MN(q+1−p+r),N(q+1+r)(C) , (2.58)

with the B`,j ’s defined in (2.47). With such definitions, it is a simple exercise torewrite the resolvent equation (2.48) as an induction relation for the augmentedvector Wj := (Wj+q, . . . ,Wj−r) ∈ CN (q+1+r), j ≥ 1. This induction relation takesthe form

Wj+1 = M(z) Wj + Fj , j ≥ 1 ,

B(z) W1 = G ,(2.59)

where the new source terms (Fj),G in (2.59) are given by:

Fj := (Ap(z)−1 Fj+q+1−p, 0, . . . , 0) , G := (g0, . . . , g1−r, F1, . . . , Fq+1−p) .

We can then obtain a result that is analogous to Proposition 4. The result is justslighlty more complicated because of the definition (2.58) of the matrix B(z) butthe proof follows exactly the same arguments.

Proposition 5 (Characterization of strong stability). Let Assumption 1 besatisfied, and let us assume q ≥ p. Then the scheme (2.24) is strongly stable in thesense of Definition 4 if and only if for all R ≥ 2, there exists a constant CR > 0such that for all z ∈ U with |z| ≤ R, for all (Fj) ∈ `2 and for all G ∈ CN(q+1−p+r),the equation (2.59) has a unique solution (Wj) ∈ `2 and this solution satisfies

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +|z||z| − 1

|GII |2 + |GI |2 ,

(2.60)where we use the decomposition G = (GI ,GII), GI ∈ CNr, GII ∈ CN(q+1−p).

Proof (Proof of Proposition 5). • Let us assume that the scheme (2.24) is stronglystable, so we can use the result of Theorem 5. Let R ≥ 2, let z ∈ U with |z| ≤ R,and let (Fj) ∈ `2, G ∈ CN(q+1−p+r). The source terms are decomposed as Fj =

(F (1)j , . . . ,F (q+1+r)

j ), GI = (G (0), . . . ,G (1−r)) ∈ CNr, GII = (G (1), . . . ,G (q+1−p)) ∈CN(q+1−p). We are looking for a solution Wj = (W (1)

j , . . . ,W (q+1+r)j ) in `2 to (2.59).

Using the notation Wj := W (q+1+r)j+r , j ≥ 1− r, we should necessarily have


∀ ` = −r, . . . , q , ∀ j ≥ 1 , W (q+1−`)j = Wj+` −

`−1∑k=−r

F (q+1−k)j+`−1−k . (2.61)

Moreover, the sequence (Wj)j≥1−r should be a solution to (2.48) with source terms(Fj), g1−r, . . . , g0 defined by

∀ j ≥ q + 2− p , Fj :=

p+r∑`=0

Ap−`(z)q−`∑k=−r

F (q+1−k)j+p−1−`−k , (2.62)

∀ j = 1, . . . , q + 1− p , Fj := G (j) +

p∑`=−r

A`(z)j+`−2∑k=−r

F (q+1−k)j+`−1−k , (2.63)

and

∀j = 1− r, . . . , 0, gj := G (j) +

j−2∑k=−r

F (q+1−k)j−1−k −

q∑`=0

B`,j(z)`−1∑k=−r

F (q+1−k)`−k . (2.64)

The relations (2.62), (2.63), (2.64) define a sequence (Fj) ∈ `2 and vectors g1−r, . . . , g0

such that, for a given constant M that does not depend on z, there holds

∑j≥1

|Fj |2 ≤M

∑j≥1

|Fj |2 + |GII |2 ,

0∑j=1−r

|gj |2 ≤M

∑j≥1

|Fj |2 + |GI |2 . (2.65)

Applying Theorem 5, we know that there exists a unique solution (Wj) ∈ `2 to (2.48)with the source terms defined in (2.62), (2.63), (2.64), and that for some constantCR independent of z, there holds

|z| − 1

|z|∑j≥1−r

|Wj |2 +

p∑j=1−r

|Wj |2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

The relation (2.61) then defines a sequence (Wj)j≥1 ∈ `2 which is a solution to(2.59). Combining (2.61), (2.65) and the estimate of (Wj), we get

|z| − 1

|z|∑j≥1

|Wj |2 +

p−1∑`=−r

∣∣∣W (q+1−`)1

∣∣∣2

≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +|z||z| − 1

|GII |2 + |GI |2 .

In order to complete the proof, it only remains to estimate the sum

q∑`=p

∣∣∣W (q+1−`)1

∣∣∣2 .


This is done with an induction argument based on the relations

∀ j = 0, . . . , q − p ,p∑

`=−r

A`(z) W (q+1−`−j)1 = G (1+j)

1 ,

and the fact that Ap(z) is invertible for all z ∈ U . The details are left to the reader.Eventually, we obtain an estimate of the form

|W1|2 =

q∑`=−r

∣∣∣W (q+1−`)1

∣∣∣2 ≤ C′R |z||z| − 1

∑j≥1

|Fj |2 +|z||z| − 1

|GII |2 + |GI |2 .

The uniqueness of the solution (Wj) ∈ `2 to (2.59) is proved by entirely similararguments to those used in the proof of Proposition 4. We feel free at this point toskip the details.• Let us now assume that (2.59) has a unique solution in `2 for all source terms

(Fj) ∈ `2 and G together with the estimate (2.60). Let R ≥ 2, let z ∈ U with|z| ≤ R, and let us consider some source terms (Fj) ∈ `2, g1−r, . . . , g0 ∈ CN for(2.29). We define the vectors

Fj := (Ap(z)−1 Fq+1−p+j , 0, . . . , 0) , G := (g0, . . . , g1−r, F1, . . . , Fq+1−p) .

The assumption yields the existence of a sequence (Wj) ∈ `2 to (2.59), satisfying

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 +|z||z| − 1

|GII |2 + |GI |2 .

The definition of the source terms (Fj),G gives

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ C′R

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Using the decomposition Wj = (W (1)j , . . . ,W (q+1+r)

j ) as well as the notation Wj :=

W (q+1+r)j+r , j ≥ 1− r, (Wj) ∈ `2 is a solution to (2.29) that satisfies

|z| − 1

|z|∑j≥1−r

|Wj |2 +

q+1∑j=1−r

|Wj |2 ≤ C′R

|z||z| − 1

∑j≥1

|Fj |2 +

0∑j=1−r

|gj |2 .

Such a solution in `2 to (2.29) is necessarily unique, and Theorem 5 completes theargument.

Remark 7. Unlike what happened in the case q < p with the definition (2.50) of

the matrix B(z), it is no longer clear that the matrix B(z) in (2.58) has maximalrank (this was uncorrectly claimed in [Cou09]). However, the result of Proposition

5 shows that if the scheme (2.24) is strongly stable, then B(z) should have maximalrank for all z ∈ U (use Proposition 5 with Fj = 0 for all j and an arbitrary G ). Arefined version of this result is stated in the following paragraph.


2.3.4 Characterization of strong stability: the main result

Up to now, we have characterized strong stability in terms of an estimate for theresolvent equation (2.29), or for the equivalent formulations (2.51) or (2.59). We havealso seen that a necessary condition for strong stability is the so-called Godunov-Ryabenkii condition of Corollary 2, which is an analogue of the Lopatinskii conditionfor hyperbolic initial boundary value problems. In this paragraph, we make a littlemore precise this necessary condition for strong stability. It will turn out that thisrefined necessary condition will also be sufficient for strong stability. Readers who arefamiliar with the theory of hyperbolic initial boundary value problems will recognizethe gap between the Lopatinskii condition and the uniform Lopatinskii condition,see [BG07, chapter 4]. The gap here between the Godunov-Ryabenkii conditionand what we shall call the uniform Kreiss-Lopatinskii condition below is entirelyanalogous.

Let us begin with a fundamental property of the matrices M(z) in (2.49) and

M(z) in (2.57). We recall that the operators Qσ that appear in (2.24) and whose ex-pression is given in (2.25) correspond to a discretization of the hyperbolic operator.According to the analysis of Section 2.2, see in particular Propositions 1 and 2, sta-bility for the discrete Cauchy problem is encoded in the uniform power boundednessof the amplification matrix A (ei η), η ∈ R. To encompass both situations s = 0 ands ≥ 1, we shall always refer to the discrete Cauchy problem as to problem (2.14),with the operators Qσ as in (2.25) or (2.15). The amplification matrix A is thendefined in (2.16) as a (block) companion matrix. When s equals 0, this definitionreduces to (2.11). The fundamental property of M(z) is stated as follows.

Lemma 15 (Stable eigenvalues of M(z) [Kre68]). Let Assumption 1 be satisfied,and let us assume that the discretization of the Cauchy problem (2.14) is stable inthe sense of Definition 2. Then for all z ∈ V , the eigenvalues of the matrix M(z) in(2.49) are those κ ∈ C \ 0 such that

det (A (κ)− z I) = 0 .

In particular for all z ∈ U , M(z) has no eigenvalue on the unit circle S1 and thenumber of eigenvalues in D equals N r (eigenvalues are counted with their algebraicmultiplicity).

We emphasize that there is no condition on the integer q in Lemma 15 becausethe definition of M(z) is independent of q, see Remark 6.

Proof (Proof of Lemma 15). The matrix M(z) in (2.49) is defined on the openneighborhood V = z ∈ C , |z| > 1 − ε0 of U . On V , both matrices A−r(z)and Ap(z) are invertible thanks to Assumption 1. Let now z ∈ V , and let X =(X1, . . . , Xp+r) ∈ CN(p+r) belong to the kernel of M(z). Using the expression (2.49)of M(z), we get

X1 = · · · = Xp+r−1 = 0 , Ap(z)−1 A−r(z)Xp+r = 0 ,

so the kernel of M(z) is reduced to 0. In particular, the eigenvalues of M(z)are nonzero. We are now going to obtain some more precise information on theseeigenvalues.


Applying some standard rules for determinants of block companion matrices(use Schur’s complement formula, see e.g. [Ser10]), we obtain for all z ∈ V and allκ 6= 0:

det(M(z)− κ I) = (−1)N(p+r−1) det

[−

p∑`=−r

κ`+r Ap(z)−1 A`(z)− κp+r I

]

= (−1)N(p+r) κNr detAp(z)−1 det

[p∑

`=−r

κ` A`(z)

]. (2.66)

In the same way, we compute

det(A (κ)− z I) = (−1)Ns det

[s∑

σ=0

zs−σ Qσ(κ)− zs+1 I

]

= (−1)N(s+1) zN(s+1) det

[p∑

`=−r

κ` A`(z)

],

where the amplification matrix A is defined in (2.16). In other words, for z ∈ Vand κ 6= 0, det(M(z)− κ I) and det(A (κ)− z I) vanish simultaneously. This provesthe first part of Lemma 15.

Let now z ∈ U . Let us assume that κ ∈ S1 is an eigenvalue of M(z). Thenz is an eigenvalue of A (κ). However, stability for the discrete Cauchy problem(2.14) implies that the von Neumann condition is satisfied, see Corollary 1, so thespectral radius of A (κ) is not larger than 1. We are led to a contradiction. Bya continuity/connectedness argument, the number of eigenvalues of M(z) in D isindependent of z ∈ U . We are now going to show that this number equals N r. Theidea is to study the behavior of eigenvalues of M(z) as z tends to infinity.

Let us first show that as z tends to infinity, the eigenvalues of M(z) which belongto D converge to 0. For otherwise, there would exist ε > 0, a sequence (zn)n≥1 with|zn| > n, and a sequence (κn)n≥1 such that

∀n ≥ 1 , ε ≤ |κn| < 1 , κn ∈ sp(M(zn)) .

Applying the formula (2.66), we have

∀n ≥ 1 , det

[p∑

`=−r

κ`n A`(zn)

]= 0 . (2.67)

Up to extracting a subsequence, we can assume that (κn) converges towards κ∞which satisfies ε ≤ |κ∞| ≤ 1 (in particular, κ∞ 6= 0). Recalling the definition (2.46)and passing to the limit in (2.67), we obtain det I = 0 which is a contradiction. Wehave thus proved that for large |z|, the eigenvalues of M(z) which belong to D arearbitrarily close to 0.

To complete the proof, we introduce the function

D(κ, Z) := det

[p∑

`=−r

κr+` A`(1/Z)

].


According to the definition (2.46) of the matrices A`, D is a polynomial functionof (κ, Z). Moreover, we have D(κ, 0) = κNr. This shows that for all Z 6= 0 suf-ficiently small, the polynomial D(·, Z) has exactly N r roots (counted with theirmultiplicity) which are close to 0. (This is a direct application of Rouche’s Theoremfor holomorphic functions.) Then the formula (2.66) shows that for large |z|, M(z)has N r eigenvalues which are close to 0. Since all eigenvalues of M(z) in D must beclose to 0, we have proved that for all z ∈ U , M(z) has exactly N r eigenvalues inD.

The eigenvalues of M(z) in D are called stable eigenvalues since they correspondto geometrically decreasing sequences (hence in `2) that are solutions to the induc-tion relation

Wj+1 = M(z) Wj , j ≥ 1 .

At the opposite, eigenvalues of M(z) in U will be called unstable eigenvalues sincethey correspond to sequences whose norm diverges geometrically.

Our proof of Lemma 15 follows [Kre68] where the same result is proved in thecase s = 0. Unlike what is stated in [GKS72], the number of eigenvalues of M(z) inD has nothing to do with the boundary conditions in (2.24). As a matter of fact, thedefinition of M(z) only involves the matrices A`,σ and is completely independent of

the matrices B`,j,σ, see (2.49). In the same way, the definition (2.57) of M(z) onlyinvolves the matrices A`,σ.

The matrix M(z) defined in (2.57) and used to rewrite the resolvent equation inthe case q ≥ p satisfies analogous properties to those stated in Lemma 15.

Lemma 16 (Stable eigenvalues of M(z)). Let Assumption 1 be satisfied, let usassume q ≥ p and let us further assume that the discretization of the Cauchy problem(2.14) is stable in the sense of Definition 2. Then for all z ∈ V , the eigenvalues of

M(z) are 0 - with algebraic multiplicity N (q + 1 − p) - and the eigenvalues of thematrix M(z).

In particular for all z ∈ U , M(z) has no eigenvalue on the unit circle S1 and thenumber of eigenvalues in D equals N (q + 1 − p + r) (eigenvalues are counted withtheir algebraic multiplicity).

Proof (Proof of Lemma 16). With the result of Lemma 15, the proof is now straight-forward (we recall that Lemma 15 holds independently of q). Indeed, for z ∈ V andκ ∈ C, we compute

det(M(z)− κ I)

= (−1)N(q+1+r) det

[p+r∑`=1

κq+1+r−` Ap(z)−1 Ap−`(z) + κq+1+r I

]

= (−1)N(q+1+r) detAp(z)−1 κN(q+1−p) det

[p∑

`=−r

κr+` A`(z)

].

Since A−r(z) is invertible, the latter equality shows that 0 is a root with multiplicity

N (q+ 1− p) of the characteristic polynomial of M(z). Moreover, the relation (2.66)

shows that the nonzero eigenvalues of M(z) are exactly the eigenvalues of M(z) andthe algebraic multiplicities coincide. The result of Lemma 16 follows.


The results of Lemma 15 and Lemma 16 imply the following necessary conditionsfor strong stability in the cases q < p and q ≥ p.

Corollary 3 (The uniform Kreiss-Lopatinskii condition in the case q < p).Let Assumption 1 be satisfied, let us assume q < p and let us further assume thatthe discretization of the Cauchy problem (2.14) is stable in the sense of Definition2. If the scheme (2.24) is strongly stable in the sense of Definition 4, then for allR ≥ 2, there exists a constant CR > 0 such that for all z ∈ U with |z| ≤ R, thereholds

∀W ∈ Es(z) , |W | ≤ CR |B(z) W | , (2.68)

where Es(z) denotes the generalized eigenspace of the matrix M(z) associated witheigenvalues in D, and where the matrix B(z) is defined in (2.50).

In other words, if the scheme (2.24) is strongly stable, then the mapping

Φ(z) : W ∈ Es(z) 7−→ B(z) W ∈ CNr ,

is an isomorphism for all z ∈ U . Moreover for all R ≥ 2, the inverse Φ(z)−1 isuniformly bounded with respect to z ∈ U , |z| ≤ R.

Proof (Proof of Corollary 3). The proof is very easy. Let R ≥ 2, and let z ∈ Uwith |z| ≤ R. According to the assumptions, we can apply both Propositions 4 andLemma 15. Let W ∈ Es(z). The sequence (Wj)j≥1 defined by

Wj+1 = M(z) Wj , j ≥ 1 ,

W1 := W ,

belongs to `2 (it converges towards 0 geometrically as j tends to +∞) and it is asolution to

Wj+1 = M(z) Wj , j ≥ 1 ,

B(z) W1 = B(z) W .

Then the estimate (2.52) for solutions to (2.51) yields (2.68). Lemma 15 shows thatthe stable subspace Es(z) has dimension N r so the linear mapping Φ(z) defined inCorollary 3 is an isomorphism (it is injective and the spaces have equal dimension).The estimate (2.68) shows that the norm of Φ(z)−1 remains uniformly bounded asz ∈ U approaches the unit circle.

From Corollary 3, we see that the scheme (2.24) could not have been stronglystable if B(z) had not had maximal rank. Hopefully, this maximal rank property isobvious here, see Remark 4.

There is of course a similar result in the case q ≥ p. We feel free to skip theproof.

Corollary 4 (The uniform Kreiss-Lopatinskii condition in the case q ≥ p).Let Assumption 1 be satisfied, let us assume q ≥ p and let us further assume thatthe discretization of the Cauchy problem (2.14) is stable in the sense of Definition

2. Let us decompose the matrix B(z) in (2.58) as

∀ z ∈ C \ 0 , B(z) =

(B](z)B[(z)

), B](z) ∈MNr,N(q+1+r)(C) ,

B[(z) ∈MN(q+1−p),N(q+1+r)(C) .


If the scheme (2.24) is strongly stable in the sense of Definition 4, then for all R ≥ 2,there exists a constant CR > 0 such that for all z ∈ U with |z| ≤ R, there holds

∀W ∈ Es(z) ∩Ker B[(z) , |W | ≤ CR |B](z) W | , (2.69)

where Es(z) denotes the generalized eigenspace of the matrix M(z) associated witheigenvalues in D.

It is not very hard to show that the space Es(z)∩Ker B[(z) is isomorphic to thestable subspace Es(z) of M(z) and thus has dimension N r for all z ∈ U . Moreover,the matrix B](z) has rank N r for all z ∈ C \ 0. Hence the estimate (2.69) is notruled out by obvious dimensions reasons (for instance if the rank of B](z) had beensmaller than N r).

Let us also observe that if the estimate (2.69) holds, then the mapping

Φ(z) : W ∈ Es(z) 7−→ B(z) W ∈ CN(q+1−p+r) ,

is injective, so it is an isomorphism. In particular, B(z) has maximal rank for all z ∈U . Again, this maximal rank property is a necessary condition for strong stability.

Remark 8. We do not know whether the terminology “uniform Kreiss-Lopatinskiicondition” is really standard in the context of finite difference schemes (probably“uniform Godunov-Ryabenkii condition” might be more appropriate). Our goal hereis to emphasize the link between this condition and the analogous necessary condi-tion for well-posedness for hyperbolic initial boundary value problems.

As we shall see below, the vector space Es(z) varies continuously - and evenholomorphically - with respect to z ∈ U . Another way to rephrase Corollary 3 istherefore: for all z ∈ U , Es(z)∩Ker B(z) = 0, that is, CN(p+r) = Es(z)⊕Ker B(z).Moreover, for all 1 < R1 ≤ R2, the quantity

supR1≤|z|≤R2

supW ∈Es(z)\0

|W ||B(z) W | ,

remains bounded as R1 tends to 1 and R2 remains fixed.The Godunov-Ryabenkii condition shows that the latter quantity is finite for all

1 < R1 ≤ R2, but it does not give any information on how this quantity varies as R1

approaches 1. Some examples for which the uniform Kreiss-Lopatinskii condition isnot satisfied show that this quantity may be unbounded as R1 tends to 1 (see lateron in these notes for the case of the Lax-Friedrichs and leap-frog schemes).

The estimate (2.68), or (2.69), is a necessary condition for strong stability. Theinjectivity of the linear mapping Φ(z) in Corollary 3 can be tested by first de-termining a basis (e1(z), . . . , eNr(z)) of Es(z), and by computing the associated(Lopatinskii) N r ×N r determinant

∆(z) := det[B(z) e1(z) . . . B(z) eNr(z)

].

The vanishing of ∆(z) is independent of the choice of the basis. The Godunov-Ryabenkii condition holds true if and only if ∆ does not vanish on U . Some examplesof computations of such determinants are given a little further in these notes for theLax-Friedrichs and leap-frog schemes with various choices of numerical boundary


conditions. However, the reader will understand that computing such determinantsis not always possible from a practical point of view. For instance, one numericalscheme based on the Runge-Kutta method and presented in Appendix A correspondsto r = 8, and it becomes impossible to compute stable eigenvalues in this case.Numerical strategies are necessary to compute “approximately” the stable subspaceand the Lopatinskii determinant.

In the spirit of [GKS72], our main result shows that the uniform Kreiss-Lopatinskii condition (meaning the fulfillment of the estimate (2.68) or (2.69) ac-cording to the sign of q − p) is not only a necessary condition for strong stabilitybut is also a sufficient condition. Our result requires however a structural assump-tion on the operators Qσ, namely the property of geometric regularity introducedin Section 2.2. More precisely, our main result in the case q < p reads as follows.

Theorem 6 (Main result for q < p). Let Assumption 1 be satisfied, let us assumeq < p and let us further assume that the discretization of the Cauchy problem (2.14)is stable in the sense of Definition 2 and that the operators Qσ are geometricallyregular in the sense of Definition 3. For all z ∈ U , we let Es(z) denote the generalizedeigenspace of the matrix M(z) in (2.49) associated with eigenvalues in D.

Then the scheme (2.24) is strongly stable in the sense of Definition 4 if andonly if for all R ≥ 2, there exists a constant CR > 0 such that for all z ∈ U with|z| ≤ R, the estimate (2.68) holds with the matrix B(z) defined in (2.50).

Our main result in the case q ≥ p is similar.

Theorem 7 (Main result for q ≥ p). Let Assumption 1 be satisfied, let us assumeq ≥ p and let us further assume that the discretization of the Cauchy problem (2.14)is stable in the sense of Definition 2 and that the operators Qσ are geometricallyregular in the sense of Definition 3. For all z ∈ U , we let Es(z) denote the generalized

eigenspace of the matrix M(z) in (2.57) associated with eigenvalues in D.Then the scheme (2.24) is strongly stable in the sense of Definition 4 if and

only if for all R ≥ 2, there exists a constant CR > 0 such that for all z ∈ U with|z| ≤ R, the estimate (2.69) holds with B](z),B[(z) as in Corollary 4.

We shall give later on a more practical version of Theorems 6 and 7, where thefulfillment of the estimates (2.68) or (2.69) will be replaced by a purely algebraiccondition (see Proposition 6 below). However, this new formulation will rely on thecontinuous extension of the stable subspace Es(z) to S1, which is still not known.Let us now give a few details on the strategy of the proof.

The proof of Theorems 6 and 7 relies on the construction of symmetrizers forthe equivalent forms (2.51) or (2.59) of the resolvent equation (2.29). A symmetrizeris a matrix S(z) such that when one multiplies (2.51) or (2.59) by W ∗j+1 S(z) and usesummation by parts (also known as Abel’s transformation), one more or less endsup with the estimate (2.52) or (2.60). A precise definition of symmetrizers is givenbelow (see Definitions 6 and 7). The crucial point is to understand the constructionof the symmetrizer when z ∈ U is close to S1. In particular, a crucial issue in theconstruction is to understand how the stable subspace Es(z), or Es(z), behaves as zapproaches S1. The geometric regularity condition will first enable us to prove thatEs(z) has a limit as z ∈ U tends to a point of S1. We shall then be able to rephrasethe uniform Kreiss-Lopatinskii condition in a more convenient way (Proposition 6)and to construct a symmetrizer which depends smoothly on z.


In order to clarify the proof of Theorem 6, we first devote some paragraphs tothe proof of several results that will be intermediate steps for the whole proof. Eachstep may have its own interest, so we feel that cutting the proof into several “small”pieces is more appropriate. It also clarifies where the assumptions of Theorem 6 areneeded. There are more or less four main steps in the proof of Theorem 6 (the proofof Theorem 7 follows exactly the same strategy):

(1) Reducing the matrix M(z) in (2.49) to a convenient block diagonal form, thatis, showing that M(z) satisfies the so-called discrete block structure conditiondefined below (see Definition 5). The analysis closely follows [Met00] and [Met05,appendix C]. This step is a refined version of the analysis in [GKS72].

(2) Constructing a symmetrizer for each block in the reduction of M(z). This partof the proof requires the analysis of quite many cases, which correspond to thepossible behaviors of eigenvalues for the amplification matrix associated withgeometrically regular operators. This is where the analysis and the examplesof Section 2.2 and Appendix A will be useful (this is actually the main reasonwhy we have given so many examples of numerical schemes in Section 2.2 andAppendix A). This part of the proof is the main novelty compared with [GKS72]since we are able to cover here all the possible cases while only two of them wereallowed in [GKS72]. In particular, the theory developed in [GKS72] could notcover the singular behaviors displayed in Figures 2.6 and 2.8.

(3) Showing that the existence of a symmetrizer implies that the stable subspaceextends continuously to z ∈ S1, and thus reformulating the uniform Kreiss-Lopatinskii condition. This part of the proof is inspired from [Met04].

(4) Proving energy estimates for the equivalent formulation (2.51) of the resolventequation. This part of the proof already appeared in [GKS72] and there is nomodification here.

In what follows, we shall deal with the three first steps of the proof as if theywere independent problems. The main reason for doing so is to clarify which as-sumptions are needed for each part of the analysis in view of a future extension tomultidimensional problems. To avoid repeating many arguments, we shall only givethe proof of Theorem 6 and leave the proof of Theorem 7 to the interested reader.Most of the arguments are the same, in particular the reduction to the discreteblock structure and the construction of symmetrizers. Minor modifications need tobe done in the final derivation of the a priori estimate and we hope that the readerwill be thrilled to find these subtelties hy himself/herself.

2.4 Characterization of strong stability: proof of themain results

2.4.1 The discrete block structure condition

The aim of this paragraph is to understand to which extent the resolvent equation(2.51), resp. (2.59), can be “diagonalized”. The goal is more or less to reduce to aset of scalar equations but this is unfortunately not always possible as we shall seebelow. We begin with the following


Definition 5 (Discrete block structure condition). Let M be a holomorphicfunction defined on some open neighborhood of U with values in Mm(C) for someinteger m. Then M is said to satisfy the discrete block structure condition if the twofollowing conditions are satisfied:

1. for all z ∈ U , sp(M(z)) ∩ S1 = ∅,2. for all z ∈ U , there exists an open neighborhood O of z in C, and there exists

an invertible matrix T (z) that is holomorphic with respect to z ∈ O such that

∀ z ∈ O , T (z)−1 M(z)T (z) = diag (M1(z), . . . ,ML(z)) ,

where the number L of diagonal blocks and the size ν` of each block M` do notdepend on z ∈ O, and where each block satisfies one of the following properties:• there exists δ > 0 such that for all z ∈ O, M`(z)

∗M`(z) ≥ (1 + δ) I,• there exists δ > 0 such that for all z ∈ O, M`(z)

∗M`(z) ≤ (1− δ) I,• ν` = 1, z and M`(z) belong to S1, and zM ′`(z)M`(z) ∈ R \ 0,• ν` > 1, z ∈ S1 and M`(z) has the form

M`(z) = κ`

1 1 0 0

0. . .

. . . 0...

. . .. . . 1

0 . . . 0 1

, κ` ∈ S1 .

Moreover the lower left coefficient m` of M ′`(z) is such that for all θ ∈ Cwith Re θ > 0, and for all complex number ζ such that ζν` = κ`m` z θ, thenRe ζ 6= 0.

We refer to the blocks M` in the reduction of M as being of the first, second, thirdor fourth type.

The discrete block structure condition is more precise than the normal form of[GKS72, Theorem 9.1]. Definition 2 clarifies the structure of the blocks associatedwith eigenvalues in S1. Such blocks are either scalar, which was not clear in [GKS72],or have a “Jordan structure” (blocks of the fourth type). This clarification willsimplify the construction of symmetrizers in the following paragraph. Our goal hereis to prove the following

Theorem 8 (Characterization of the discrete block structure condition[Cou09]). Let Assumption 1 be satisfied. Then M defined by (2.49) satisfies thediscrete block structure condition if and only if the operators Qσ in (2.25) are ge-ometrically regular and the discretization (2.14) is stable in the sense of Definition2.

Theorem 8 is the analogue for finite difference schemes of Theorem C.3 in[Met05]. The assumptions of Theorem 8 allow more general situations than thecases covered by [GKS72]. In particular, we show that assumptions 5.2 and 5.3 in[GKS72] are not necessary to reduce M to the discrete block structure. Before prov-ing Theorem 8, we recall the basic observation that was already discussed in Section2.2: the geometric regularity of the operators Qσ is not a consequence of the stabilityof (2.14) (except in some very specific situations, see Lemma 7). However, we haveseen that many finite difference schemes used to discretize hyperbolic equations sat-isfy this geometric regularity condition. We therefore believe that Theorem 8 appliesmore or less to all “reasonable” finite difference discretizations of the form (2.24).


Proof (Proof of Theorem 8). • Let us start with the “easy” part of the Theorem. Weassume here that M defined by (2.49) satisfies the discrete block structure condition.Let us first show that the amplification matrix satisfies the von Neumann condition.Let κ ∈ S1 and let z ∈ sp(A (κ)). Let us assume z ∈ U . Recalling the definition(2.16), we obtain (the argument is the same as in the proof of Lemma 15)

0 = det(A (κ)− z I) = (−1)Ns det

[s∑

σ=0

zs−σ Qσ(κ)− zs+1 I

]

= (−1)N(s+1) zN(s+1) det

[p∑

`=−r

κ` A`(z)

]. (2.70)

Since κ is nonzero, the relation (2.66) shows that κ ∈ S1 is an eigenvalue of M(z),and z ∈ U . This is ruled out by the discrete block structure condition (see condition(1) in Definition 5). In other words, the eigenvalues of A (κ) belong to D or S1, sothe von Neumann condition (2.18) is satisfied.

We are now going to prove that the operators Qσ are geometrically regular.Let κ ∈ S1 and let us assume that z ∈ S1 is an eigenvalue of A (κ) with algebraicmultiplicity α. The same argument as above based on relation (2.66) shows thatκ is an eigenvalue of M(z). We apply property (2) of the discrete block structurecondition at the point z: there exists an open neighborhood O of z in C, and thereexists an invertible matrix T (z) that depends holomorphically on z ∈ O such that

∀ z ∈ O , T (z)−1 M(z)T (z) = diag (M1(z), . . . ,ML(z)) , (2.71)

where, for some integer µ ≥ 1, there holds

κ ∈ sp(M`(z))⇐⇒ 1 ≤ ` ≤ µ .

Moreover, the blocks M1, . . . ,ML are of the first, second, third or fourth type. Sincewe have κ ∈ S1, it is not difficult to check that the blocks M1, . . . ,Mµ in (2.71) canonly be of the third or fourth type6. For all (κ, z) sufficiently close to (κ, z), we have

det(M(z)− κ I) = ϑ(κ, z)

µ∏`=1

det(M`(z)− κ I) , ϑ(κ, z) 6= 0 ,

and ϑ is a holomorphic function of (κ, z) near (κ, z). Using the relations (2.70)and (2.66), which are both valid for (κ, z) close to (κ, z), we obtain (for a possiblydifferent function ϑ which is still denoted ϑ)

det(z I −A (κ)) = ϑ(κ, z)

µ∏`=1

det(M`(z)− κ I) , ϑ(κ, z) 6= 0 . (2.72)

We now examine each determinant det(M`(z) − κ I) in (2.72). We recall thatM`, 1 ≤ ` ≤ µ, is either a block of the third or fourth type, and κ is the uniqueeigenvalue of M`(z). If M` is a block of the third type, then we have

6 The eigenvalues of a block of the first type necessarily belong to U , and theeigenvalues of a block of the second type belong to D, see Lemma 17 a littlefurther for a refined statement.


det(M`(z)− κ I) = M`(z)− κ ∈ C ,

and7

∂(M`(z)− κ)

∂z

∣∣(κ,z)

= M′`(z) 6= 0 .

If M` is a block of the fourth type, then we have

M`(z)− κ I = κ

0 1 0 0

0. . .

. . . 0...

. . .. . . 1

0 . . . 0 0

, (2.73)

and therefore (we use the notation of Definition 5 for blocks of the fourth type)

∂ det(M`(z)− κ I)

∂z

∣∣∣(κ,z)

= (−1)ν`−1 κν`−1 m` 6= 0 .

Applying the Weierstrass preparation Theorem, for which we refer to [Hor90], foreach ` = 1, . . . , µ, there exists a holomorphic function β` defined on a suitableneighborhood of κ and that satisfies

∀ ` = 1, . . . , µ , det(M`(z)− κ I) = ϑ(κ, z) (z − β`(κ)) , β`(κ) = z ,

ϑ(κ, z) 6= 0 . (2.74)

Using the latter factorization in (2.72), we obtain

det(z I −A (κ)) = ϑ(κ, z)

µ∏`=1

(z − β`(κ)) , ϑ(κ, z) 6= 0 .

Evaluating at κ = κ, we find that µ equals the multiplicity of z as a root of the char-acteristic polynomial of A (κ), hence µ = α. Going back to Definition 3 of geomet-rically regular operators, we see that it only remains to construct some eigenvectorse`(κ) of A (κ) associated with the eigenvalues β`(κ) and that depend holomorphi-cally on κ.

We now go back to the reduction (2.71). In what follows, Tj(z) denotes the j-thcolumn vector of the matrix T (z). Let ` ∈ 1, . . . , α. If M`(z) is a block of the thirdtype, we define

E`(κ) := Tj`+1(β`(κ)) , j` :=

`−1∑`′=1

ν`′ ,

where we use the same notation as in Definition 5, that is, νk denotes the sizeof the block Mk in (2.71) (this size is independent of z). We also recall that thefunction β` satisfies (2.74). Since Tj`+1(z) is an eigenvector of M(z) associated withthe eigenvalue M`(z), we obtain the relation

M(β`(κ))E`(κ) = κE`(κ) ,

which holds for all κ close to κ, and E`(κ) depends holomorphically on κ. Let us nowconsider the case when M`(z) is a block of the fourth type. Using the factorization

7 Recall from Definition 5 that for a block of the third type, M′`(z) can not be zero.


(2.74), we know that the matrix M`(β`(κ)) − κ I is singular for all κ close to κ.Moreover, the rank of M`(z)−κ I equals ν`−1, see (2.73), so the rank of M`(β`(κ))−κ I is at least ν` − 1 for all κ. Consequently, the kernel of M`(β`(κ)) − κ I is one-dimensional for all κ close to κ, and the last row of M`(β`(κ)) − κ I is a linearcombination of the first ν` − 1 rows. We can then construct a vector e`(κ) ∈ Cν`that depends holomorphically on κ and such that8

e`(κ) =(1 0 . . . 0

), (M`(β`(κ))− κ I) e`(κ) = 0 .

It is now not difficult to construct a vector E`(κ) that depends holomorphically onκ, that satisfies

M(β`(κ))E`(κ) = κE`(κ) , E`(κ) = Tj`+1(z) , j` :=

`−1∑`′=1

ν`′ . (2.75)

Indeed, if we write the vector e`(κ) as (γ1(κ), . . . , γν`(κ)), it is sufficient to define

E`(κ) := γ1(κ)Tj`+1(β`(κ)) + · · ·+ γν`(κ)Tj`+ν`(β`(κ)) .

Eventually, for all ` = 1, . . . , α, we have constructed a vector E`(κ) satisfying (2.75)and that depends holomorphically on κ. Relation (2.75) shows that the E`(κ)’s arelinearly independent eigenvectors of M(z) associated with the eigenvalue κ.

We decompose the vectors E`(κ) as E`(κ) = (E1,`(κ) . . . Ep+r,`(κ)), where eachEk,` belongs to CN . Using (2.75), we find

E`(κ) =(κp+r−1 Ep+r,`(κ) . . . κEp+r,`(κ) Ep+r,`(κ)

),

p∑j=−r

κj Aj(β`(κ))Ep+r,`(κ) = 0 .

In particular, the vectors Ep+r,`(κ), ` = 1, . . . , α, are linearly independent in CN .From the definitions (2.46) and (2.16), we obtain(

β`(κ)s+1 I −s∑

σ=0

β`(κ)s−σ Qσ(κ)

)Ep+r,`(κ) = 0 .

Consequently, the vectors of CN(s+1) defined by

∀ ` = 1, . . . , α , e`(κ) :=(β`(κ)sEp+r,`(κ) . . . β`(κ)Ep+r,`(κ) Ep+r,`(κ)

),

satisfy∀ ` = 1, . . . , α , A (κ) e`(κ) = β`(κ) e`(κ) .

It is straightforward to check that the vectors e`(κ) are linearly independent, so thevectors e`(κ) remain linearly independent for κ close to κ. We have thus proved that

8 To construct e`(κ), it is sufficient to take 1 as its first coordinate, and to determinethe last coordinates by solving the linear system formed by the first ν`−1 rows inthe system (M`(β`(κ)) − κ I) e`(κ) = 0. The last row will be automatically zeroas a linear combination of the other rows.


the operators Qσ are geometrically regular. Proposition 3 shows that the discretiza-tion (2.14) is stable in the sense of Definition 2 (because the von Neumann conditionis satisfied).• From now on, we assume that the operators Qσ are geometrically regular

and that the discretization (2.14) of the Cauchy problem is stable. In particular,Proposition 2 shows that the matrix A (κ) is uniformly power bounded for κ ∈ S1.Our goal is to show that the matrix M(z) defined by (2.49) satisfies the discreteblock structure condition of Definition 5. Since the proof is quite long, we split it inseveral steps.

Step 1. First of all, condition (1) of Definition 5 follows from Lemma 15. Thisproperty immediately implies that the discrete block structure condition is satisfiedin the neighborhood of any z ∈ U . More precisely, let z ∈ U . In a small neigh-borhood O of z, the generalized eigenspace associated with eigenvalues of M(z) inD and the generalized eigenspace associated with eigenvalues of M(z) in U bothdepend holomorphically on z ∈ O (this follows from the Dunford-Taylor formula forprojectors, see the proof of Lemma 6). We can then reduce M(z) to a block diagonalform

T (z)−1 M(z)T (z) = diag (M[(z),M](z)) ,

M[(z) ∈MNr(C) , M](z) ∈MNp(C) ,

where the eigenvalues of M[(z) belong to D and the eigenvalues of M](z) belong toU . The dimension of each block follows from Lemma 15. The invertible matrix T (z)depends holomorphically on z ∈ O. Then we use the following classical result.

Lemma 17. Let M ∈Mm(C). Then the spectrum of M is included in D if and onlyif there exists an invertible matrix P and a positive constant δ such that

(P−1 M P )∗ (P−1 M P ) ≤ (1− δ) I .

Similarly, the spectrum of M is included in U if and only if there exists an invertiblematrix P and a positive constant δ such that

(P−1 M P )∗ (P−1 M P ) ≥ (1 + δ) I .

Proof (Proof of Lemma 17). Let M ∈Mm(C) be such that there exists an invertiblematrix P and a positive constant δ satisfying

(P−1 M P )∗ (P−1 M P ) ≤ (1− δ) I .

Let µ be an eigenvalue of M , and let us consider an eigenvector that we write underthe form P X, with X ∈ Cm, |X| = 1. Then we have P−1 M P X = µX, and

|µ|2 = |(P−1 M P )X|2 ≤ 1− δ < 1 ,

so the spectrum of M is included in D.Let now M ∈ Mm(C) have its spectrum included in D. Let us first choose an

invertible matrix P that reduces M to its Jordan form

P−1 M P =

µ1 θ1 0 0

0. . .

. . . 0...

. . .. . . θm−1

0 . . . 0 µm

,


with µj ∈ D and θj ∈ 0, 1. Introducing Pε := diag (1, ε, . . . , εm−1), ε > 0, we have

P−1ε P−1 M P Pε =

µ1 ε θ1 0 0

0. . .

. . . 0...

. . .. . . ε θm−1

0 . . . 0 µm

.

Since the matrix I − diag (|µ1|2, . . . , |µm|2) is positive definite, the matrix

I − (P−1ε P−1 M P Pε)

∗ (P−1ε P−1 M P Pε)

is positive definite for ε > 0 sufficiently small and the result follows. The analysis inthe case of eigenvalues in U instead of D is similar.

Up to a constant change of basis (which modifies T (z) but keeps the holomor-phy), we can thus achieve the inequalities

M[(z)∗M[(z) ≤ (1− 2 δ) I , M](z)

∗M](z) ≥ (1 + 2 δ) I ,

for some positive constant δ. Thanks to a continuity argument, we can conclude thatthe discrete block structure condition is satisfied in a sufficiently small neighborhoodO of z ∈ U . The reduction only involves one block of the first type and one blockof the second type.

Step 2. We now turn to the case z ∈ S1. If M(z) has no eigenvalue in S1 then weare reduced to the preceeding case. We thus assume that M(z) has some eigenvaluesin S1. More precisely, let κ1, . . . , κk denote the elements of sp(M(z)) ∩ S1, andlet α1, . . . , αk denote the corresponding algebraic multiplicities of these eigenvalues.The generalized eigenspace Ker(M(z)−κj I)αj is denoted Kj . For z sufficiently closeto z, we also let Kj(z) denote the generalized eigenspace of M(z) associated withits αj eigenvalues that are close to κj . The space Kj(z) depends holomorphicallyon z (same argument as in Lemma 6) and satisfies Kj(z) = Kj . Then for z in asmall neighborhood O of z, we can perform a block diagonalization of M(z) with aholomorphic change of basis:

T (z)−1 M(z)T (z) = diag (M[(z),M](z),M1(z), . . . ,Mk(z)) ,

where the eigenvalues of M[(z) belong to D, the eigenvalues of M](z) belong toU , and for all j = 1, . . . , k, the αj eigenvalues of Mj(z) ∈ Mαj (C) belong to asufficiently small neighborhood of κj . As in the preceeding case, we can alwaysachieve the inequalities

∀ z ∈ O , M[(z)∗M[(z) ≤ (1− δ) I , M](z)

∗M](z) ≥ (1 + δ) I ,

for some constant δ > 0, so from now on we focus on the blocks Mj(z). For thesake of clarity, we shall only deal with the first block M1(z). This is only to avoidoverloaded notations with many indeces. Of course, the analysis below is valid forany of the blocks Mj(z). We are going to show that in a convenient holomorphicbasis of K1(z), the block M1(z) reduces to a block diagonal form with blocks of thethird or fourth type. The proof follows the analysis of [Met00, Met05].


Step 3. Following [Met00], we first study the characteristic polynomial of M1(z).For z close to z, the α1 eigenvalues of M1(z) are close to κ1. Combining the relations(2.66) and (2.70), we obtain

det(M1(z)− κ I) = ϑ(κ, z) det (z I −A (κ)) , (2.76)

where ϑ is holomorphic with respect to (κ, z) and does not vanish on a small neigh-borhood of (κ1, z). We know that z ∈ S1 is an eigenvalue of A (κ1) so we canuse the geometric regularity of the operators Qσ. For (κ, z) in a sufficiently smallneighborhood of (κ1, z), (2.76) reads

det(M1(z)− κ I) = ϑ(κ, z)

α∏j=1

(z − βj(κ)

), (2.77)

where α is a fixed integer (not necessarily equal to α1), and the βj ’s are holomorphicfunctions on a neighborhood W of κ1 satisfying βj(κ1) = z. Thanks to the uniformpower boundedness of the matrices A (κ) for κ ∈ S1, we know that |βj(κ)| ≤ 1 forκ ∈ S1 ∩W . Using the Taylor expansion∣∣βj(κ1 ei ξ)

∣∣2 =∣∣z + i κ1 β

′j(κ1) ξ + o(ξ)

∣∣2 = 1 + 2 Re (i z κ1 β′j(κ1)) ξ + o(ξ) ,

for ξ ∈ R close to 0, we obtain that there exists a real number σj such that

κ1 β′j(κ1) = σj z , σj ∈ R . (2.78)

Thanks to (2.77), we can see that κ1 is a root of finite multiplicity of the holo-morphic function z−βj(·). (For otherwise, the function z−βj(κ) would be identicallyzero for all κ close to κ1, and this is ruled out by (2.77).) Consequently there existsan integer νj ≥ 1 such that

∀ ν = 1, . . . , νj − 1 , β(ν)j (κ1) = 0 , β

(νj)

j (κ1) 6= 0 . (2.79)

We can apply the Weierstrass preparation Theorem to the holomorphic functionz − βj(κ). For all j = 1, . . . , α, there exists Pj(κ, z) that is a unitary polynomialfunction in κ with degree νj , such that for (κ, z) close to (κ1, z), there holds

z − βj(κ) = ϑ(κ, z)Pj(κ, z) , Pj(κ, z) = (κ− κ1)νj , ϑ(κ1, z) 6= 0 . (2.80)

Using (2.80), (2.77) reduces to

det(M1(z)− κ I) = ϑ(κ, z)

α∏j=1

Pj(κ, z) .

For z close to z, the polynomial Pj(·, z) has νj roots, and these roots are close to κ1.Consequently, the size of the block M1(z) equals ν1 + · · · + να. We also know thatthe size of this block equals α1, the algebraic multiplicity of κ1 as an eigenvalue ofM(z). Up to reordering the terms, there exists an integer µ (possibly zero) such that

ν1 = · · · = νµ = 1 , νµ+1, . . . , να ≥ 2 .

For j = 1, . . . , µ, we have β′j(κ1) 6= 0, see (2.79), or equivalently σj 6= 0 in (2.78).Therefore βj is a biholomorphic homeomorphism from a neighborhood W of κ1 to


a neighborhood O of z. We let mj denote its (holomorphic) inverse. With suchnotation, we obtain Pj(κ, z) = κ−mj(z) for all j = 1, . . . , µ.

Using the relation (2.80), we also obtain ∂zPj(κ1, z) 6= 0. Then Puiseux’s ex-pansions theory shows that for z close to z and z 6= z, the νj roots of Pj(·, z) aresimple, see for instance [Bau85]. More precisely, Puiseux’s expansions theory showsthat the νj roots of Pj(·, z) behave asymptotically, at the leading order in (z− z) asthe roots of

(κ− κ1)νj + ∂zPj(κ1, z) (z − z) = 0 ,

when z is close to z.Step 4. For each eigenvalue βj(κ), j = 1, . . . , α and κ close to κ1, we know that

A (κ) has a holomorphic eigenvector ej(κ) ∈ CN(s+1). Using the definition (2.16) ofA , we find that ej(κ) reads

∀ j = 1, . . . , α , ej(κ) =

βj(κ)s ej(κ)

...βj(κ) ej(κ)

ej(κ)

, ej(κ) ∈ CN ,

p∑`=−r

κ` A`(βj(κ)) ej(κ) = 0 .

The vectors e1(κ1), . . . , eα(κ1) are linearly independent in CN because the vectorse1(κ1), . . . , eα(κ1) are linearly independent in CN(s+1). Therefore when κ is close toκ1, e1(κ), . . . , eα(κ) remain linearly independent. We define

∀ j = 1, . . . , α , Ej(κ) :=

κp+r−1 ej(κ)

...κ ej(κ)ej(κ)

∈ CN(p+r) .

These vectors depend holomorphically on κ, they are linearly independent in CN(p+r)

for κ close to κ1, and Ej(κ) is an eigenvector of M(βj(κ)) associated with theeigenvalue κ:

∀ j = 1, . . . , α ,(M(βj(κ))− κ I

)Ej(κ) = 0 . (2.81)

In particular, for j = 1, . . . , µ and for z in a neighborhood O of z, we have

∀ j = 1, . . . , µ , ∀ z ∈ O ,(M(z)−mj(z) I

)Ej(mj(z)) = 0 . (2.82)

Let us recall that mj is the holomorphic inverse of βj for j = 1, . . . , µ, that iswhen β′j(κ1) 6= 0. For all j = 1, . . . , µ, we have thus constructed a holomorphiceigenvalue mj(z) and a holomorphic eigenvector Ej(mj(z)) of M(z). Moreover, wehave m′j(z) = 1/β′j(κ1) so we get

∀ j = 1, . . . , µ , mj(z) = κ1 ∈ S1 , z m′j(z)mj(z) =1

σj∈ R \ 0 .

Step 5. We now turn to the most difficult case j = µ+ 1, . . . , α (that is, σj = 0).We start from the relation (2.81), differentiate this relation νj−1 times with respectto κ, and evaluate the result at κ = κ1. This yields

2 Stability of finite difference schemes for boundary value problems 163(M(z)− κ1 I

)Ej(κ1) = 0 ,

− Ej(κ1) +(M(z)− κ1 I

)E′j(κ1) = 0 ,

...

− (νj − 1)E(νj−2)

j (κ1) +(M(z)− κ1 I

)E

(νj−1)

j (κ1) = 0 .

Then for all j = µ+ 1, . . . , α, we define the following vectors:

(Ej,1, . . . , Ej,νj

):=(Ej(κ1),

κ1

1!E′j(κ1), . . . ,

κνj−1

1

(νj − 1)!E

(νj−1)

j (κ1)), (2.83)

that satisfy(M(z)−κ1 I

)Ej,1 = 0 , ∀ ν = 2, . . . , νj ,

(M(z)−κ1 I

)Ej,ν = κ1 Ej,ν−1 . (2.84)

Using the relations (2.82) and (2.84), we can show that the vectors

E1(κ1), . . . , Eµ(κ1), Eµ+1,1, . . . , Eµ+1,νµ+1, . . . , Eα,1, . . . , Eα,να ,

are linearly independent. Moreover, these α1 vectors span the generalized eigenspaceK1 of M(z) associated with the eigenvalue κ1 (they all belong to this space and theyare linearly independent so they form a basis). So far we have thus obtained a basisof K1 in which the block M1(z) reads

M1(z) = diag(κ1, . . . , κ1,Mµ+1, . . . ,Mα

),

M j := κ1

1 1 0 0

0. . .

. . . 0...

. . .. . . 1

0 . . . 0 1

∈Mνj (C) .

In the next step of the analysis, we are going to extend the definition of the vectorsEj,ν to a neighborhood of z.

Step 6. Let us recall that for all j = 1, . . . , α, the polynomial Pj(·, z) is definedby (2.80). We can choose r > 0 such that for z in a neighborhood O of z, the νjroots of Pj(·, z) belong to the disc of center κ1 and radius r/2. Then for all z ∈ O,for all j = µ + 1, . . . , α and for all ν = 1, . . . , νj , we define a vector Ej,ν(z) by theformula

Ej,ν(z) :=κν−1

1 (νj − ν)!

2 i π νj !

∫|κ−κ1|=r

∂νκPj(κ, z)

Pj(κ, z)Ej(κ) dκ .

Cauchy’s formula shows that for z = z, Ej,ν(z) coincides with the vector Ej,ν definedby (2.83). Moreover, Ej,ν(z) depends holomorphically on z ∈ O. In particular wecan choose the neighborhood O such that for all z ∈ O, the vectors

E1(m1(z)), . . . , Eµ(mµ(z)) ,

Eµ+1,1(z), . . . , Eµ+1,νµ+1(z), . . . , Eα,1(z), . . . , Eα,να(z),

are linearly independent. We are now going to show that these vectors span theinvariant subspace K1(z), and that in this basis of K1(z), the matrix M1(z) is in


block diagonal form with blocks of the third and fourth type (the proof will bealmost finished then !).

For z close to z and j = µ + 1, . . . , α, we let Fj(z) denote the vector spacespanned by the linearly independent vectors Ej,1(z), . . . , Ej,νj (z). For j = 1, . . . , µ,we let Fj(z) denote the one-dimensional vector space spanned by Ej(mj(z)). Thenfor all j, the dimension of Fj(z) is νj . Moreover the sum of the Fj(z) is direct and hasdimension α1. We already know that for j = 1, . . . , µ, Ej(mj(z)) is an eigenvectorof M(z) for the eigenvalue mj(z), see (2.82). Consequently, Fj(z) is stable by thematrix M(z) and Fj(z) ⊂ K1(z) for j = 1, . . . , µ. We are now going to show that thesame properties hold true for j = µ+ 1, . . . , α.

For z = z, thanks to (2.84), we know that Fj(z) is stable by M(z) and Fj(z) ⊂ K1.From now on we thus consider a fixed z ∈ O \ z. For all j = µ + 1, . . . , α, we letκj,1, . . . , κj,νj denote the νj disctinct roots of the polynomial Pj(·, z). (We recall thatthese roots are distinct thanks to Puiseux’s expansions theory.) These roots belongto the disc of center κ1 and radius r/2. Therefore, using the residue Theorem, weobtain

Ej,ν(z) =

νj∑m=1

ωj,ν,mEj(κj,m) ,

for some suitable complex numbers ωj,ν,m. Therefore Fj(z) is contained in the vector

space Fj(z) spanned by the vectors Ej(κj,1), . . . , Ej(κj,νj ). Because the dimension of

Fj(z) is νj , we can conlude that the dimension of Fj(z) is also νj and Fj(z) = Fj(z).Let us now show that Fj(z) is stable by M(z). We know that Pj(κj,m, z) = 0 soz = βj(κj,m). Using (2.81) we see that Ej(κj,m) is an eigenvector of M(z) for the

eigenvalue κj,m that is close to κ1. Consequently the vector space Fj(z) is stable

by M(z) and Fj(z) ⊂ K1(z). Since Fj(z) = Fj(z), we have proved that for all j =1, . . . , α, Fj(z) is stable by M(z) and Fj(z) ⊂ K1(z). Using a dimension argument,we have obtained

K1(z) = F1(z)⊕ · · · ⊕ Fα(z) ,

and each Fj(z) is a stable vector space for M(z). Moreover, the characteristic poly-nomial of the restriction of M(z) to Fj(z) is Pj(·, z). We have thus constructed aholomorphic basis of K1(z) in which the matrix M1(z) reads

M1(z) = diag(m1(z), . . . ,mµ(z),Mµ+1(z), . . . ,Mα(z)

).

We also know that the characteristic polynomial of Mj(z) is Pj(·, z) for j = µ +1, . . . , α, and Mj(z) is the Jordan block M j defined above (same expression as inDefinition 5). The size of each block in the reduction of M1(z) is independent of z.

Step 7. The only remaining task is to obtain the property stated in Definition5 for the lower left corner coefficient mj of M ′j(z), j = µ + 1, . . . , α. We know thatPj(κ, z) is the characteristic polynomial of Mj(z), and (2.80) gives ∂zPj(κ1, z) 6= 0.According to the form of Mj(z) = M j , we also have

∂zPj(κ1, z) = det

? −κ1 0... 0

. . .

?...

. . . −κ1

−mj 0 . . . 0

= −κνj−1

1 mj .


Hence mj is not zero. Let θ ∈ C satisfy Re θ > 0. For ε > 0, we define zε :=z (1 + ε θ) ∈ U . The eigenvalues of Mj(zε) are the roots of Pj(·, zε). Accordingto Puiseux’s expansions theory, the eigenvalues κ1(ε), . . . , κνj (ε) of Mj(zε) have anasymptotic expansion of the form

κν(ε) = κ1

(1 + ε1/νj ζν +O(ε2/νj )

), (2.85)

where the complex numbers ζν are such that

0 = Pj(κν(ε), zε) = (κν(ε)− κ1)νj − κνj−1

1 mj (zε − z) + o(ε)

=(κνj1 ζ

νjν − κ

νj−1

1 mj z θ)ε+ o(ε) .

In other words, the ζν ’s are the roots of the equation

ζνj = κ−11 mj z θ ,

and the νj roots of this equation are simple. Our goal is to show that none of theseroots is purely imaginary. Let us argue by contradiction and let us therefore assumethat, say, ζ1 is purely imaginary. We write ζ1 = i ξ1. Then some simple Taylorexpansions (recall (2.85)) yield

κ1(ε)

κ1

− ei ξ1 ε1/νj

= O(ε2/νj ) ,

∀ ν = 2, . . . , νj ,κν(ε)

κ1

− ei ξ1 ε1/νj

= O(ε1/νj ) ,

and we get∣∣∣∣det(Mj(zε)− κ1 ei ξ1 ε

1/νjI)∣∣∣∣ =

νj∏ν=1

∣∣∣∣κν(ε)− κ1 ei ξ1 ε1/νj

∣∣∣∣ = O(ε1+1/νj

). (2.86)

To complete the proof, we need the following

Lemma 18 ([GKS72]). Let Assumption 1 be satisfied, and let us assume that thediscretization (2.14) is stable in the sense of Definition 2. Then there exists a con-stant C > 0 such that for all z ∈ U and for all κ ∈ S1, there holds

|(M(z)− κ I)−1| ≤ C |z||z| − 1

.

Let us assume for the moment that Lemma 18 holds. Then using the blockdiagonalization of M(z) in the neighborhood of z ∈ S1, we find that there exists aconstant C > 0 and a neighborhood O of z such that for all z ∈ O ∩U and for allκ ∈ S1, there holds

|(T (z)−1 M(z)T (z)− κ I)−1| ≤ C

|z| − 1.

In particular, for all ε > 0 sufficiently small, and all κ ∈ S1, there holds (recallzε = z (1 + ε θ) and Re θ > 0)

|(Mj(zε)− κ I)−1| ≤ C

ε.


This inequality is uniform with respect to κ, so we can use it for κ = κ1 ei ξ1 ε1/νj

.Using (2.86), and the classical formula P−1 = Com(P )T / det(P ) for an invertiblematrix P , we obtain that the comatrix of Mj(z) − κ1 I vanishes. However, this isimpossible because the rank of Mj(z)− κ1 I is νj − 1. We have thus obtained thatall the roots ζν have nonzero real part.

Proof (Proof of Lemma 18). We first apply Proposition 2 and the Kreiss matrix The-orem (Theorem 1): since the amplification matrix A (κ) is uniformly power boundedfor κ ∈ S1, there exists a constant C > 0 such that

∀κ ∈ S1 , ∀ z ∈ U , |(A (κ)− z I)−1| ≤ C

|z| − 1. (2.87)

Let z ∈ U , κ ∈ S1, and let Y = (y, 0, . . . , 0) ∈ CN(s+1) with y ∈ CN . We are go-ing to compute the vector (A (κ)−z I)−1 Y . Indeed, let us denote X = (x0, . . . , xs) ∈CN(s+1) the unique solution to the linear system (A (κ)− z I)X = Y . We have

∀σ = 0, . . . , s, xσ = zs−σ xs ,(I −

s∑σ=0

z−σ−1 Qσ(κ)

)xs = −z−s−1 y .

The inequality (2.87) gives (|z| − 1) |X| ≤ C |y| so in particular, we have (|z| −1) |x0| ≤ C |y|. Using the relation x0 = zs xs, we get the estimate

|xs| ≤C |z|−s

|z| − 1|y| , where xs = −z−s−1

(I −

s∑σ=0

z−σ−1 Qσ(κ)

)−1

y .

The latter matrix is invertible for otherwise, A (κ) − z I would have a nontrivialkernel. Taking the supremum over y ∈ CN , we obtain that there exists a constantC > 0 such that

∀κ ∈ S1 , ∀ z ∈ U ,

∣∣∣∣∣(I −

s∑σ=0

z−σ−1 Qσ(κ)

)−1∣∣∣∣∣ ≤ C |z||z| − 1

.

Using the relation (this relation already appeared earlier in the proof of Theorem 8)

I −s∑

σ=0

z−σ−1 Qσ(κ) =

p∑`=−r

κ` A`(z) ,

we have just proved that there exists a constant C > 0 such that

∀κ ∈ S1 , ∀ z ∈ U ,

∣∣∣∣∣(

p∑`=−r

κ` A`(z)

)−1∣∣∣∣∣ ≤ C |z||z| − 1

. (2.88)

We now consider a vector b = (bp, . . . , b1−r) ∈ CN(p+r) and we let X =(xp−1, . . . , x−r) denote the unique solution to the linear system (M(z)− κ I)X = b(Lemma 15 shows that the matrix M(z) − κ I is invertible). From the definition(2.49), we obtain the relations


∀ ` = 1− r, . . . , p− 1, x` = κr+` x−r +

`+r−1∑j=0

κj b`−j ,

κr(

p∑`=−r

κ` A`(z)

)x−r = −b(κ, z) ,

with a vector b(κ, z) defined by

b(κ, z) := Ap(z) bp +

p−1∑`=1−r

A`(z)`+r−1∑j=0

κj b`−j + κAp(z)p+r−2∑j=0

κj bp−1−j .

For z ∈ U and κ ∈ S1, we have a uniform bound

|b(κ, z)| ≤ C0 |b| ,

because the matrices A`(z) are uniformly bounded for z ∈ U , see (2.46). We thenuse the estimate (2.88) to obtain the upper bound

|x−r| ≤ C|z||z| − 1

|b| ,

with a constant C that is uniform with respect to κ ∈ S1 and z ∈ U . The othercomponents x1−r, . . . , xp−1 of x are easily estimated in terms of x−r and b. We havethus proved that there exists a constant C > 0 such that for all z ∈ U and for allκ ∈ S1, we have

|(M(z)− κ I)−1 b| ≤ C |z||z| − 1

|b| .

The proof of Lemma 18 is complete.

Theorem 8 shows that under the assumptions of Theorem 6, the matrix M(z)satisfies the discrete block structure condition. We are now interested in constructinga symmetrizer for M(z). Rather than working on M(z) directly, we shall work onthis partially diagonalized form of M(z) and eventually go back to M(z) by changingbasis.

2.4.2 The construction of symmetrizers

The following terminology was borrowed from [Met04] and adapted to the contextof finite difference schemes in [Cou09].

Definition 6 (K-symmetrizer). Let z ∈ U , and let M be a function defined onsome neighborhood O of z with values in Mm(C) for some integer m. Then M issaid to admit a K-symmetrizer at z if there exists a decomposition

Cm = Es ⊕ Eu ,

with associated projectors (πs, πu), such that for all K ≥ 1, there exists a neighbor-hood OK of z, there exists a C∞ function SK on OK with values in Hm, and thereexists a constant cK > 0 such that the following properties hold for all z ∈ OK ∩U :

• M(z)∗ SK(z)M(z)− SK(z) ≥ cK (|z| − 1)/|z| I,


• for all W ∈ Cm, W ∗ SK(z)W ≥ K2 |πuW |2 − |πsW |2.

If M is a function defined on a neighborhood O of U with values in Mm(C)for some integer m, then M is said to admit a K-symmetrizer if it admits a K-symmetrizer at all points of U .

We recall that in Definition 6, Hm denotes the set of Hermitian matrices of sizem.

A few remarks should be made. In the decomposition as a direct sum of Cm,Es should be thought of as the stable subpsace of M(z), meaning the generalizedeigenspace associated with eigenvalues in D, and Eu should be thought of as the un-stable subpsace of M(z), meaning the generalized eigenspace associated with eigen-values in U , see Lemma 19 below. The main difficulty arises when there are alsoeigenvalues on S1 so that one needs to determine whether such neutral eigenvaluesshould be counted as stable or unstable.

The goal of the symmetrizer is basically to make the matrix M∗ SKM − SKpositive definite by putting a large positive weight K2 on the unstable componentsand the negative weight −1 on the stable components. As explained below, this israther easy when stable and unstable eigenvalues decouple. This decoupling occurseither when M(z) has no eigenvalue on S1 or more generally when there is no “sin-gular” crossing of stable and unstable eigenvalues on S1. The construction of thesymmetrizer becomes much more involved when M(z) has at least one eigenvalueon S1 that corresponds to such a crossing, because then one needs a precise descrip-tion of how the spectrum of M(z) behaves when z is close to z. For the stabilityanalysis of finite difference schemes, the reduction of M to the discrete block struc-ture (Theorem 8) was precisely performed in order to give the information requiredfor this construction.

Before stating the main result of this paragraph, which is Theorem 9 below, letus give a rather elementary result which explains some necessary properties for theexistence of a K-symmetrizer.

Lemma 19. Let z ∈ U , and let M be a function defined on some neighborhoodO of z with values in Mm(C) for some integer m. If M admits a K-symmetrizerat z, then M(z) has no eigenvalue on S1. Furthermore, the vector space Es in thedecomposition of Cm contains the generalized eigenspace associated with eigenvaluesof M(z) in D.

Lemma 19 shows that in the “interior” case z ∈ U there is more or less no choicefor Es in the decomposition of Cm. For dimension reasons, the vector space Es willbe chosen to be exactly the generalized eigenspace associated with eigenvalues in D(stable eigenvalues). There is more freedom in the choice of Eu but the most naturalchoice will be the generalized eigenspace associated with eigenvalues in U (unstableeigenvalues). The limit case z ∈ S1 will be analyzed by a continuity argument.

Proof (Proof of Lemma 19). Under the assumption of the Lemma, we know (ap-ply Definition 6 with K = 1) that there exists a Hermitian matrix S such thatM(z)∗ SM(z)− S is positive definite. Here we have used the assumption |z| > 1. IfX is an eigenvector for M(z) associated with an eigenvalue κ ∈ S1, we have

X∗ (M(z)∗ SM(z)− S)X = (|κ|2 − 1)X∗ S X = 0 .


Since M(z)∗ SM(z)−S is positive definite, this implies X = 0. Hence M(z) has noeigenvalue on S1.

Let us now consider a vector W in the generalized eigenspace of M(z) associatedwith eigenvalues in D. We then define a sequence (Wj) ∈ `2 by the iterative formula

W1 := W , Wj+1 = M(z)Wj , j ≥ 1 .

For K ≥ 1, the point z belongs to the set OK on which the mapping SK is defined.For all j ≥ 1, there holds

W ∗j(M(z)∗ SK(z)M(z)

)Wj =

(M(z)Wj

)∗SK(z)M(z)Wj

= W ∗j+1 SK(z)Wj+1 .

We thus get the following relations for all integer J ≥ 1:

0 =

J∑j=1

W ∗j(M(z)∗ SK(z)M(z)

)Wj −W ∗j+1 SK(z)Wj+1

=W ∗1 SK(z)W1 −W ∗J+1 SK(z)WJ+1

+

J∑j=1

W ∗j

(M(z)∗ SK(z)M(z)− SK(z)

)Wj .

Observing that the matrix M(z)∗ SK(z)M(z)− SK(z) is positive definite and thatWJ+1 tends to 0 as J tends to infinity, we can pass to the limit with respect to Jand obtain

W ∗1 SK(z)W1 ≤ 0 .

We now use the second property of the symmetrizer SK , see Definition 6, and wehave thus obtained

|πuW | ≤ 1

K|πsW | .

Since the latter inequality holds for all K ≥ 1, and the vector W as well as theprojectors are independent of K, we can pass to the limit and obtain W ∈ Es. Theproof of Lemma 19 is complete.

Our main result in this paragraph reads as follows. This result was partlyachieved in [Cou09] and completed in [Cou11a].

Theorem 9 (Existence of a K-symmetrizer [Cou09, Cou11a]). Let Assump-tion 1 be satisfied, and let M defined by (2.49) satisfy the discrete block structureassumption. Then M admits a K-symmetrizer and at each point z ∈ U , the dimen-sion of the vector space Es in the decomposition of CN(p+r) equals N r.

We emphasize that at this stage, no assumption on the numerical boundaryconditions has been made. More precisely, Theorem 8 characterizes the block struc-ture condition by means of some properties of the operators Qσ used in the dis-cretization of the hyperbolic operator. According to Theorem 9, the existence ofa K-symmetrizer is completely independent of the numerical boundary conditionsused in (2.24). It is a property of the discretized hyperbolic operator only. In thefollowing paragraphs, we shall see how the result of Theorem 9 can be used to obtainthe existence of a Kreiss symmetrizer (the terminology is introduced below). As in


[Met04], the Kreiss symmetrizer is the main tool in showing strong stability for thenumerical scheme (2.24). It will be obtained by using the result of Theorem 9 with alarge enough parameter K, provided that the uniform Kreiss-Lopatinskii conditionholds (see the following paragraphs for more details). We shall explain later on whythese two different kinds of symmetrizers are useful.

Proof (Proof of Theorem 9). We start the proof of Theorem 9 by showing two ratherelementary results, the proof of which relies on some manipulations of Definition 6.

Lemma 20. Let z ∈ U , and let M1, resp. M2, be a function defined on some neigh-borhood O of z with values in Mm1(C), resp. Mm2(C), for some integer m1, resp.m2. Assume that both M1 and M2 admit a K-symmetrizer at z with correspondingvector spaces Es1,E

s2 of dimension µ1, µ2.

Then the block diagonal matrix diag(M1,M2) ∈ Mm1+m2(C) admits a K-symmetrizer at z with a vector space Es of dimension µ1 + µ2.

Proof (Proof of Lemma 20). For all vector W ∈ Cm1+m2 , we let W1 ∈ Cm1 denotethe vector formed by the m1 first coordinates of W and W2 ∈ Cm2 the vector formedby the m2 last coordinates of W . Then we set

Es := W ∈ Cm1+m2 / (W1,W2) ∈ Es1 × Es2 ,

Eu := W ∈ Cm1+m2 / (W1,W2) ∈ Eu1 × Eu2 .

It is straightforward to check that Es and Eu are complementary vector spaces inCm1+m2 and that Es has dimension µ1 + µ2. The projectors πs, πu satisfy

∀W ∈ Cm1+m2 , πsW =

(πs1 W1

πs2 W2

), πuW =

(πu1 W1

πu2 W2

).

Let K ≥ 1, and let OK denote a neighborhood of z on which both mappingsSK,1, SK,2 respectively symmetrizing M1, M2, are defined. For z ∈ OK , we defineSK(z) := diag(SK,1(z), SK,2(z)) ∈Hm1+m2 , and it is now a simple exercise to checkthat SK satisfies all the properties required for a symmetrizer. The proof of Lemma20 is therefore complete.

Lemma 21. Let z ∈ U , and let M be a function defined on some neighborhood O ofz with values in Mm(C) for some integer m. Assume that there exists a C∞ functionT defined on O with values in Glm(C) such that T−1 M T admits a K-symmetrizer

at z with a vector space Es

of dimension µ.Then M admits a K-symmetrizer at z with a vector space Es of dimension µ.

Proof (Proof of Lemma 21). The proof is slightly more subtle than the proof ofLemma 20 but remains quite simple. First of all, since T is smooth, there is no lossof generality (up to restricting O) in assuming that there exists a constant c > 0such that for all z ∈ O, there holds

∀W ∈ Cm , c |W | ≤ |T (z)−1 W | ≤ 1

c|W | . (2.89)

We define the complementary vectors spaces

Es := T (z) Es, Eu := T (z) E

u,


where Es, E

uare the complementary vector spaces given by the existence of a K-

symmetrizer for T−1 M T .Let now K ≥ 1. We fix K ≥ 1 such that

1

2c4 K2 ≥ K2 +

1

2. (2.90)

For such a K, that only depends on K, there exist a neighborhood OK of z, aconstant cK > 0 and a C∞ mapping SK defined on OK with values in Hm suchthat

∀ z ∈ OK ∩U , (T−1 M T )(z)∗ SK(z) (T−1 M T )(z)− SK(z) ≥ cK|z| − 1

|z| I ,

∀W ∈ Cm , W ∗SK(z)W ≥ K2 |πuW |2 − |πsW |2 .

For z ∈ OK , we define

SK(z) :=c2

2(T−1(z))∗ SK(z)T−1(z) ,

and we are going to show that SK symmetrizes M . Let W ∈ Cm be decomposed asW = W s + Wu according to the decomposition Cm = Es ⊕ Eu. Then T−1(z)W s

and T−1(z)Wu are the components of the vector T−1(z)W according to the de-

composition Cm = Es⊕ E

u. Consequently, we have

W ∗ SK(z)W =c2

2(T−1(z)W )∗ SK(z)T−1(z)W

≥ c2

2K2 |πu T−1(z)W |2 − c2

2|πs T−1(z)W |2

=c2

2K2 |T−1(z)Wu|2 − c2

2|T−1(z)W s|2 .

Using the estimate (2.89), we end up with

W ∗ SK(z)W ≥ c4

2K2 |Wu|2 − 1

2|W s|2 ≥

(K2 +

1

2

)|Wu|2 − 1

2|W s|2 ,

where in the end we have used the inequality (2.90). By continuity, up to restrictingthe neighborhood OK , there holds

W ∗ SK(z)W ≥ K2 |Wu|2 − |W s|2 ,

for all z ∈ OK , and therefore for all z ∈ OK ∩ U . Let us now check the secondproperty for SK . If z ∈ OK ∩U , we have

M(z)∗ SK(z)M(z)− SK(z)

=c2

2

(M(z)∗ (T−1(z))∗ SK(z)T−1(z)M(z)− (T−1(z))∗ SK(z)T−1(z)

)=c2

2T−1(z)∗

((T−1 M T )(z)∗ SK(z) (T−1 M T )(z)− SK(z)

)T−1(z)

≥ c2 cK2

|z| − 1

|z| T−1(z)∗ T−1(z) ≥ c4 cK2

|z| − 1

|z| I ,

where we have used (2.89) again. The proof of Lemma 21 is thus complete.


We now turn to the proof of Theorem 9. First of all, Lemma 21, combined withLemma 20, shows that it is sufficient to construct a K-symmetrizer for each blockof the first, second, third or fourth type arising in the discrete block structure, seeDefinition 5. If we wish the corresponding vector space Es to have dimension N r,it is sufficient to show that for each block M`, the corresponding vector space Esàrising in the K-symmetrizer decomposition has a dimension equal to the numberof stable eigenvalues of the block. More precisely, let us consider a block M`(z)defined in the neighborhood of z ∈ U and occurring in the discrete block structureof M(z). There is no restriction in assuming that M` is defined on the open diskB(z, r) centered at z and of radius r. In particular, the set B(z, r)∩U is connected.On B(z, r) ∩U , M`(z) has no eigenvalue in S1 so there is no ambiguity in definingan integer µ` equal to the number of eigenvalues of M`(z) in D when z belongs toB(z, r)∩U (this number is independent of z). The number µ` is called the numberof stable eigenvalues of the block M`, and is made explicit below for each type ofblock. Lemma 15 shows that the sum of the µ`’s equals N r.• Blocks of the first type. Let z ∈ U , and let us consider a block M`(z) of size m`

defined on a neighborhood O of z and satisfying M`(z)∗M`(z) ≥ (1 + δ) I for some

constant δ > 0 that is independent of z. Lemma 17 shows that all eigenvalues ofM`(z) belong to U so the number of stable eigenvalues of such a block equals zero.Let K ≥ 1, and let us define Es` := 0, Eu` := Cm` . (Observe that the dimensionof Es` equals the number of stable eigenvalues of the block.) We also define thesymmetrizer SK as SK(z) := K2 I independently of z. With these definitions, therelation

W ∗ SK(z)W = K2 |W |2 = K2 |πu` W |2 − |πs`W |

2 , (2.91)

is obvious. Moreover, there holds

M`(z)∗ SK(z)M`(z)− SK(z) = K2 (M`(z)

∗M`(z)− I)≥ K2 δ I ≥ K2 δ

|z| − 1

|z| I .

We have thus shown the existence of a K-symmetrizer at z for a block M` of thefirst type.• Blocks of the second type. Let z ∈ U , and let us consider a block M`(z) of size

m` defined on a neighborhood O of z and satisfying M`(z)∗M`(z) ≤ (1 − δ) I for

some δ > 0 that is independent of z. Lemma 17 shows again that all eigenvalues ofM`(z) belong to D so the number of stable eigenvalues of such a block equals m`. LetK ≥ 1, and let us define Es` := Cm` , Eu` := 0. We also define the symmetrizer SKas SK(z) := −I independently of z, and the reader can easily adapt the argumentdeveloped for blocks of the first type to show that SK satisfies all the propertiesrequired for a symmetrizer. We observe again that the dimension of Es` equals thenumber of stable eigenvalues of the block.• Blocks of the third type (part I). We recall from Definition 5 that blocks

of the third type are scalar and can only occur for z ∈ S1. We thus consider aholomorphic function M` defined on a neighborhood O of z ∈ S1 and satisfyingM`(z) ∈ S1, zM′`(z)M`(z) > 0. (According to Definition 5, zM′`(z)M`(z) is anonzero real number so we first consider the case where this number is positive.)Let us first show that there is no stable eigenvalue in that case. For ε > 0 smallenough, (1 + ε) z belongs to O ∩U and Taylor’s expansion reads


M`((1 + ε) z)

M`(z)= 1 + zM′`(z)M`(z) ε+O(ε2) .

In particular, the modulus of M`((1+ε) z) is larger than 1 for ε > 0 small enough andthere is no stable eigenvalue for such a scalar block. Unsurprisingly, we thus defineEs` := 0, Eu` := C, and SK(z) := K2 independently of z. This symmetrizer triviallysatisfies the property (2.91). Following the analysis performed above for blocks ofthe first type, the result relies on a lower bound of the quantity |M`(z)|2 − 1 forz ∈ O ∩ U . This lower bound is derived in the following Lemma which we stateseparately for the sake of clarity.

Lemma 22. Let f be a holomorphic function defined on a disk B(1, r) centered at1 and of radius r > 0, verifying f(1) = 1, Re f ′(1) > 0, and

∀ z ∈ B(1, r) ∩ S1 , |f(z)| ≥ 1 .

Then there exists a constant c > 0 such that, up to diminishing r, there holds

∀ z ∈ B(1, r) ∩U , |f(z)|2 − 1 ≥ c (|z| − 1) .

Proof (Proof of Lemma 22). For τ in a sufficiently small neighborhood of 0, wedefine:

h(τ) := ln f(eτ ) ,

where ln denotes the standard complex logarithm defined on C\R−. We have h′(0) =f ′(1), and h(τ) has nonnegative real part when τ is purely imaginary. Using thenotation τ = x+ i y, a direct Taylor expansion yields

Re h(τ) = Re h(i y) + Re (h(τ)− h(i y)) ≥ Re (h(τ)− h(i y))

= Re (h′(i y)x) + o(x)

= (Re f ′(1))x+ o(x) ,

where the last equality holds for sufficiently small r (and the smallness conditiononly depends on f). We have thus shown the estimate

Re h(τ) ≥ Re f ′(1)

2Re τ ,

for all τ of nonnegative real part close to 0. The estimate for |f(z)|2 for z ∈ B(1, r)∩U easily follows:

|f(z)|2 − 1 = (|f(z)|+ 1) (|f(z)| − 1) = (|f(z)|+ 1)(eRe h(ln z) − 1

)≥ Re f ′(1)

2Re ln z .

Remark 9. The assumption |f(z)| ≥ 1 for all z ∈ B(1, r)∩S1 is absolutely necessaryin Lemma 22, and it is no consequence of the assumption Re f ′(1) > 0. The readermay for instance consider the example

f(z) := 1 + (z − 1) +

(1

2+ i

)(z − 1)2 ,


which satisfies f(1) = 1, f ′(1) = 1. However, if one considers the points zα := 1+i α,with α > 0 small enough, there holds |f(zα)|2 − 1 < 0 and zα ∈ U . This prevents ffrom verifying the conclusion of Lemma 22.

More generally, the property |f(z)| ≥ 1 for all z ∈ B(1, r) ∩ S1 can not followfrom any information on a finite number of derivatives of f at 1. In general, thisproperty can only follow from the full series expansion of f at 1.

We can apply Lemma 22 to the function w 7→ M`(z w)/M`(z). Indeed, we knowthat M`(z) belongs to U for all z ∈ O ∩U . By continuity, this implies M`(z) ∈ Ufor all z ∈ O ∩U . We therefore obtain the estimate

M`(z)∗ SK(z)M`(z)− SK(z) = K2 (|M`(z)|2 − 1

)≥ cK2 (|z| − 1) ≥ cK2 |z| − 1

|z| ,

for all z ∈ O ∩ U sufficiently close to z. We have proved that SK satisfies all theproperties of a symmetrizer, and the dimension of Es` coincides with the number ofstable eigenvalues of the block.• Blocks of the third type (part II). We now turn to the case z ∈ S1, M`(z) ∈ S1,

zM′`(z)M`(z) < 0. Unsurprisingly, the reader will easily verify that there is onestable eigenvalue and that the symmetrizer SK can be chosen as SK(z) := −1independently of z. The argument relies on the following analogue of Lemma 22,which we feel free to use without proof.

Lemma 23. Let f be a holomorphic function defined on a disk B(1, r) centered at1 and of radius r > 0, verifying f(1) = 1, Re f ′(1) < 0, and

∀ z ∈ B(1, r) ∩ S1 , |f(z)| ≤ 1 .

Then there exists a constant c > 0 such that, up to diminishing r, there holds

∀ z ∈ B(1, r) ∩U , |f(z)|2 − 1 ≤ −c (|z| − 1) .

• Blocks of the fourth type. This is by far the most difficult case. A complete anal-ysis of the construction of the symmetrizer is performed in [Cou11a]. The analysisis unfortunately very long, and involves a generalization of the original constructionperformed in [Kre70]. In order to keep the length of these notes reasonable, we shallnot detail the construction of the symmetrizer for blocks of the fourth type andwe shall rather refer to [Cou11a, Theorem 3.4]. In particular, the dimension of thecorresponding vector space Es` equals the number of stable eigenvalues of the block.This number can be explicitly determined from the size ν` of the block and the lowerleft coefficient m` of M′`(z), see [Cou11a, Proposition 4.1] for a precise statement.

We just emphasize for the interested reader the new main difficulty comparedwith [Kre70]. In the analysis of [Kre70], which is devoted to boundary value prob-lems for hyperbolic systems of partial differential equations, the construction of thesymmetrizer relies on the fact9 that for z ∈ S1 close to z, all eigenvalues of the blockbelong to S1. This is a very strong property which implies that some coefficients in

9 We slightly adapt the result of [Kre70] to our framework but there is no difficultyto pass from one to the other thanks to the exponential function.


the matrices are either real or purely imaginary. In our framework, there is a lotmore freedom because we only know that for z = z, M`(z) has one eigenvalue on S1.When z varies on S1 close to z, the eigenvalues of M`(z) usually do not stay on S1.This phenomenon can be checked by hand on the following elementary example10:

z = κ = 1 , M(z) :=

(1 1

z − 1 1

).

Other examples of this behavior occur for discretizations of the hyperbolic operatorwhose amplification matrix displays some eigenvalues curves with singular pointson S1. Examples of such discretizations are given in Appendix A. As a matter offact, when singular points in S1 occur for eigenvalues of the amplification matrixA (κ), this gives rise in the reduction of M to blocks of the fourth type, see theproof of Theorem 8. Unless the behavior of the eigenvalues corresponds to that ofthe leap-frog scheme, see Figure 2.1, the eigenvalues of the block in the reduction ofM can have a much more complex behavior than just remaining on S1 for z ∈ S1.This led us in [Cou11a] to introducing an integer which we called the dissipationindex and that gave a description of the singularity for the eigenvalue curve for A .The construction of the symmetrizer for a block of the fourth type depends bothon the size of the block and of the dissipation index (there are approximately tencases to deal with). Even though we shall not reproduce the complete analysis here,we strongly encourage the reader to go through [Cou11a] since we believe that thisnew construction is basically the first step towards a full treatment of the analogousmultidimensional problem. This extension is postponed to a future work.

The K-symmetrizer construction performed in this paragraph will be crucial forthe proof of Theorems 6 and 7. However, before giving the proof of Theorem 6, weneed one last technical - though crucial - point about the behavior of the stablesubspace Es(z) when z ∈ U tends to a point of S1.

2.4.3 Extending the stable subspace

The main result of this paragraph is the following.

Theorem 10 (Continuous extension of the stable subspace [Cou09]). LetAssumption 1 be satisfied, and let us assume that the discretization of the Cauchyproblem (2.14) is stable in the sense of Definition 2. Let us also assume that thematrix M defined by (2.49) admits a K-symmetrizer where, at each point z ∈ U ,the dimension of the vector space Es in the decomposition of CN(p+r) equals N r.

Then the stable subspace Es(z) of M(z), which is well-defined for z ∈ U accordingto Lemma 15, defines a holomorphic vector bundle over U that can be extended ina unique way as a continuous vector bundle over U .

In all what follows, we shall let Es(z) denote the continuous extension of thestable subspace for z ∈ S1(= ∂U ). In general, for z ∈ S1, the matrix M(z) may haveeigenvalues on S1, so the number of eigenvalues in D can be less than N r. As wasalready pointed out in the proof of Theorem 9, the difficulty consists in determining

10 On this example, the reader can check that the eigenvalues of M(ei ε), ε > 0small, do not belong to S1.


whether eigenvalues on S1 should count as stable or unstable eigenvalues, and thisis determined by a perturbation argument, that is by slightly moving z towards theopen set U and by studying whether the eigenvalues move towards D or towardsU . The cases of the Lax-Friedrichs and leap-frog schemes are detailed below.

Proof (Proof of Theorem 10). Lemma 15 shows that the stable subspace Es(z) ofM(z) has constant dimension N r for all z ∈ U . The holomorphic dependence ofM(z) on z implies that Es(z) also varies holomorphically with z on U . (Here weuse the same arguments as in the proof of Lemma 6 and Theorem 8: the spectralprojector on Es(z) is given by the Dunford-Taylor formula, which shows that theprojector depends holomorphically on z. We can then construct a basis of Es(z)that depends holomorphically on z in the neighborhood of any point of U . In otherwords, Es defines a holomorphic vector bundle over U .)

Let z ∈ S1 and let us first show that Es(z) has a limit as z ∈ U tends to z.We consider the decomposition CN(p+r) = Es ⊕ Eu given by the existence of a K-symmetrizer at z. From the assumption of Theorem 10, we know that the dimensionof Es equals N r. Let now K > 2, and let us consider a neighborhood OK of z anda symmetrizer SK defined on OK and satisfying the properties given in definition 6.Let z ∈ OK ∩U and let W ∈ Es(z). We define the sequence:

W1 := W , Wj+1 = M(z)Wj , j ≥ 1 .

Using the exact same method as in the proof of Lemma 19, we end up with theinequality W ∗1 SK(z)W1 ≤ 0, which in turn yields:

∀ z ∈ OK ∩U , ∀W ∈ Es(z) , K |πuW | ≤ |πsW | .

The rest of the analysis follows [Met04]. Writing πsW = W −πuW , we get (usethe triangle inequality)

∀ z ∈ OK ∩U , ∀W ∈ Es(z) , (K − 1) |πuW | ≤ |W | . (2.92)

The estimate (2.92) shows that the mapping

Φ(z) : Es(z) −→ Es

W 7−→ πsW ,

which is defined for z ∈ OK ∩ U , is injective. (If W belongs to the kernel of Φ(z),then W belongs to Es(z) ∩ Eu and (2.92) gives (K − 1) |W | ≤ |W | so W is zerobecause K is larger than 2.) Since the dimensions of Es(z) and Es are the same,Φ(z) is an isomorphism. We can write the inverse mapping Φ(z)−1 in the followingway

Φ(z)−1 : Es −→ Es(z)W 7−→W + ϕ(z)W ,

where ϕ(z) is a linear mapping from Es to Eu. This may look suprising but we onlydecompose the vector Φ(z)−1 W along the direct sum Es ⊕ Eu and we observe thatthe component on Es equals W itself (use the definition of Φ(z)). Using (2.92) onceagain, we obtain

∀ z ∈ OK ∩U , ∀W ∈ Es , |ϕ(z)W | ≤ 1

K − 2|W | . (2.93)


Indeed, (2.92) shows that for all W ∈ Es, there holds

(K − 1) |ϕ(z)W | = (K − 1) |πu (W + ϕ(z)W )|≤ |W + ϕ(z)W | ≤ |W |+ |ϕ(z)W | ,

and (2.93) follows (use K > 2).We now have all the ingredients in order to show that Es(z) tends to Es as z ∈ U

tends to z. We consider a basis (e1, . . . , eNr) of Es and we fix ε > 0. Let us chooseK > 2 such that |ej |/(K − 2) ≤ ε for all j = 1, . . . , N r. The above analysis showsthat the estimate (2.93) holds for all z ∈ OK ∩U . In particular, we have

∀ z ∈ OK ∩U , ∀ j = 1, . . . , N r , |ej − Φ(z)−1 ej | ≤ ε .

We know that Φ(z)−1 is an isomorphism so (Φ(z)−1 e1, . . . , Φ(z)−1 eNr) is a basisof Es(z). We have thus proved that for z ∈ U sufficiently close to z, there existsa basis of Es(z) whose elements are ε-close to the elements of a basis of Es. Inother words, we have shown that Es(z) tends to Es as z ∈ U tends to z. Thismeans that the vector bundle Es can be extended to U , and it remains to showthat this extended bundle is continuous over U . This is not straightforward becausecontinuity at z ∈ S1 now requires to consider the limit of Es(z) when z ∈ U tendsto z, while before we have only studied the limit of Es(z) when z ∈ U tends to z.

Let us observe that the above argument shows that for z ∈ S1, the vector spaceEs of dimension N r in the decomposition of CN(p+r) is necessarily unique since itis the limit of Es((1 + ε) z) as ε > 0 tends to 0.

Let us now prove that the bundle Es, which has been extended to ∂U , is con-tinuous over U . It is obviously continuous over U since it is holomorphic, and wethus only check the continuity of Es at any point of S1. We follow [Met04] againand perform more or less the same analysis as above. We use the convention intro-duced above and let Es(z) denote the continuous extension of the stable subspacefor z ∈ S1(= ∂U ). Let z ∈ S1, and let K > 2. With the above argument, we alreadyhave the estimate (2.92). Furthermore, there is no loss of generality in assuming thatthe neighborhood OK of z is an open disk B(z, rK), rK > 0.

Let us consider a point z′ ∈ OK ∩ S1. Since OK is an open neighborhood of z′,there exists a sequence (zn) in OK ∩ U that converges towards z′. In particular,the above analysis shows that Es(zn) converges towards Es(z′). This means thatany element W ′ ∈ Es(z′) can be written as the limit - in CN(p+r) - of a sequence(Wn) where for each integer n, Wn belongs to Es(zn). Applying (2.92) and passingto the limit as n tends to infinity, we get the inequality (K − 1) |πuW ′| ≤ |W ′| forall W ′ ∈ Es(z′). In other words, we have obtained

∀ z ∈ OK ∩U , ∀W ∈ Es(z) , (K − 1) |πuW | ≤ |W | . (2.94)

(Observe the slight, though important, difference between (2.92) and (2.94).) At thispoint, the exact same argument as above shows that Es(z) tends to Es(z) as z ∈ Utends to z. The only difference is that we are now allowed to consider some z ∈ OKthat belong to S1 and use (2.94) while before we were only allowed to consider somez ∈ OK that belonged to U and use (2.92). Eventually, we have proved that Es iscontinuous at any point of S1.

Remark 10. Here we have followed the approach of [Met04], which gives an “analyt-ical” and somehow simple proof of the continuous extension of the stable bundle.


As observed in [Met04], the nice point is that constructing a symmetrizer of somekind seems to be necessary to deal with the derivation of a priori estimates for so-lutions to the resolvent equation. In the original approach by Kreiss [Kre70], seealso the books [BG07, Cha82], the first step for deriving a priori estimates consistedin showing through mostly “algebraic” arguments that the stable subspace couldbe continuously extended, and the second step consisted in constructing a suitable”Kreiss symmetrizer”. The alternative approach introduced in [Met04] bypasses thealgebraic part of the proof and focuses on the K-symmetrizer construction. Thecontinuous extension of the stable bundle appears as a corollary of the existence ofa K-symmetrizer (which itself relies on the block structure). Furthermore, the orig-inal construction of a Kreiss symmetrizer (see Definition 7 below) also appears as acorollary of the existence of a K-symmetrizer and of the fulfillment of the UKLC.

From our point of view, this alternative approach clarifies one of the main tech-nical and difficult points of the theory. The main remaining difficulties are the (i)reduction of the symbol M to the discrete block structure and (ii) the construc-tion of the K-symmetrizer. This technical simplification gives us hope to deal withmultidimensional problems in a near future.

2.4.4 Proof of Theorem 6

We first give a new formulation of the Uniform Kreiss-Lopatinskii Condition in theframework of Theorem 6.

Proposition 6 (Reformulation of the UKLC). Under the assumptions of The-orem 6, the UKLC holds if and only if

∀ z ∈ U , Ker B(z) ∩ Es(z) = 0 ,

where Es(z) denotes the generalized eigenspace of M(z) associated with eigenvaluesin D, which is defined in Lemma 15 for z ∈ U and is continuously extended toz ∈ S1.

We observe again that the UKLC is compatible with the dimensions of thevector spaces: Es(z) has dimension N r, while B(z) ∈ MNr,N(p+r)(C) has maximalrank (see the expression (2.50)) so its kernel has dimension N p. Hence there is noobstruction for Ker B(z) and Es(z) to be complementary in CN(p+r).

Proof (Proof of Proposition 6). Let us first verify that the stable subspace Es can becontinuously extended to the boundary S1 of U . Applying first Theorem 8, we knowthat the matrix M defined by (2.49) satisfies the discrete block structure condition.We can then apply Theorem 9: M admits a K-symmetrizer where, at each point ofU , the dimension of the vector space Es in the decomposition of CN(p+r) equalsN r. Eventually Theorem 10 shows that the stable subspace extends continuouslyto S1, and the extended bundle is continuous over U .• We now prove the result of Proposition 6. We first assume that the UKLC is

satisfied, meaning that for all R ≥ 2, there exists a constant CR > 0 such that forall z ∈ U with |z| ≤ R, the estimate (2.68) holds with the matrix B(z) defined in(2.50). We let C2 denote the corresponding constant for R = 2. It is already clearthat Es(z) does not intersect the kernel of B(z) for z ∈ U (this is the Godunov-Ryabenkii condition). We thus consider z0 ∈ S1. The space Es(z0) is the limit, as


ε > 0 tends to 0, of Es((1 + ε) z0). Any vector W ∈ Es(z0) can thus be written asthe limit, as ε > 0 tends to 0, of a sequence of vectors Wε ∈ Es((1 + ε) z0). Passingto the limit in the inequality

∀ ε ∈ ]0, 1] , |Wε| ≤ C2 |B((1 + ε) z0) Wε| ,

we obtain the inequality |W | ≤ C2 |B(z0) W | for all W ∈ Es(z0). This propertyimplies that Es(z) does not intersect the kernel of B(z) for all z ∈ S1.• We now assume that Es(z) does not intersect the kernel of B(z) for all z ∈ U

and we are going to show that the UKLC holds. Let R ≥ 2. For z ∈ U with |z| ≤ R,we consider the quantity

m(z) := infW ∈Es(z),|W |=1

∣∣B(z) W∣∣ .

The quantity m(z) is positive for all z, and m depends continuously on z becauseboth the vector space Es(z) and the matrix B(z) depend continuously on z. Since theannulus z ∈ C , 1 ≤ |z| ≤ R is compact, m is bounded from below by a positiveconstant cR > 0 on this annulus. In other words, we have shown the inequality

∀W ∈ Es(z) , |W | ≤ 1

cR|B(z) W | ,

as long as 1 ≤ |z| ≤ R. Consequently the UKLC is satisfied.

We introduce the following terminology.

Definition 7 (Kreiss symmetrizer). Let M be defined by (2.49), and let B bedefined by (2.50). The pair (M,B) is said to admit a Kreiss symmetrizer if for allR ≥ 2, there exists a constant cR > 0 and there exists a C∞ function S on theannulus z ∈ C , 1 ≤ |z| ≤ R with values in HN(p+r) such that the followingproperties hold for all z in the annulus:

• M(z)∗ S(z)M(z)− S(z) ≥ cR (|z| − 1)/|z| I,• for all W ∈ CN(p+r), W ∗ S(z) W ≥ cR |W |2 − c−1

R |B(z) W |2.

Remark 11. In contrast with a K-symmetrizer, a Kreiss symmetrizer depends bothon the discretized hyperbolic operator (through the matrix M), and on the numericalboundary conditions (through the matrix B(z)). However, the existence of a Kreisssymmetrizer does not seem sufficient for showing that the stable subspace extendscontinuously to U , and this is the reason why we have first proved the existence ofa K-symmetrizer from which everything follows (as long as the UKLC is satisfied).The reader can compare with the approach in [BG07, Cha82].

We can now prove a refined version of Theorem 6.

Theorem 11 (Existence of a Kreiss symmetrizer and strong stability). LetAssumption 1 be satisfied, let us assume q < p and let us further assume that thediscretization of the Cauchy problem (2.14) is stable in the sense of Definition 2 andthat the operators Qσ are geometrically regular in the sense of Definition 3.

If the UKLC holds, then the pair (M,B) admits a Kreiss symmetrizer and thescheme (2.24) is strongly stable.


The assumptions of Theorem 11 are exactly the same as the assumptions ofTheorem 6. It should be rather clear at this point that Theorem 11 yields the resultof Theorem 6. Indeed, Theorem 11 shows that the UKLC is a sufficient conditionfor strong stability (it even shows that the UKLC is a sufficient condition for theexistence of a Kreiss symmetrizer). In the meantime, Corollary 3 shows that theUKLC is a necessary condition for strong stability. We thus focus on the proof ofTheorem 11.

Proof (Proof of Theorem 11). • We first show that under the assumptions of Theo-rem 11, the pair (M,B) admits a Kreiss symmetrizer. Following the same argumentsas in the proof of Proposition 6, we already know that M admits a K-symmetrizerwhere, at each point of U , the dimension of the vector space Es in the decompo-sition of CN(p+r) equals N r. Lemma 19 and Lemma 15 show that at each pointz ∈ U , the vector space Es in the decomposition of CN(p+r) coincides with Es(z).Furthermore, the proof of Theorem 9 shows that this property holds true also onthe boundary S1 of U . Summarizing, M admits a K-symmetrizer in the sense ofDefinition 6 where, at each point z ∈ U , the vector space Es in the decompositionof CN(p+r) equals Es(z).

Let R ≥ 2, and let z ∈ U with |z| ≤ R. We are going to show that the pair(M,B) admits a Kreiss symmetrizer in the neighborhood of z. More precisely, sincethe UKLC holds, Proposition 6 shows that there exists a constant c > 0 such that

∀W ∈ Es(z) , c |W | ≤ |B(z) W | .

We fix a parameter K ≥ 1 by choosing K2 := 1 + 4 |B(z)|2/c2. Applying Theorem9, we know that M admits a K-symmetrizer at z so there exists a neighborhood Oof z, a constant c > 0, and a C∞ function S on O with values in HN(p+r) such that

for all z ∈ O ∩U , there holds

M(z)∗ S(z)M(z)− S(z) ≥ c (|z| − 1)/|z| I , (2.95)

and∀W ∈ CN(p+r) , W ∗ S(z) W ≥ K2 |πu W |2 − |πs W |2 .

In particular, we have

W ∗ S(z) W ≥ |πs W |2 +K2 |πu W |2 − 2 | πs W︸︷︷︸∈Es(z)

|2

≥ |πs W |2 +K2 |πu W |2 − 2

c2∣∣B(z) (W − πu W )

∣∣2≥ |πs W |2 +K2 |πu W |2 − 4

c2

(|B(z) W |2 + |B(z)πu W |2

).

With our choice of the parameter K, we get

W ∗ S(z) W ≥ |πs W |2 + |πu W |2 − 4

c2|B(z) W |2 ≥ 1

2|W |2 − 4

c2|B(z) W |2 .

In particular, the matrix S(z) + 4 c−2 B(z)∗ B(z) − I/4 is positive definite so, by acontinuity argument, for all z sufficiently close to z, there holds

∀W ∈ CN(p+r) , W ∗ S(z) W ≥ c |W |2 − 1

c|B(z) W |2 , (2.96)


with a suitable constant c > 0 that is independent of z. To summarize, we haveproved that for all z in the annulus z ∈ C , 1 ≤ |z| ≤ R, there exists a neighbor-hood O of z, there exists a constant c > 0, and there exists a C∞ function S on Owith values in HN(p+r) such that (2.95) and (2.96) hold for all z ∈ O∩U . (Actually,

the reader may observe that (2.96) holds not only for z ∈ O ∩U but for all z ∈ O,but this will not play any role in what follows.)

We now make the construction of the Kreiss symmetrizer “global” by a com-pactness argument. The annulus z ∈ C , 1 ≤ |z| ≤ R is covered by a finite numberO1, . . . ,OJ of such neighborhoods. We consider a partition of unity χ1, . . . , χJ thatis subordinated to this covering. In other words, χj is a nonnegative C∞ functionwith support in Oj for every j, and there holds

∀ z ∈ U , |z| ≤ R ,J∑j=1

χj(z) = 1 .

We define

∀ z ∈ U , |z| ≤ R , S(z) :=

J∑j=1

χj(z)Sj(z) ∈HN(p+r) .

If cj denotes the constant associated with the neighborhood Oj and if c > 0 denotesthe minimum of the cj ’s, then it is not so difficult to check the property

∀ z ∈ U , |z| ≤ R , M(z)∗ S(z)M(z)− S(z) ≥ c (|z| − 1)/|z| I ,

(just multiply (2.95) on Oj by χj(z) and sum with respect to j), as well as

∀ z ∈ U , |z| ≤ R , ∀W ∈ CN(p+r) , W ∗ S(z) W ≥ c |W |2 − 1

c|B(z) W |2 .

In other words, the pair (M,B) admits a Kreiss symmetrizer.•We now show that the existence of a Kreiss symmetrizer is a sufficient condition

for strong stability. Let R ≥ 2, and let us consider a Kreiss symmetrizer S on theannulus z ∈ C , 1 ≤ |z| ≤ R. We consider a point z in this annulus and a sequence(Wj) ∈ `2. The source terms (Fj), G are defined such that (2.51) holds. The apriori estimate of (Wj) follows from computations that are rather similar to whatwe have already done. More precisely, we multiply the induction relation in (2.51)by (S(z) Wj+1)∗ and use the fact that S(z) is hermitian to obtain

J∑j=1

Re W ∗j+1 S(z)M(z) Wj −J+1∑j=2

W ∗j S(z) Wj +

J∑j=1

Re W ∗j+1 S(z) Fj = 0 .

Using the induction relation again and substituting the expression of Wj+1, we get

W ∗1 S(z) W1 −W ∗J+1 S(z) WJ+1 +

J∑j=1

W ∗j(M(z)∗ S(z)M(z)− S(z)

)Wj

= −Re

J∑j=1

(Wj+1 + M(z) Wj)∗ S(z) Fj .


We let J tend to +∞ and use the properties of the Kreiss symmetrizer, which yields

cR|z| − 1

|z|∑j≥1

|Wj |2 + cR |W1|2 −1

cR|G |2

≤ −Re

J∑j=1

(Wj+1 + M(z) Wj)∗ S(z) Fj .

Using some uniform bounds for S(z) and M(z) on the annulus and the Cauchy-Schwarz inequality, we end up with

|z| − 1

|z|∑j≥1

|Wj |2 + |W1|2 ≤ CR

|z||z| − 1

∑j≥1

|Fj |2 + |G |2 ,

with a constant CR > 0 that does not depend on z ∈ U , |z| ≤ R.It remains to show that the resolvent equation (2.51) admits a unique solution

in `2 for all source terms (up to now we have only proved an a priori estimatefor the solution). This final part of the proof follows from applying Lemma 11 andLemma 12 again. More precisely, Lemma 11 shows that the resolvent equation (2.29)is uniquely solvable for |z| large enough. There is no difficulty to show that theequivalent formulation (2.51) is also uniquely solvable for |z| large enough. Then wecan apply Lemma 12 on every annulus z ∈ C , 1 + 2−ν ≤ |z| ≤ 2ν, ν ∈ N largeenough. Eventually, Proposition 4 shows that the scheme (2.24) is strongly stable.

2.4.5 Some examples: the Lax-Friedrichs and leap-frog schemes

The aim of this paragraph is to show how the theory developed in the proof ofTheorem 7 applies in the case of some elementary numerical schemes. We shall testvarious discretized boundary conditions and compute the associated Lopatinskiideterminants. For simplicity, we restrict in this paragraph to the case of a singlescalar transport equation

∂tu+ a ∂xu = F (t, x) , (t, x) ∈ R+ × R+ , u|t=0 = 0 . (2.97)

For a < 0, there is no boundary condition to prescribe on x = 0, while for a > 0the transport equation (2.97) should be supplemented with a Dirichlet boundarycondition on x = 0.

The Lax-Friedrichs scheme

The Lax-Friedrichs discretization of the transport equation is given by (2.20) (hereN = 1 and A = a is a real number). We have seen in Section 2.2 that this schemeis stable in the sense of Definition 1 if and only if λ |a| ≤ 1, and the correspondingoperator QLF is geometrically regular. From the general definition (2.46), we obtain

A−1(z) = −1 + λa

2 z, A0(z) = 1 , A1(z) = −1− λa

2 z.

Consequently, Assumption 1 holds if and only if λ |a| < 1, which we assume fromnow on. It is not so surprising that the limit case λ |a| = 1 is excluded by the


theory because in that case the Lax-Friedrichs scheme “degenerates” and becomesthe upwind scheme which does not involve the same number of grid points (eitherp or r is zero if λ |a| = 1, while p = r = 1 when λ |a| < 1).

The matrix M(z) in (2.49) reads11

M(z) =

2 z

1− λa −1 + λa

1− λa1 0

,

and we are going to check in an easy and direct way that M satisfies the discreteblock structure condition. The eigenvalues of M(z) are the roots to the polynomialequation

κ2 − 2 z

1− λa κ+1 + λa

1− λa = 0 .

In particular, the matrix M(2) has two real eigenvalues: one belongs to the interval]0, 1[ and the other one belongs to ]1,+∞[. Moreover, M(z) has an eigenvalue onS1 if and only if z belongs to the curve cos η − i λ a sin η , η ∈ R. Since λ |a| < 1,the latter curve is included in the closed unit disk and its contact points with S1

are ±1 (η ∈ Zπ). Applying a continuity/connectedness argument, we are led to thefollowing conclusion: for all z ∈ U \ ±1, the matrix M(z) has a unique eigenvalueκs(z) in D and a unique eigenvalue in U . The eigenvalue κs depends holomorphicallyon z near any point of U \ ±1, and M is holomorphically diagonalizable near anypoint of U \ ±1.

For z ∈ U \ ±1, the stable subspace Es(z) of M(z) has dimension 1 - this iscompatible with Lemma 15 because N = r = 1 in this example - and is given by

∀ z ∈ U \ ±1 , Es(z) = Span

(κs(z)

1

).

In particular, the continuous extension of Es to S1 proved in Theorem 6 is trivialhere (it is even a holomorphic extension !), except possibly at the points ±1 whichwe examine right now. From the expression of Es, we see that Es(z) will have a limitat ±1 if we can prove that the eigenvalue κs has a limit at ±1.

The eigenvalues of M(1) are 1 and (1 + λa)/(1 − λa). In the case a < 0, thereholds (1+λa)/(1−λa) ∈ ]0, 1[, so this is another trivial case of continuous extensionof the stable eigenvalue and we have κs(1) = (1 + λa)/(1− λa). In the case a > 0,there holds (1 + λa)/(1 − λa) ∈ U so the only possible extension of κs at thepoint 1 is 1. For z close to 1, M(z) has a unique eigenvalue close to 1 that dependsholomorphically on z. If we consider the points zε := 1+ε ∈ U , ε > 0 small enough,the expansion of the eigenvalue of M(zε) close to 1 reads

1− 1

λaε+ o(ε) ,

so this eigenvalue belongs to D for ε > 0 small enough. By uniqueness of the stableeigenvalue, we can conlude that κs extends holomorphically to a whole neighborhoodof 1 and κs(1) = 1 ∈ S1 when a > 0. The situation at z = −1 is examined in exactlythe same way and we obtain the following conclusion: κs admits a holomorphic

11 Observe that in this special case, M is a holomorphic function on C and not onlyon a neighborhood of U .


extension to a whole neighborhood of −1, and κs(−1) = −(1 + λa)/(1− λa) ∈ D ifa < 0, κs(−1) = −1 if a > 0.

The discrete block structure condition is very easy to verify because of the spec-tral splitting satisfied by the matrix M: M has two distinct eigenvalues at everypoint of U and is therefore diagonalizable (with a holomorphic change of basis) inthe neighborhood of any point of U . The reduction near any point of U \ ±1involves one (scalar) block of the first type and one (scalar) block of the secondtype. If a < 0, the reduction near ±1 involves one (scalar) block of the second typeand one (scalar) block of the third type. If a > 0, the reduction near ±1 involvesone (scalar) block of the first type and one (scalar) block of the third type.

Let us now verify whether the UKLC is satisfied for various types of discretizedboundary conditions. If we consider an incoming transport equation (a > 0), it seemsnatural to prescribe the Dirichlet boundary condition for the numerical scheme. Inthe outgoing case, there is no obvious choice since the continuous problem does notprescribe anything. We slightly anticipate and first choose to prescribe Dirichletboundary condition. This might look surprising but one needs to do something !(The reader is referred to [GKO95, chapter 13] and [God96, chapter V] and toreferences therein for more accurate choices of numerical boundary conditions.) Forthe numerical Dirichlet boundary condition, the approximation to (2.97) reads

Un+1j =

Unj−1 + Unj+1

2− λa

2(Unj+1 − Unj−1) +∆tFnj , j ≥ 1 , n ≥ 0 ,

Un+10 = gn+1 , n ≥ 0 ,

U0j = 0 , j ≥ 0 .

In this case, one has q = 0, B0,0 = B0,−1 = 0 and the matrix B(z), whose abstractdefinition is (2.50), reads

∀ z ∈ C \ 0 , B(z) =(0 1).

It is easily checked that the UKLC is satisfied, whatever the sign of a (even inthe outgoing case !). Indeed, the intersection of Es(z) with Ker B(z) is non-trivialprovided that the Lopatinskii determinant

∆(z) := B(z)

(κs(z)

1

),

vanishes. Here this determinant equals 1 for all z ∈ U so the UKLC is satisfied. Froma practical point of view, it is interesting to test the Dirichlet boundary conditionfor an outgoing transport equation since it may be the most surprising case. Letus therefore consider the transport equation (2.97) with a < 0 and F = 0. In thatcase, the solution to (2.97) is identically 0. To approximate this solution, we use thenumerical scheme

Un+1j =

Unj−1 + Unj+1

2− λa

2(Unj+1 − Unj−1) , j ≥ 1 , n ≥ 0 ,

Un+10 = gn+1 , n ≥ 0 ,

U0j = 0 , j ≥ 0 ,

with a nonzero source terme (gn) on the boundary. The numerical computations arerun with a = −1, λ = 0.9, and gn = 1 for all n ≥ 1. The result of the computation


is shown in Figure 2.2 at time t = 1/2 and time t = 1. The space interval is [0, 1]and the number of grid points is 100. By finite speed of propagation, we know thatboth the exact solution and the numerical solution vanish at the right end of thecomputation interval, so we impose a homogeneous Dirichlet condition at 1. Thisis relevant provided that the computations are run up to a certain number of timesteps (up to time 1 at least). The observed numerical solution is very small, whichis predicted by Theorem 6 and our verification of the UKLC. This means that theperturbation introduced at the boundary is not propagated into the interior by thenumerical scheme. Similarly, prescribing homogeneous Dirichlet boundary conditionwill almost not reflect any outgoing wave if one prescribes a nonzero initial condition.

Fig. 2.2. The Lax-Friedrichs scheme for an outgoing transport equation with anon-homogeneous boundary condition gn = 1 at time t = 1/2 (left) and time t = 1(right). The numerical scheme should approximate the solution zero. The solutionis represented on a log scale, and the space interval is [0, 1].

We now turn to the Neumann boundary condition. The corresponding numericalscheme reads

Un+1j =

Unj−1 + Unj+1

2− λa

2(Unj+1 − Unj−1) +∆tFnj , j ≥ 1 , n ≥ 0 ,

Un+10 = Un+1

1 + gn+1 , n ≥ 0 ,

U0j = 0 , j ≥ 0 .

For this scheme, we still have q = 0, and in the notation of (2.24), B0,0 = 0,B0,−1 = T0. The corresponding matrix B(z) reads

B(z) =(−1 1

),

so the Lopatinskii determinant reads

∆(z) = 1− κs(z) .

If a < 0, we have seen that κs(z) belongs to D for all z ∈ U . In particular, κs(z) 6= 1and the UKLC holds. When one wishes to discretize the outgoing transport equa-tion (2.97), for which no boundary condition is required, one can therefore use thestronlgy stable (and consistent !) scheme

186 Jean-Francois CoulombelUn+1j =

Unj−1 + Unj+1

2− λa

2(Unj+1 − Unj−1) +∆tF (n∆t, j ∆x) , j, n ≥ 0,

Un+10 = Un+1

1 , n ≥ 0 ,

U0j = 0 , j ≥ 0 .

(2.98)To observe the strong stability of the latter numerical scheme, one can use the sametest as the one reported in Figure 2.2 for the Dirichlet boundary condition (thatis, no source term in the interior and a constant source term equal to 1 on theboundary). The numerical results are entirely similar with either the Dirichlet orthe Neumann boundary condition.

If a > 0, we know that κs(z) belongs to D for all z ∈ U \ ±1 so ∆ does notvanish on this set. In particular, the Godunov-Ryabenkii condition is satisfied. Sinceκs(±1) = ±1, we also find that ∆ vanishes at 1 and does not vanish at −1. In theincoming case a > 0, the Neumann boundary condition does not satisfy the UKLCand the corresponding numerical scheme is not strongly stable. What can we observeand conclude in such a situation ? We report on a very simple numerical test whichshows that the violation of strong stability is a serious obstacle for convergence of thenumerical solution. We consider the incoming transport equation (2.97) with a = 1and F = 0. We impose the homogeneous Dirichlet boundary condition u(t, 0) = 0so the exact solution to the transport equation is identically 0. Since u(t, 0) = 0,we have ∂tu|x=0 = 0 and (2.97) gives ∂xu|x=0 = 0. At the continuous level, thismay suggest using a homogeneous Neumann condition at the boundary rather thanthe homogeneous Dirichlet boundary condition. At the discrete level, we considerthe numerical scheme (2.98). When the source term (Fnj ) vanishes, the numericalsolution is 0 and it reproduces the exact solution. We perturb this situation bychoosing F 0

1 = 1/∆t and all other Fnj vanish12. The solution is represented in Figure2.3 at time t = 1/2 and time t = 1, where we have chosen λ = 0.9 again. The numberof grid points is 1000 and the space interval is [0, 1]. The numerical solution is somekind of traveling wave propagating to the right and connecting a state U > 0 to0. In particular, the exact boundary condition is not approximated at all by thenumerical scheme (even though the homogeneous discrete Neumann condition wasimposed!). Though we have not pushed any rigorous investigation further than a fewnumerical tests, we believe that this specific choice of source term may be a goodcandidate for showing rigorously that the energy estimate (2.26) is not satisfied.

The leap-frog scheme

We consider the leap-frog approximation (2.22) for the transport equation (2.97).We still restrict to the scalar case N = 1, A = a ∈ R. For this scheme, there holdsp = r = 1, and the definition (2.46) reads

12 This perturbation is a classical test for stability. First, it is easy to use since itis localized on a single mesh of the grid, and even though its L∞-norm is large,the L2

t,x-norm of this perturbation is of order 1, independently of ∆t. The secondreason why it is useful is that because of space localization, its Fourier transformtriggers more or less all frequencies (like a Dirac mass) so if one frequency isamplified by the scheme, there is a reasonable chance to observe this phenomenonwith this perturbation.


Fig. 2.3. The Lax-Friedrichs scheme for an incoming transport equation with ahomogeneous Neumann boundary condition at time t = 1/2 (left) and time t = 1(right). The interior source term vanishes except F 0

1 which we choose equal to 1/∆t.

A−1(z) = −λaz, A0(z) = 1− 1

z2, A1(z) =

λa

z.

Assumption 1 is thus satisfied as long as a 6= 0. (When a = 0, the scheme degeneratesand involves only one point.) We have seen in Section 2.2 that both stability in thesense of Definition 2 and geometric regularity hold as long as λ |a| < 1. We thusassume 0 < λ |a| < 1 from now on.

The matrix M(z) in (2.49) reads

M(z) =

1− z2

λa z1

1 0

,

so the eigenvalues of M(z) are the roots to the polynomial equation

κ2 +z2 − 1

λa zκ− 1 = 0 .

The matrix M(2) has two real eigenvalues: one of them belongs to the interval] −∞,−1[ and the second one belongs to ]0, 1[. Moreover, M(z) has no eigenvalueon S1 when z belongs to U so we can conclude, as in Lemma 15, that M(z) has aunique eigenvalue κs(z) ∈ D and a unique eigenvalue in U for all z ∈ U . Of course,κs depends holomorphically on z ∈ U . The stable subspace Es(z) has dimension 1and is given by

∀ z ∈ U , Es(z) = Span

(κs(z)

1

).

This is exactly the same expression as for the Lax-Friedrichs scheme, which is notsurprising because M is still a companion matrix13. Our goal is now to study thecontinuous extension of the stable eigenvalue κs to the boundary S1 of U andto verify that M satisfies the discrete block structure condition. The situation isslightly more complicated but in some sense much more interesting than for theLax-Friedrichs scheme.

13 The reader will find in the following paragraph an extension of this remark wherethe structure of M will be fully used. This will help us proving the so-calledGoldberg-Tadmor Lemma.


Computing the discriminant of the characteristic polynomial of M(z), we firstobserve that M has a double eigenvalue if and only if z is one of the points±(√

1− λ2 a2 + i λ a) or their conjugates. These four points are located on S1, andunsurprisingly they correspond exactly to the singular points of the eigenvaluescurves for the leap-frog scheme (see the right picture in Figure 2.1). We can alreadyconclude that M can be holomorphically diagonalized in the neighborhood of anypoints z ∈ S1 which is not one of these four points and that the stable eigenvalue κsadmits a holomorphic extension to the neighborhood of any such “non-exceptional”point. The continuous - and even holomorphic extension - of the stable subspace isclear in this case. Let us now focus on the points where M has a double eigenvalue.We consider for instance the point z :=

√1− λ2 a2 + i λ a (the three other cases are

entirely similar). There holds

M(z) =

(−2 i 1

1 0

),

so M(z) is similar to a Jordan block with the eigenvalue −i. More precisely, if weintroduce the invertible matrix

T :=

(1 1i 0

),

we have

T−1 M(z)T = −i(

1 10 1

).

In view of Definition 5, the constant matrix T is a good candidate for reducing Mto the discrete block structure condition. Let us check this property in full details.We compute

T−1 M(z)T =

−i −i1− z2

λa z+ 2 i

1− z2

λa z+ i

. (2.99)

In order to check that the discrete block structure condition holds, we only need tocompute the derivative at z = z of the lower left coefficient of the matrix on theright-hand side of (2.99). We obtain

∂

∂z

(1− z2

λa z+ 2 i

) ∣∣∣z=z

= −2√

1− λ2 a2

λa z.

Let now θ ∈ C with Re θ > 0. We consider the roots ζ to the equation

ζ2 = (−i) −2√

1− λ2 a2

λa zz θ = −2 i

√1− λ2 a2

λaθ .

The roots ζ cannot be purely imaginary, for otherwise i θ would be a real number.According to Definition 5, the derivative of the lower left coefficient in (2.99) satisfiesthe property required in the definition of the discrete block structure condition. Thisreduction involves a single 2× 2 block of the fourth type. We have even shown thatthe change of basis can be chosen to be independent of z in the neighborhood ofz (this is highly non-generic, and it is due to the fact that we consider a scalarequation). The continuous extension of κs, and therefore of Es, to z follows fromthe continuity of the roots of the characteristic polynomial of M(z).


Let us now check whether the UKLC is satisfied for various types of numericalboundary conditions. As before, we first consider the Dirichlet boundary conditions.In other words, we consider the numerical scheme

Un+1j = Un−1

j − λa (Unj+1 − Unj−1) +∆tFnj , j ≥ 1 , n ≥ 1 ,

Un+10 = gn+1 , n ≥ 1 ,

U1j = U0

j = 0 , j ≥ 0 .

The matrix B(z) defined in (2.50) reads

B(z) =(0 1),

and the associated Lopatinskii determinant equals 1 for all z ∈ U . This shows, asfor the Lax-Friedrichs scheme, that the Dirichlet boundary condition satisfies theUKLC for the leap-frog scheme. We emphasize that this result is independent of thesign of a. Numerical tests as the one reported in Figure 2.2 can be performed andgive rather good results in the outgoing case (meaning that the numerical solutionremains close to the exact solution even in the case of non-homogeneous Dirichletboundary conditions).

The reader can also check that the leap-frog scheme combined with the Neumanncondition at the boundary always satisfies the Godunov-Ryabenkii condition, butalways violates the UKLC. Again, this result is independent of the sign of a. Ifone performs the same kind of test as the one reported in Figure 2.3, the numericalsolution has similar features, meaning that it looks like a traveling wave propagatingto the right and connecting some state U > 0 to 0.

We now study another type of discrete boundary condition which is obtained byusing backward integration along the characteristics. More precisely, for a < 0, thetransport equation (2.97) is outgoing. On the boundary mesh of index j = 0, weapply the so-called upwind scheme, which amounts to considering the scheme

Un+1j = Un−1

j − λa (Unj+1 − Unj−1) +∆tFnj , j ≥ 1 , n ≥ 1 ,

Un+10 = Un0 − λa (Un1 − Un0 ) + gn+1 , n ≥ 1 ,

U1j = U0

j = 0 , j ≥ 0 .

(2.100)

This numerical procedure seems to be a somehow reasonable discretization for a < 0since we use a stable approximation of the Cauchy problem in the interior domainand a rather precise approximation of the solution at the boundary. It seems muchless reasonable in the case a > 0 for in that case, the upwind discretization “onthe right” is known to be unstable for the Cauchy problem (one should use thediscretization “on the left”). We are going to examine the strong stability of (2.100)according to the sign of a.

The careful reader may have observed that the discrete boundary conditionin (2.100) involves not only Un1 but also Un0 , which does not exactly fall into theframework of (2.24). However, we could have equally considered boundary operatorsBj,σ in (2.24) of the form

Bj,−1 =

q∑`=0

B`,j,−1 T` , j = 1− r, . . . , 0 ,

Bj,σ =

q∑`=−r−j

B`,j,σ T` , j = 1− r, . . . , 0 , σ = 0, . . . , s .


For such boundary operators, the reader can verify that the values Un+1j , j =

1 − r, . . . , 0, are obtained as linear combinations of some Un−sj , . . . , Unj , which are

already known from the previous iteration steps, and of some Un+1j , j ≥ 1, which

are also known because they are obtained from the “interior” discretization. Hencethe numerical scheme is explicit and well-defined. There is a slight difference in thedefinition of the matrix B(z) in (2.50), and we leave as an exercise to the reader togo through the derivation of the resolvent equation (2.51) in the case of (2.100). Theassociated matrix B(z) is

B(z) =

(λa

z1− 1 + λa

z

),

and with the above parametrization of the stable subspace, the Lopatinskii deter-minant reads

∆(z) =λa

zκs(z) + 1− 1 + λa

z.

Our goal is therefore to determine whether there exists some z ∈ U such that

z − 1 = λa (1− κs(z)) , (2.101)

knowing that κs(z) satisfies the relation

κs(z)(z2 − 1

)= λa z

(1− κs(z)2) . (2.102)

If z ∈ U \ 1, the only possibility for ∆(z) to vanish is to have κs(z) 6= 1 and wecan then divide both left and right hand side terms in (2.102) by the correspondingexpression in (2.101). We then obtain κs(z) = z. In other words, for z 6= 1, theonly possibility for ∆(z) to vanish is to have κs(z) = z but then (2.101) givesλa = −1. This is obviously in contradiction with our stability assumption for thediscrete Cauchy problem. Hence ∆ can only vanish at the point 1. In particular, theGodunov-Ryabenkii condition holds for (2.100) whatever the sign of a. Moreover,the above expression of ∆ yields ∆(1) = λa (κs(1) − 1) so ∆ vanishes at 1 if andonly if κs(1) equals 1. The eigenvalues of M(1) are ±1 so it is not clear at first sightwhether κs(1) equals 1 or −1. Considering the sequence of points zε := 1 + ε withε > 0 going to 0, we can compute the asymptotic expansions of both eigenvalues ofM(zε). We then obtain κs(1) = 1 if a > 0 and κs(1) = −1 if a < 0. Consequently,we find ∆(1) = 0 if a > 0 and ∆(1) 6= 0 if a < 0. The numerical scheme (2.100)satisfies the UKLC and is strongly stable if a < 0, while it is not strongly stable ifa > 0. We can go a little further. In the previous paragraph, when we have shownthat the Lax-Friedrichs scheme with the Neumann condition on the boundary is notstrongly stable, we have shown that 1 is a root of the Lopatinskii determinant. Inthat case, the reader can check that ∆ extends holomorphically to a neighborhoodof 1 and that 1 is a simple root of ∆. The situation is a little more singular for(2.100) when a > 0: the Lopatinskii determinant ∆ also extends holomorphicallyto a neighborhood of 1, but here 1 is at least a double root of ∆. Indeed, we candifferentiate ∆ with respect to z and obtain

∆′(1) = λaκ′s(1) + 1 + λa (κs(1)− 1) = λaκ′s(1) + 1 .

In the meantime, we can differentiate (2.102) with respect to z, use κs(1) = 1 (herewe use a > 0), and get κ′s(1) = −1/(λa). In other words, ∆′(1) vanishes and 1 is atleast a double root of ∆.


We report on the numerical simulation of (2.100) in the unstable case a = 1. Thespace interval is [0, 1], we choose 1000 grid points, λ = 0.9, the source term gn onthe boundary equals zero for all n ≥ 2, while Fnj = 0 for all j, n except F 1

1 = 1/∆t.The numerical solution is represented at time t = 1/2 (left) and time t = 1 (right)in Figure 2.4. The instability is of a different kind than the one reported in thecase of the Lax-Friedrichs scheme with Neumann boundary condition, but it is notas violent as an exponential growth. Anyway, the exact boundary condition is notapproximated at all since Un0 seems to grow linearly in n. The exact same numericaltest can be performed in the strongly stable case a = −1 (we do not change anyother parameter). The results are shown on Figure 2.5 on a log scale: the numericalsolution remains small, as predicted by the strong stability estimate.

Fig. 2.4. The leap-frog scheme (2.100) for an incoming equation (a = 1) withbackward integration along the characteristic at the boundary. The interior sourceterm vanishes except F 1

1 which we choose equal to 1/∆t.

Fig. 2.5. The leap-frog scheme (2.100) for an outgoing equation (a = −1) withbackward integration along the characteristic at the boundary. The interior sourceterm vanishes except F 1

1 which we choose equal to 1/∆t. The numerical solution isrepresented on a log scale.


2.4.6 Goldberg-Tadmor’s Lemma for Dirichlet boundaryconditions

The aim of this paragraph is to understand why in all above examples the Dirichletboundary conditions lead to strongly stable numerical schemes. This result is firstdue to Goldberg and Tadmor [Gol81] and we show that it holds in our more generalframework. The result is the following.

Proposition 7 (Goldberg-Tadmor). Let us consider the scalar case N = 1, witha numerical scheme (2.14) that is stable for the discrete Cauchy problem. Then thenumerical scheme

Un+1j =

s∑σ=0

Qσ Un−σj +∆tFnj , j ≥ 1 , n ≥ s ,

Un+1j = gn+1

j , j = 1− r, . . . , 0 , n ≥ s ,Unj = 0 , j ≥ 1− r , n = 0, . . . , s ,

(2.103)

is strongly stable in the sense of Definition 4.

Proposition 7 shows that in the scalar case, there exists at least one way toimpose numerical boundary conditions and to obtain strong stability. The readermay observe that this is far from clear when one considers the characterization inTheorems 6 and 7. What may look surprising at first glance is that, in general,the Dirichlet boundary condition is not consistent in the L∞-norm (just think ofan outgoing transport equation with a bump propagating towards the left, whichdoes not satisfy the homogeneous Dirichlet boundary condition at all !). From anumerical point of view, the Dirichlet boundary condition may give rise to bound-ary layers, and one way to reformulate Proposition 7 is to say that in the scalarcase, such numerical boundary layers are stable. We emphasize that Proposition 7is independent of the underlying transport equation that is approximated by theoperators Qσ, meaning that these operators may be obtained by discretizing eitheran incoming or an outgoing transport equation.

Before proving Proposition 7, we state and prove two preliminary results thatwill be useful later on.

Lemma 24. Let M ∈Mm(C) and let λ be an eigenvalue of M with algebraic multi-plicity p. If Ker (M−λ I) has dimension 1, then for all k = 1, . . . , p, Ker (M−λ I)k

has dimension k.

Proof (Proof of Lemma 24). There is nothing to prove if p equals 1, so we assumep ≥ 2. The result is proved by induction on k. Let us assume that the result holds upto the index k. If k = p then the proof is complete, so we further assume k ≤ p− 1.We already know that Ker (M −λ I)k+1 contains Ker (M −λ I)k. The dimension ofKer (M − λ I)k+1 can not be equal to k for otherwise, there would hold Ker (M −λ I)k = Ker (M − λ I)k+1 and this implies Ker (M − λ I)k = Ker (M − λ I)k+j forall integer j. In particular, Ker (M − λ I)p would have dimension k < p and this isimpossible.

Let us now assume that the dimension Ker (M−λ I)k+1 equals at least k+2. Inparticular, there exist two linearly independent vectors X1, X2 in Ker (M−λ I)k+1 \Ker (M − λ I)k. Since (M − λ I)kXi, i = 1, 2 belong to the one-dimensional space


Ker (M − λ I), there exists a non-trivial linear combination µ1 X1 + µ2 X2 thatbelongs to Ker (M − λ I)k but this is excluded by the construction of X1, X2. Weare led to a contradiction. The only remaining possibility is to have Ker (M−λ I)k+1

of dimension k + 1.

As a matter of fact, Lemma 24 is a particular case of a more general fact.More precisely, it is known that for any eigenvalue λ of a matrix M , the sequence(dim Ker (M−λ I)k)k≥1 is concave. The proof of this fact uses similar arguments tothose developed in the proof of Lemma 24. The following Lemma is a generalizationof Lemma 8.

Lemma 25. Let M ∈Mm(C) be a companion matrix, that is

M =

µ1 . . . . . . µm1 0 . . . 0

0. . .

. . ....

0 0 1 0

.

Let λ be a nonzero eigenvalue of M with algebraic multiplicity p. Then for all k =1, . . . , p, there holds

Ker (M − λ I)k =

(P (m− 1)λm−1, . . . , P (1)λ, P (0))T , P ∈ Ck−1[X].

We warn the reader that Lemma 25 is not true in general for block companionmatrices.

Proof (Proof of Lemma 25). The proof is performed by induction on k. The resultis clear for k = 1 (see Lemma 8), and we assume that it holds up to the order k < p(otherwise the proof is already complete). Combining Lemma 8 and Lemma 24, wealready know that Ker (M − λ I)k+1 has dimension k + 1. Since λ is nonzero, thelinear map

P ∈ Ck[X] 7−→ (P (m− 1)λm−1, . . . , P (1)λ, P (0))T ∈ Cm

is an injection (here we use k+ 1 ≤ p ≤ m). It therefore only remains to prove thatthe image of this linear map is included in Ker (M−λ I)k+1. Since we already knowthat the image of any polynomial of degree ≤ k − 1 belongs to Ker (M − λ I)k, weonly need to find one polynomial of degree equal to k and whose image by the latterlinear map belongs to Ker (M − λ I)k+1. We define

Q(X) :=

k−1∏j=0

(X − j) ,

whose degree equals k, and we define Y := (Q(m − 1)λm−1, . . . , Q(1)λ,Q(0))T .Using the definition of the companion matrix M , we compute

(M − λ I)Y =

y

(Q(m− 1)−Q(m− 2))λm−1

...(Q(1)−Q(0))λ

,

y :=

m∑`=1

µ`Q(m− `)λm−` −Q(m− 1)λm .


Let us define the polynomial R(X) := Q(X + 1)−Q(X), which has degree k− 1. Ifwe can show that the above complex number y equals R(m − 1)λm, then we shallhave (M − λ I)Y ∈ Ker (M − λ I)k by the induction assumption and the proof willbe complete. Let us therefore show y = R(m − 1)λm. We know that λ is a root ofmultiplicity p ≥ k + 1 to the characteristic polynomial of M , hence

dk

dXk

(Xm −

m∑`=1

µ`Xm−`

)∣∣∣X=λ

= 0 .

Since λ is nonzero, we have

∀ j ∈ N , dk

dXk

(Xj)∣∣

X=λ= Q(j)λj−k ,

so we get

Q(m)λm−k −m∑`=1

µ`Q(m− `)λm−`−k = 0 .

Combining with the above definition of y, we end up with y = (Q(m) − Q(m −1))λm = R(m−1)λm which is the relation we were aiming at. The proof of Lemma25 is now complete.

We now turn to the proof of Proposition 7.

Proof (Proof of Proposition 7). We first recall the result of Lemma 7 which showsthat, under the assumptions of Proposition 7, the operators Qσ are geometricallyregular. Theorem 8 shows that the matrix M associated with (3.7.3) satisfies thediscrete block structure condition, and Theorem 9 then shows that M admits a K-symmetrizer with a vector space Es of dimension r. Eventually, Theorem 10 showsthat the stable bundle Es of M extends continuously from U to U . The proof ofProposition 7 then splits into two steps.• Let z ∈ U , and let κ1, . . . , κK denote the eigenvalues of M(z) with correspond-

ing algebraic multiplicities α1, . . . , αK . For z ∈ U close to z, we know from Lemma15 that the number of stable eigenvalues of M(z) close to κk is independent of z.We let µk denote this number, which can be computed for instance by counting thestable eigenvalues of M((1 + ε) z), 0 < ε 1. Our first goal is to show that Es(z)can be decomposed as

Es(z) = ⊕Kk=1Ker (M(z)− κk I)µk . (2.104)

Let us first observe that (2.104) is trivial when z ∈ U because in that case, theeigenvalues κk either belong to D (the stable ones) or to U (the unstable ones). Wetherefore have µk = αk if κk ∈ D, and µk = 0 if κk ∈ U , which clearly implies(2.104). We thus turn to the more delicate case z ∈ S1. There is no loss of generalityin assuming that the eigenvalues are ordered in such a way that κ1, . . . , κK1

belongto D (stable eigenvalues), κK1+1, . . . , κK2

belong to U (unstable eigenvalues), and

κK2+1, . . . , κK belong to S1 (neutral eigenvalues). Of course, we set µk = αk for1 ≤ k ≤ K1, and µk = 0 for K1 + 1 ≤ k ≤ K2. Let ε > 0 be so small thatthe disks centered at κ1, . . . , κK and of radius ε are pairwise disjoint. For n ∈ Nsufficiently large, the matrix M((1 + 2−n) z) has exactly µk stable eigenvalues in the


disk centered at κk and of radius ε. We let κ(n)k,1 , . . . , κ

(n)k,µk

denote these eigenvalues.

The eigenvalues κ(n)k,j tend to κk as n tends to infinity, so we have

limn→+∞

µk∏j=1

(M((1 + 2−n) z)− κ(n)

k,j I)

=(M(z)− κk I

)µk .Let now X ∈ Es(z), and let Xn ∈ Es((1 + 2−n) z) denote a sequence that convergestowards X. Such a sequence exists since we already know that the whole vectorspace Es((1 + 2−n) z) converges towards Es(z). Using (2.104) for every n, we have

Xn ∈ ⊕Kk=1Ker

µk∏j=1

(M((1 + 2−n) z)− κ(n)

k,j I)

= Ker

K∏k=1

µk∏j=1

(M((1 + 2−n) z)− κ(n)

k,j I).

Passing to the limit, we obtain

X ∈ Ker

K∏k=1

(M(z)− κk I

)µk = ⊕Kk=1Ker (M(z)− κk I)µk .

This relation shows that Es(z) is contained in the vector space on the right hand-side of (2.104). We also know that Es(z) has dimension r. Furthermore, M(z) is acompanion matrix so we can apply Lemma 8 and Lemma 24 which show that eachvector space Ker (M(z) − κk I)µk has dimension µk. Since the sum of all the µk’sequals r, we have obtained (2.104) for all z ∈ U .• The resolvent equation for (3.7.3) reads (2.51) with

∀ z ∈ C \ 0 , B(z) = B :=

0 . . . 0 1 0...

.... . .

0 . . . 0 0 1

∈Mr,p+r(C) . (2.105)

We recall that we consider the case of scalar problems so N equals 1 here. ApplyingProposition 6, we need to show that the kernel of the constant matrix B does notcontain any element of Es(z) for all z ∈ U . Consequently, let z ∈ U . By the non-characteristic discrete boundary assumption, we know that the companion matrixM(z) does not have 0 as an eigenvalue. We recall (2.104) and use Lemma 25 to com-pute a basis of Es(z). Up to reordering the eigenvalues, we can assume that µk > 0for all k = 1, . . . ,K and do not consider the other eigenvalues of M(z) anylonger.For each k, we define the polynomials

Pk,1(X) := 1 , Pk,2(X) :=1

κkX , . . . , Pk,µk (X) :=

1

κµk−1k

µk−2∏j=0

(X − j) .

(2.106)It is clear that the polynomials Pk,`, 1 ≤ ` ≤ µk, span Cµk−1[X] and Lemma 25shows that the vectors


Ek,1 :=

Pk,1(p+ r − 1)κp+r−1

k

...Pk,1(1)κkPk,1(0)

. . . Ek,µk :=

Pk,µk (p+ r − 1)κp+r−1

k

...Pk,µk (1)κkPk,µk (0)

,

span Ker (M(z) − κk I)µk . Using the decomposition (2.104), we wish to show thatthe vectors BEk,j , 1 ≤ k ≤ K, 1 ≤ j ≤ µk, are linearly independent. This isindeed equivalent to showing that the kernel of B does not intersect Es(z). Applyingthe matrix B in (2.105) to a vector of Cp+r amounts to keeping only the last rcoordinates of the vector. Therefore, showing that the kernel of B does not intersectEs(z) amounts to proving that the matrix

P1,1(r − 1)κr−11 . . . P1,1(1)κ1 P1,1(0)

......

P1,µ1(r − 1)κr−11 . . . P1,µ1(1)κ1 P1,µ1(0)

......

PK,1(r − 1)κr−1K . . . PK,1(1)κK PK,1(0)

......

PK,µK (r − 1)κr−1K . . . PK,µK (1)κK PK,µK (0)

(2.107)

is invertible. (In (2.107), the first µ1 rows are (BE1,1)T , . . . , (BE1,µ1)T and so on.)Before going on, let us observe that when all the µk’s equal 1, then K equals r andthe latter matrix coincides with the Vandermonde matrixκ

r−11 . . . κ1 1...

......

κr−1r . . . κr 1

,

which is known to be invertible (the κk’s are pairwise ditinct). Let us go back to thegeneral case and assume that the vector (cr−1, . . . , c0)T belongs to the kernel of thematrix in (2.107). We define the polynomial

P(X) := c0 + · · ·+ cr−1 Xr−1 .

For all j = 1, . . . , µ1, there holds

r−1∑`=0

c` P1,j(`)κ`1 = 0 .

From the definition (2.106) of the polynomials P1,j , we have

P1,j(`) =

κ1−j1

`!

(`+ 1− j)! , if j ≤ `+ 1,

0 , otherwise.

We therefore obtain

∀ j = 1, . . . , µ1 ,

r−1∑`=j−1

c``!

(`+ 1− j)! κ`+1−j1 = 0 ,


or equivalently∀ j = 1, . . . , µ1 , P(j−1)(κ1) = 0 .

The same analysis can be done for all the κk’s, and we find that P can be factorizedby

K∏k=1

(X − κk)µk .

Since the sum of the µk’s equals r, and the degree of P does not exceed r − 1,we can conlude that P equals 0, or equivalently that the kernel of the matrix in(2.107) is trivial. We have thus shown that the Uniform Kreiss-Lopatinskii Conditionis satisfied by the Dirichlet boundary conditions and Theorem 6 shows that thenumerical scheme (3.7.3) is strongly stable.

2.5 Fully discrete initial boundary value problems:semigroup stability

The goal of this section is to understand how one can incorporate nonzero initialdata in the numerical scheme (2.24). Of course, one can always consider initialconditions (f0), . . . , (fs) in (2.24), and the numerical scheme is still well-defined.The main problem is to understand how one can control the numerical solution (Unj )in `∞n (`2j ). In particular, if one can show a bound of the form ‖Un‖`2j ≤ C

n ‖U0‖`2j ,

C > 1, this would correspond for the continuous problem to a bound of the form‖u(t)‖L2(R) ≤ Ct/∆t ‖u|t=0‖L2(R), which would be useless in the limit ∆t → 0.

Basically, we are looking for an energy estimate of the solution in `∞n (`2j ) that iscompatible, in the limit ∆t→ 0, with an energy estimate for the continuous problem.

2.5.1 A simple but unsufficient argument

As we have seen in Section 2.2, it is very easy to incorporate initial conditions for theCauchy problem and to obtain `∞n (`2j ) bounds thanks to Fourier transform. Usingthe linearity of (2.24), we can thus try to decompose the solution (Unj ) as the sumUnj = V nj + Wn

j , where (V nj ) is a solution to a Cauchy problem and (Wnj ) is a

solution to a problem of the form (2.24) with zero initial data. This strategy givesthe following result.

Proposition 8. Let us assume that the numerical scheme (2.14) is stable for theCauchy problem (in the sense of Definition 2) and that (2.24) is strongly stable inthe sense of Definition 4. Then there exists a constant C > 0 such that for all γ ≥ 1and for all ∆t ∈ ]0, 1], the solution to (2.24) satisfies


supn≥0

e−2 γ n∆t∑j≥1−r

∆x |Unj |2 +γ

γ ∆t+ 1

∑n≥0

∑j≥1−r

∆t∆x e−2 γ n∆t |Unj |2

+∑n≥0

p∑j=1−r

∆t e−2 γ n∆t |Unj |2 ≤C

∆t2

∑j≥1−r

∆x(|f0j |2 + · · ·+ |fsj |2

)+γ ∆t+ 1

γ

∑n≥s

∑j≥1

∆t∆x e−2 γ (n+1)∆t |Fnj |2

+∑n≥s+1

0∑j=1−r

∆t e−2 γ n∆t |gnj |2 . (2.108)

Proof (Proof of Proposition 8). •We first extend the initial conditions (f0), . . . , (fs)and the interior source term (Fnj ) by zero for j ≤ −r. We also decompose the solution(Unj ) to (2.24) as Unj = V nj +Wn

j , where (V nj ) is a solution toVn+1j =

s∑σ=0

Qσ Vn−σj +∆tFnj , j ∈ Z , n ≥ s ,

V nj = fnj , j ∈ Z , n = 0, . . . , s ,

(2.109)

and (Wnj ) is a solution to

Wn+1j =

s∑σ=0

QσWn−σj , j ≥ 1 , n ≥ s ,

Wn+1j =

s∑σ=−1

Bj,σWn−σ1 + gn+1

j , j = 1− r, . . . , 0 , n ≥ s ,

Wnj = 0 , j ≥ 1− r , n = 0, . . . , s ,

(2.110)

with

∀ j = 1− r, . . . , 0 , ∀n ≥ s , gn+1j := gn+1

j − V n+1j +

s∑σ=−1

Bj,σ Vn−σ1 . (2.111)

This strategy will allow us to use the strong stability assumption for (2.24) on thesequence (Wn

j ) since the initial conditions for (2.110) vanish.• Our first goal is to estimate (V nj ). We start from (2.109) and apply a partial

Fourier transform with respect to the space variable (as in the proof of Proposition2). With the amplification matrix A defined in (2.16), we obtain

∀n ≥ s , ∀ ξ ∈ R ,

V n+1(ξ)

...V n+1−s(ξ)

= A (ei∆x ξ)

V n(ξ)

...

V n−s(ξ)

+∆t

Fn(ξ)...0

.

This relation yields


∀n ≥ s , ∀ ξ ∈ R ,

V n(ξ)

...

V n−s(ξ)

= A (ei∆x ξ)n−s

V s(ξ)

...

V 0(ξ)

+∆t

n−1∑m=s

A (ei∆x ξ)n−1−m

Fm(ξ)...0

.

Using the uniform bound for the amplification matrix A (here we use the stabilityassumption for the Cauchy problem), we obtain, for a given numerical constant C0,

∀n ≥ s , ∀ ξ ∈ R ,∣∣V n(ξ)

∣∣+ · · ·+∣∣V n−s(ξ)∣∣

≤ C0

(∣∣fs(ξ)∣∣+ · · ·+∣∣f0(ξ)

∣∣)+ C0 ∆t

n−1∑m=s

∣∣Fm(ξ)∣∣ . (2.112)

It only remains to “integrate” (2.112) with respect to n. For the sake of clarity, westate this kind of Gronwall inequality separately (the proof is a simple applicationof the `1 ? `2 convolution inequality and we leave it as an exercise for the interestedreader).

Lemma 26. Let s be an integer, and let C1, C2 be some nonnegative constants. Let(an)n≥s and (bn)n≥s denote some sequences of nonnegative numbers that satisfy

∀n ≥ s , an ≤ C1 as + C2

n−1∑m=s

bm .

Then for all γ > 0 and all ∆t ∈ ]0, 1], there holds

supn≥s

e−2 γ n∆t a2n +

γ

1 + γ ∆t

∑n≥s+1

∆t e−2 γ n∆t a2n

≤ 2C21 e−2 γ s∆t a2

s + 2C2

2

∆t21 + γ ∆t

γ

∑n≥s

∆t e−2 γ (n+1)∆t b2n .

We apply Lemma 26 to (2.112), and obtain

supn≥s

e−2 γ n∆t∣∣V n(ξ)

∣∣2 +γ

1 + γ ∆t

∑n≥s+1

∆t e−2 γ n∆t∣∣V n(ξ)

∣∣2≤ C

(e−2 γ s∆t

(∣∣f0(ξ)∣∣2 + · · ·+

∣∣fs(ξ)∣∣2)+

1 + γ ∆t

γ

∑n≥s

∆t e−2 γ (n+1)∆t∣∣Fn(ξ)

∣∣2≤ C

∣∣f0(ξ)∣∣2 + · · ·+

∣∣fs(ξ)∣∣2 +1 + γ ∆t

γ

∑n≥s

∆t e−2 γ (n+1)∆t∣∣Fn(ξ)

∣∣2 ,


with an appropriate numerical constant C. We integrate the latter inequality withrespect to ξ, use Plancherel’s and Fubini’s Theorems, and obtain our first mainestimate for the sequence (V nj ):

supn≥s

e−2 γ n∆t∑j∈Z

∆x |V nj |2 +γ

1 + γ ∆t

∑n≥s+1

∑j∈Z

∆t∆x e−2 γ n∆t |V nj |2

≤ C

∑j≥1−r

∆x(|f0j |2 + · · ·+ |fsj |2

)

+1 + γ ∆t

γ

∑n≥s

∑j≥1−r

∆t∆x e−2 γ (n+1)∆t |Fnj |2 . (2.113)

Observe that in the right hand-side of (2.113), the sums with respect to j only startat j = 1 − r since the initial conditions and the interior source term vanish forj ≤ −r.

To make the estimates below easier to read, we define the quantity

Source :=∑j≥1−r

∆x(|f0j |2 + · · ·+ |fsj |2

)+

1 + γ ∆t

γ

∑n≥s

∑j≥1−r

∆t∆x e−2 γ (n+1)∆t |Fnj |2

+∑n≥s+1

0∑j=1−r

∆t e−2 γ n∆t |gnj |2 ,

which gives a measure of the source terms with some appropriate weights. With thisdefinition, the inequality (2.113) reads

supn≥s


∆x |V nj |2 +γ

1 + γ ∆t

∑n≥s+1

∑j∈Z

∆t∆x e−2 γ n∆t |V nj |2

≤ C Source .

If we add some terms on the left hand-side that are obviously smaller than the righthand-side, we get

supn≥0


∆x |V nj |2 +γ

1 + γ ∆t

∑n≥0

∑j∈Z

∆t∆x e−2 γ n∆t |V nj |2

≤ C Source .

We then easily deduce (here we use γ ≥ 1):

∑n≥0

max(p,q+1)∑j=1−r

∆t e−2 γ n∆t |V nj |2 ≤ C1 + γ ∆t

γ ∆xSource

≤ C 1 +∆t

∆xSource ≤ 2C λ

∆tSource .

Combining with (2.113), we have already derived the inequality


supn≥0


∆x |V nj |2 +γ

1 + γ ∆t

∑n≥0

∑j≥1−r

∆t∆x e−2 γ n∆t |V nj |2

+∑n≥0


∆t e−2 γ n∆t |V nj |2 ≤C

∆tSource . (2.114)

with a new numerical constant that is still denoted C. The inequality (2.114) re-spresents “half” of (2.108). More precisely, it is now sufficient to prove a similarestimate to (2.114) for the sequence (Wn

j ) and the combination of both estimateswill give (2.108).•We recall the definition (2.111) of the source term gnj , n ≥ s+1. The operators

Bj,σ are defined in (2.25). In particular, there exists a numerical constant C suchthat

∀ j = 1− r, . . . , 0 , ∀n ≥ s+ 1 ,∣∣gnj ∣∣ ≤ |gnj |+ |V nj |+ C

s+1∑σ=0

q+1∑`=1

|V n−σ` | .

We then obtain∑n≥s+1

0∑j=1−r

∆t e−2 γ n∆t∣∣gnj ∣∣2 ≤ ∑

n≥s+1

0∑j=1−r

∆t e−2 γ n∆t |gnj |2

+∑n≥0

q+1∑j=1−r

∆t e−2 γ n∆t |V nj |2

≤ C

∆tSource ,

where we have used (2.114) in the end to estimate the traces of (V nj ) on j = 1, . . . , q+1. We now use the fact that (2.110) is strongly stable and get

γ

1 + γ ∆t

∑n≥s+1

∑j≥1−r

∆t∆x e−2 γ n∆t |Wnj |2

+∑n≥s+1

p∑j=1−r

∆t e−2 γ n∆t |Wnj |2 ≤

C

∆tSource .

Adding zero to the left hand-side (the initial conditions in (2.110) vanish), we obtain

γ

1 + γ ∆t

∑n≥0

∑j≥1−r

∆t∆x e−2 γ n∆t |Wnj |2 +

∑n≥0

p∑j=1−r

∆t e−2 γ n∆t |Wnj |2

≤ C

∆tSource . (2.115)

Using (2.115), we derive the `∞n (`2j ) estimate (we use the same type of inequalitiesas above):


∆x |Wnj |2 ≤

1 + γ ∆t

γ ∆t

C

∆tSource ≤ 2C

∆t2Source .

Combining with (2.115), we end up with


supn≥0


∆x |Wnj |2 +

γ

1 + γ ∆t

∑n≥0

∑j≥1−r

∆t∆x e−2 γ n∆t |Wnj |2

+∑n≥0

p∑j=1−r

∆t e−2 γ n∆t |Wnj |2 ≤

C

∆t2Source . (2.116)

Summing (2.116) and (2.114), we complete the proof of Proposition 8.

Of course, the result of Proposition 8 is not satisfactory because it does notgive any information in the limit ∆t → 0. Nevertheless, the proof of Proposition 8gave us the opportunity to introduce some major tools in the derivation of so-calledsemigroup estimates (meaning estimates in `∞n (`2j ) for the solution). The first maintool is to introduce an auxiliary problem that takes care of the initial condition. Bylinearity of the problem, we are reduced to the case of zero initial data for (2.24).There are two important steps in the estimates of the solution, and at each of thesesteps we have lost one (large) factor ∆t−1 in the proof of Proposition 8. The firstcrucial point is to obtain trace estimates for the solution to the auxiliary problem.These trace estimates should be obtained for a solution to a numerical scheme forwhich the initial conditions do not vanish (consequently it does not seem possible toexploit the results of Section 2.3 to derive these estimates). There is no clear reasonwhy the solution to the Cauchy problem should satisfy a trace estimate uniformlyin ∆t, so our strategy in Proposition 8 looks a little hopeless. The second crucialpoint is to obtain semigroup estimates for the solution to (2.24) with zero initialdata. Without any additional information, this step yields a factor ∆t−1, so a newstrategy is needed.

As far as the choice of the auxiliary problem is concerned, we can try to followRauch’s method [Rau72]. The most simple strategy is to find some kind of “strictlydissipative” numerical boundary conditions. This strategy is the main guideline of[Wu95] and was also used in [Cou11b] to extend the result of [Wu95] to multidimen-sional problems.

2.5.2 Wu’s argument

From now on, we consider numerical schemes with only one time step, meaningthat s = 0 in (2.24). Furthermore, in this paragraph, and this paragraph only, weconsider scalar problems, meaning that N = 1. The numerical scheme thus reads

Un+1j = QUnj , j ≥ 1 , n ≥ 0 ,

Un+1j =

∑q`=0 b`,j,−1 U

n+11+` + b`,j,0 U

n1+` , j = 1− r, . . . , 0 , n ≥ 0 ,

U0j = fj , j ≥ 1− r ,

(2.117)

where the operator Q is given by

Q =

p∑`=−r

a` T` , (a−r, . . . , ap) ∈ Rp+r+1 ,

and the b`,j,−1, b`,j,0 are real numbers. The integer r and p in Q are fixed by theconditions a−r 6= 0, ap 6= 0. The unknown (Unj ) in (2.117) is a sequence of realnumbers. Let us first observe that the amplification matrix A associated with Q is


a complex number, see (2.11). Consequently, if the numerical scheme for the Cauchyproblem is stable in the sense of Definition 1, then one necessarily has (this is thevon Neumann condition)

∀ η ∈ R ,∣∣A (ei η)

∣∣ ≤ 1 ,

and this implies∀ v ∈ `2(Z) , ‖Qv‖`2(Z) ≤ ‖v‖`2(Z) . (2.118)

In other words, we are in the case of strong stability for the discrete Cauchy problem.The following Lemma is proved in [Wu95] and states that there exists at least one

choice of numerical boundary conditions for which one can perform energy estimates“by hand” and incorporate nonzero initial data.

Lemma 27 ([Wu95]). Let either r ≥ 1 or let r = 0 and a−r 6= 1. Let us furtherassume that the operator Q in (2.117) satisfies (2.118). Then there exists a choiceof real numbers b1,aux, . . . , bp+1,aux such that the solution to

V n+1j = QV nj , j ≥ 1 , n ≥ 0 ,

V n+1j = 0 , j = 2− r, . . . , 0 , n ≥ 0 ,

V n+11−r =

∑p`=0 b1+`,aux V

n+11+` , n ≥ 0 ,

V 0j = fj , j ≥ 1− r ,

(2.119)

satisfies

supn≥0

∑j≥1−r

∆x |V nj |2 +∑n≥0

∆t

1∑j=1−r

|V nj |2 ≤ C∑j≥1−r

∆x |fj |2 . (2.120)

for all ∆t ∈ ]0, 1] with a constant C that does not depend on the initial condition(fj) in (2.119), nor on ∆t.

We refer to [Wu95, page 84], see also [GKO95, page 583], for the proof of Lemma27. The estimate (2.120) is very strong because there is even no exponential weightin the terms on the left hand-side. Of course, one trivial consequence of (2.120) isthe following estimate that looks more like what we were used to:

supn≥0


∆x |V nj |2 +∑n≥0

∆t e−2 γ n∆t1∑

j=1−r

|V nj |2

≤ C∑j≥1−r

∆x |fj |2 .

One important thing to notice in Lemma 27 compared with Proposition 8 is thatnow we have a very good control of the trace of the solution to the auxiliary problem.Lemma 27 is the building block for proving the following Theorem that answers theproblem of semigroup estimates for scalar equations and one time step schemes.

Theorem 12 (Semigroup stability for scalar problems [Wu95]). Let eitherr ≥ 1 or let r = 0 and a−r 6= 1. Let us consider the numerical scheme (2.117) withan operator Q that satisfies (2.118). Let us further assume that the scheme (2.117)


is strongly stable in the sense of Definition 4. Then there exists a constant C > 0such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution to (2.117) satisfies

supn≥0


∆x |Unj |2 ≤ C∑j≥1−r

∆x |fj |2 .

The proof of Theorem 12 is based on a decomposition U = V +W that is similarto the one used in the proof of Proposition 8. Lemma 27 gives the semigroup estimatefor the auxiliary problem as well as some trace estimates. Unfortunately, Lemma 27does not give a trace estimate for any fixed index j; it only gives a control of thetraces from j = 1−r up to j = 1. To control the traces for any index j, the argumentin [Wu95] relies on the Goldberg-Tadmor Lemma and this is the main point of theproof where it is crucial to deal with scalar equations. Deriving a semigroup estimatefor W follows from the same argument as for V since we already know that thetraces of W are controlled (this is the strong stability assumption). Since one stepin the proof of Theorem 12 heavily relies on Proposition 7 (the Goldberg-TadmorLemma), it is not clear that the result extends to multidimensional systems becausesuch systems usually do not reduce to decoupled scalar equations.

2.5.3 A more general framework for semigroup stability

Our goal in this paragraph is to propose an analogous method to that of Wu butthat can be extended to multidimensional problems. In particular, a crucial issue isto avoid using the fact that the equation is scalar, or to avoid using Proposition 7.One should perform similar calculations to those in [Wu95] but always in a vectorialframework. The main point to keep in mind is that (2.118) is a property that canhold even for non-scalar problems and this will be our starting point for the analysisof this paragraph. The results that we present here are all taken from [Cou11b]. Asin the preceeding paragraph, we restrict to one time step schemes:

Un+1j = QUnj +∆tFnj , j ≥ 1 , n ≥ 0 ,

Un+1j =

∑q`=0 B`,j,−1 U

n+11+` +B`,j,0 U

n1+` + gn+1

j , j = 1− r, . . . , 0 , n ≥ 0 ,

U0j = fj , j ≥ 1− r ,

(2.121)where the operator Q is given by

Q =

p∑`=−r

A` T` , A−r, . . . , Ap ∈MN (R) ,

and the unknown (Unj ) in (2.121) is a sequence of vectors in RN . Similarly, thematrices B`,j,−1, B`,j,0 in (2.121) belong to MN (R). We then make the followingassumption.

Assumption 2 (Strong stability for the Cauchy problem). The operator Q in(2.121) satisfies ‖Qv‖`2(Z) ≤ ‖v‖`2(Z) for all v ∈ `2(Z).

For simplicty, we shall use the following notation for the `2 norms: ∆x > 0 beingthe space step, then for all integers m1 ≤ m2, we set


‖V ‖2m1,m2:= ∆x

m2∑j=m1

|Vj |2

to denote the `2-norm on the interval [m1,m2] (m1 may equal −∞ and m2 may equal+∞). The corresponding scalar product is denoted by (·, ·)m1,m2 . Our main resultgives semigroup estimates as well as interior and trace estimates for the solution to(2.121) with arbitrary initial data in `2.

Theorem 13 ([Cou11b]). Let Assumptions 1 and 2 be satisfied, and assume thatthe scheme (2.121) is strongly stable in the sense of Definition 4. Then there exists aconstant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution to (2.121) satisfiesthe estimate

supn≥0

e−2 γ n∆t ‖Un‖21−r,+∞ +γ

γ ∆t+ 1

∑n≥0

∆t e−2 γ n∆t ‖Un‖21−r,+∞

+∑n≥0

∆t e−2 γ n∆tp∑

j=1−r

|Unj |2

≤ C

‖f‖21−r,+∞ +γ ∆t+ 1

γ

∑n≥0

∆t e−2 γ (n+1)∆t ‖Fn‖21,+∞

+∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.122)

As in [Wu95], the proof of Theorem 13 relies on the introduction of an auxiliaryproblem where, compared with (2.121), we modify the numerical boundary condi-tions. Our auxiliary problem is not the same as in [Wu95]. As a matter of fact, wedirectly show by means of the energy method that the Dirichlet boundary conditionsare what we call strictly dissipative. This is an improved version of Proposition 7since we are able to prove the strong stability estimate and also a semigroup esti-mate for the solution to the numerical scheme with Dirichlet boundary conditionsand arbitrary initial data (recall that Proposition 7 first assumes that the equationis scalar and only gives a strong stability estimate for zero initial data). Moreover,since we are able to obtain a direct proof of the Goldberg-Tadmor Lemma (with aneven stronger result), we do not need to rely anylonger on Proposition 7 and we canthus go further than the scalar case. More precisely, it is shown in [Cou11b] thatthe approach developed for proving Theorem 13 works in exactly the same way formultidimensional problems (not necessarily scalar ones). As far as we know, thisresult even gives the first examples of strongly stable schemes for genuine multi-dimensional problems (meaning problems that do not reduce to scalar equations).What remains of this paragraph is devoted to the proof of Theorem 13. We firstfocus on the case of Dirichlet boundary conditions, and we shall then see how thispreliminary result can be used to prove Theorem 13.

We therefore begin with the proof of the following refined version of Goldberg-Tadmor’s Lemma. Considering the numerical scheme

V n+1j = QV nj +∆tFnj , j ≥ 1 , n ≥ 0 ,

V n+1j = gn+1

j , j = 1− r, . . . , 0 , n ≥ 0 ,

V 0j = fj , j ≥ 1− r ,

(2.123)


we are going to show

Theorem 14 ([Cou11b]). Let Assumptions 1 and 2 be satisfied. Then there existsa constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution to (2.123)satisfies the estimate

supn≥0

e−2 γ n∆t ‖V n‖21−r,+∞ +γ

γ ∆t+ 1

∑n≥0

∆t e−2 γ n∆t ‖V n‖21−r,+∞

+∑n≥0

∆t e−2 γ n∆t


|V nj |2

≤ C

‖f‖21−r,+∞ +γ ∆t+ 1

γ

∑n≥0

∆t e−2 γ (n+1)∆t ‖Fn‖21,+∞

+∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.124)

In particular, the discretization (2.123) is strongly stable in the sense of Definition4.

Proof (Proof of Theorem 14). • For simplicity, we shall give the proof of Theorem14 in the special case where Fnj = 0 in (2.123). The argument is simpler in thiscase, and we refer to [Cou11b] for a complete treatment of the case with an interiorsource term. We decompose the operator Q as

Q := I + Q .

Assumption 2 is then equivalent to the inequality

∀V ∈ `2 , 2(V, Q V

)−∞,+∞ + ‖Q V ‖2−∞,+∞ ≤ 0 . (2.125)

We first use the relation V n+1j = (I+Q)V nj for j ≥ 1 (recall that we assume Fnj = 0

here), and derive

‖V n+1‖21,+∞ − ‖V n‖21,+∞ = 2(V n, Q V n

)1,+∞ + ‖Q V n‖21,+∞ . (2.126)

For a fixed integer n, we introduce the sequence (Wj)j∈Z such that Wj = V nj for

j ≥ 1 − r and Wj = 0 for j ≤ −r. Due to the structure of the operator Q (a

linear combination of the shifts T−r, . . . ,Tp), we have QWj = 0 if j ≤ −r− p, and

QWj = Q V nj if j ≥ 1. Using (2.125), we thus get

0 ≥2(W, QW

)−∞,+∞ + ‖QW‖2−∞,+∞

=2(W, QW

)1−r,0 + 2

(W, QW

)1,+∞

+ ‖QW‖21−r−p,−r + ‖QW‖21−r,0 + ‖QW‖21,+∞=2(V n, QW

)1−r,0 + 2

(V n, Q V n

)1,+∞

+ ‖QW‖21−r−p,−r + ‖QW‖21−r,0 + ‖Q V n‖21,+∞=2(V n, Q V n

)1,+∞ + ‖Q V n‖21,+∞

+ ‖V n + QW‖21−r,0 + ‖QW‖21−r−p,−r − ‖V n‖21−r,0 . (2.127)


We insert (2.127) into (2.126) and obtain

‖V n+1‖21,+∞ − ‖V n‖21,+∞ + ‖QW‖21−r−p,−r + ‖V n + QW‖21−r,0 ≤ ‖V n‖21−r,0 .(2.128)

At this point, two situations may occur depending on the integer p. Let us firstconsider the case p ≥ 1. Then, by Assumption 1 (with s = 0), Ap is an invertiblematrix and the following result holds.

Lemma 28. Let p ≥ 1 and let Ap be invertible. Then there exists a constant c > 0that does not depend on ∆t nor on V n such that the following estimate holds:

‖QW‖21−r−p,−r + ‖V n + QW‖21−r,0 ≥ c ‖V n‖21−r,p .

Proof (Proof of Lemma 28). Proving Lemma 28 is equivalent to proving that thequadratic form (that is independent on n)

(V n1−r, . . . , Vnp ) 7−→

−r∑j=1−r−p

|QWj |2 +

0∑j=1−r

|V nj + QWj |2 (2.129)

is positive definite. Recall that W denotes the extension of V n by zero for j ≤ −r.The quadratic form (2.129) is clearly nonnegative. Let us therefore consider somevector (V n1−r, . . . , V

np ) that satisfies

∀ j = 1− r − p, . . . ,−r, QWj = 0 , ∀ j = 1− r, . . . , 0, V nj + QWj = 0 .(2.130)

We first show by induction on j that V nj = 0 for all j = 1−r, . . . , p−r. Let us recall

that p ≥ 1, so we can write Q = Q− I in the form

Q = Ap Tp +

p−1∑`=−r

A` T` .

In particular, we have QW1−r−p = Ap Vn1−r, and V n1−r = 0 because Ap is invertible.

For j = 1− r− p, . . . ,−r, QWj equals Ap Vnj+p plus a linear combination of the V n` ,

` < j + p. Since the first term V n1−r is zero, we can proceed by induction and getV n1−r = · · · = V np−r = 0.

We now use the second set of equalities in (2.130). In particular, we have V n1−r +

QW1−r = QW1−r = Ap Vn1−r+p. Therefore, V n1−r+p = 0, and the rest of the proof

follows from another induction argument. We have thus shown that (2.130) implies(V n1−r, . . . , V

np ) = 0. Hence the quadratic form (2.129) is positive definite. The proof

of Lemma 28 is complete.

We now complete the estimate of the sequence (V nj ). Going back to (2.128) andusing Lemma 28, we have

‖V n+1‖21,+∞ − ‖V n‖21,+∞ + c∆x

p∑j=1−r

|V nj |2 ≤ ∆x0∑

j=1−r

|V nj |2 . (2.131)

The end of the proof consists in “integrating” (2.131) over N. We let γ > 0 and, forthe sake of clarity, we introduce the notation


Vn := e−2 γ n∆t ‖V n‖21,+∞ , Bn := e−2 γ n∆tp∑

j=1−r

|V nj |2 ,

Gn := e−2 γ n∆t0∑

j=1−r

|V nj |2 .

We multiply (2.131) by exp(−2 γ n∆t) to obtain

e2 γ ∆t Vn+1 − Vn +c

λ∆tBn ≤

1

λ∆tGn .

Summing this inequality from 0 to N yields

e2 γ ∆t VN+1 +e2 γ ∆t − 1

∆t

N∑1

∆tVn +c

λ

N∑0

∆tBn

≤ V0 +1

λ

N∑0

∆tGn ≤ V0 +1

λ

∑n≥0

∆tGn .

Letting N tend to +∞, we have proved

e2 γ ∆t supn≥1

Vn+γ∑n≥1

∆tVn+∑n≥0

∆tBn ≤ C

V0 +∆xG0 +∑n≥1

∆tGn

. (2.132)

The right-hand side of (2.132) is directly estimated by the right-hand side of (2.124),see the definition above for Gn and use (2.123) (recall that there is no interior sourceterm here). The constant C in (2.132) is independent of γ and ∆t.

It remains to treat the case p = 0 for which Lemma 28 does not hold anymore.In this case, we go back to (2.128) and simply ignore the nonnegative “boundaryterms” on the left hand-side

‖V n+1‖21,+∞ − ‖V n‖21,+∞ ≤ ‖V n‖21−r,0 .

We then proceed as above (with the same notation) to derive the weighted-in-timeestimate

e2 γ ∆t supn≥1

Vn + γ∑n≥1

∆tVn ≤ C

V0 +∆xG0 +∑n≥1

∆tGn

.

We have thus derived the inequality

e2 γ ∆t supn≥1

e−2 γ n∆t ‖V n‖21,+∞ + γ∑n≥1

∆t e−2 γ n∆t ‖V n‖21,+∞

≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 .

Adding some terms on the left hand-side that are obivously estimated by the righthand-side, we obtain (recall p = 0)


supn≥0

e−2 γ n∆t ‖V n‖21−r,+∞ +γ

γ ∆t+ 1

∑n≥0

∆t e−2 γ n∆t ‖V n‖21−r,+∞

+∑n≥0

∆t e−2 γ n∆tp∑

j=1−r

|V nj |2

≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.133)

• The estimate (2.133) completes the proof of Theorem 14 when q < p. We thusassume from now on q ≥ p. In that case, we need some additional trace estimates,namely we need to control

∑n≥0

∆t e−2 γ n∆tq+1∑j=p+1

|V nj |2 .

This is done by using the “shift trick” introduced in [Wu95]. More precisely, whenp ≥ 1, we define the sequence Wn

j := V nj+1 for n ≥ 0 and j ≥ 1− r, which solves thesystem (recall that Fnj equals 0 in (2.123) for the case that we consider here):

Wn+1j = QWn

j , j ≥ 1 , n ≥ 0 ,

Wn+1j = gn+1

j+1 , j = 1− r, . . . ,−1 , n ≥ 0 ,

Wn+10 = V n+1

1 , n ≥ 0 ,

W 0j = fj+1 , j ≥ 1− r .

Applying (2.133) to W , we obtain

∑n≥0

∆t e−2 γ n∆t |Wnp |2 ≤ C

‖f‖22−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=2−r

|gnj |2

+∑n≥1

∆t e−2 γ n∆t |V n1 |2 .

Using again (2.133) to estimate the last term of the right hand-wide (this is possiblebecause p ≥ 1) yields

∑n≥0

∆t e−2 γ n∆t |V np+1|2 ≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 .

We have therefore derived a trace estimate for (V np+1)n≥0. A straightforward induc-tion argument gives

∑n≥0

∆t e−2 γ n∆tq+1∑j=p+1

|V nj |2

≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.134)


The combination of (2.133) and (2.134) proves the main stability estimate (2.124)for p ≥ 1.• To complete the proof of Theorem 14, we only need to show how to pass from

(2.133) to (2.124) in the case p = 0. Since (2.133) does not give any trace estimatefor (V nj ), j ≥ 1, the shift argument of [Wu95] cannot be used anylonger. FromAssumption 1, we know that the spectral radius of A0 is strictly less than 1. Hence,there exist a positive definite symmetric matrix H and a positive number ε0 suchthat if we consider the new Euclidean norm on RD

∀X ∈ RD , |X|H :=√X∗HX ,

then we have∀X ∈ RD , |A0 X|H ≤

√1− 2 ε0 |X|H .

From the relation

V n+11 = A0 V

n1 +

−1∑`=−r

A` Vn1+` = A0 V

n1 +

0∑j=1−r

Aj−1 gnj︸︷︷︸

=:Xn

,

where we use the notation g0j := fj for j = 1− r, . . . , 0, we get

|V n+11 |2H = |A0 V

n1 |2H + 2 (A0 V

n1 )∗HXn + |Xn|2H

≤ (1− 2 ε0) |V n1 |2H + 2 (A0 Vn1 )∗HXn + |Xn|2H≤ (1− ε0) |V n1 |2H + (1 + ε−1

0 ) |Xn|2H .

By definition of Xn, this turns into

|V n+11 |2H − |V n1 |2H + ε0 |V n1 |2H ≤ C

0∑j=1−r

|gnj |2 .

Using the same summation process as earlier, we obtain(1− e−2 γ ∆t) + ε0 e−2 γ ∆t

∑n≥0

∆t e−2 γ n∆t |V n1 |2H

≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 .

The norm | · |H and the standard Euclidean norm are equivalent, so that

∑n≥0

∆t e−2 γ n∆t |V n1 |2 ≤ C

‖f‖21−r,+∞ +∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 ,

with a constant C that does not depend on γ nor on ∆t. The proof of (2.124)follows from an induction argument where we apply the above method to recoverthe estimate for the trace (V nj )n≥0, j = 2, . . . , q + 1. The proof of Theorem 14 isnow complete.


It only remains to show how Theorem 14, which is already interesting on itsown, also implies Theorem 13.

Proof (Proof of Theorem 13). •We rewrite the solution to (2.121) as Unj = V nj +Wnj ,

where (V nj ) satisfiesV n+1j = QV nj +∆tFnj , j ≥ 1 , n ≥ 0 ,

V n+1j = gn+1

j , j = 1− r, . . . , 0 , n ≥ 0 ,

V 0j = fj , j ≥ 1− r ,

and (Wnj ) satisfies

Wn+1j = QWn

j , j ≥ 1 ,

Wn+1j =

∑q`=0 B`,j,−1 W

n+11+` +B`,j,0 W

n1+` + gn+1

j , j = 1− r, . . . , 0 ,W 0j = 0 , j ≥ 1− r .

(2.135)

The source term g in (2.135) is defined by

∀ j = 1− r, . . . , 0 , ∀n ≥ 1 , gnj :=

q∑`=0

B`,j,−1 Vn1+` +B`,j,0 V

n−11+` . (2.136)

The estimate for (V nj ) is given by Theorem 14. In addition, since the discretization(2.121) is strongly stable in the sense of Definition 4 and the initial data in (2.135)is zero, (Wn

j ) satisfies

γ

γ ∆t+ 1

∑n≥0

∆t e−2 γ n∆t ‖Wn‖21−r,+∞ +∑n≥0

∆t e−2 γ n∆t0∑

j=1−r

|Wnj |2

≤ C∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 .

The defining equation (2.136) together with (2.124) allow us to control the terminvolving gnj by

∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2

≤ C

‖f‖21−r,+∞ +γ ∆t+ 1

γ

∑n≥0

∆t e−2 γ (n+1)∆t ‖Fn‖21,+∞

+∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.137)

Hence, we obtain


γ

γ ∆t+ 1

∑n≥0

∆t e−2 γ n∆t ‖Wn‖21−r,+∞ +∑n≥0

∆t e−2 γ n∆t0∑

j=1−r

|Wnj |2

≤ C

‖f‖21−r,+∞ +γ ∆t+ 1

γ

∑n≥0

∆t e−2 γ (n+1)∆t ‖Fn‖21,+∞

+∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 . (2.138)

The combination of (2.138) for (Wnj ) and of (2.124) for (V nj ) proves a first part of

Theorem 13. To complete the proof, it only remains to control the `∞n (`2j ) norm of(Wn

j ).• We start from (2.135) and apply the strategy of the proof of Theorem 14.

Since the derivation of the inequality (2.128) only relies on Assumption 2 and not onthe numerical boundary conditions, we have (just ignore the nonnegative boundaryterms on the left hand-side of (2.128))

‖Wn+1‖21,+∞ − ‖Wn‖21,+∞ ≤ ‖Wn‖21−r,0 .

We multiply this inequality by exp(−2 γ n∆t) and use the summation process as inthe proof of Theorem 14. Since the initial data for (2.135) vanish, this yields

supn≥0

e−2 γ n∆t ‖Wn‖21,+∞ ≤ C∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|Wnj |2 .

We now use the strong stability of (2.135) and the above estimate for the sourceterm (gnj ) to derive

supn≥0

e−2 γ n∆t ‖Wn‖21,+∞

≤ C

‖f‖21−r,+∞ +γ ∆t+ 1

γ

∑n≥0

∆t e−2 γ (n+1)∆t ‖Fn‖21,+∞

+∑n≥1

∆t e−2 γ n∆t0∑

j=1−r

|gnj |2 .

Summing the latter inequality with (2.138) and the estimate (2.124) for (V nj ), wecomplete the proof of the estimate (2.122).

2.5.4 The Lax-Friedrichs scheme

The above analysis applies to the Lax-Friedrichs scheme (2.20) provided that wecan check Assumption 2. More precisely, let us consider the numerical scheme (2.20)with a real symmetric matrix A. The amplification matrix ALF satisfies the vonNeumann condition if λ ρ(A) ≤ 1, see (2.21). Moreover, when A is symmetric, theamplification matrix ALF is a normal matrix. Hence its norm equals its spectralradius and we can conlude that Assumption 2 is satisfied. We can now state ourmain result for the Lax-Friedrichs scheme with general boundary conditions:

2 Stability of finite difference schemes for boundary value problems 213Un+1j =

Unj−1 + Unj+1

2− λA

2(Unj+1 − Unj−1) +∆tFnj , j ≥ 1 , n ≥ 0 ,

Un+10 =

∑q`=0 B`,−1 U

n+11+` +B`,0 U

n1+` + gn+1 , n ≥ 0 ,

U0j = fj , j ≥ 0 .

(2.139)

Theorem 15. Let A be a real symmetric matrix and let λ > 0 satisfy λ ρ(A) < 1.If the numerical scheme (2.139) is strongly stable in the sense of Definition 4, thenthere exists a constant C > 0 such that for all γ > 0 and all ∆t ∈ ]0, 1], the solutionto (2.139) satisfies the estimate (2.122).

If all eigenvalues of A are negative, we have seen that the Neumann bound-ary condition Un+1

0 = Un+11 + gn+1 yields a strongly stable scheme, so Theorem

15 applies. Of course, the result is not very spectacular for such simple numericalschemes, but for schemes that involve many grid points (as in the case of Runge-Kutta schemes detailed in Appendix A), it can become very complicated to verifyan estimate like (2.122). As observed in numerous places in these notes, our futuregoal is to extend all the results presented here to multidimensional problems and wehope that our future results may bring more significant progress in this direction.

2.6 A partial conclusion

In these notes, we have tried to make a general and complete presentation of thederivation of stability estimates for fully discretized hyperbolic initial boundary valueproblems. The theory involves quite many arguments that we briefly summarize.

(i) The stability theory for the discretized Cauchy problem gives rise to the well-known von Neumann condition. The latter is a necessary condition for stability.In the class of geometrically regular operators, it turns out to be also a sufficientcondition for stability.

(ii) The stability theory for discretized initial boundary value problems deals firstwith problems with zero initial data. In that case, an appropriate notion ofstability was introduced in [GKS72] and is referred to as strong stability. Usingthe Laplace transform, strong stability is first shown to be equivalent to anestimate for the resolvent equation. This preliminary reduction shows that theso-called Godunov-Ryabenkii condition is necessary for strong stability to hold. Arefined and more quantitative version of the Godunov-Ryabenkii condition arisesfor strongly stable schemes and was referred to in these notes as the UniformKreiss-Lopatinskii condition.

(iii)The difficult part of the theory is to show that the UKLC is not only necessarybut also sufficient for strong stability. The main technical points for doing so areto reduce the symbol M of the resolvent equation to the discrete block structureand then to construct a Kreiss symmetrizer. Reducing the symbol to the discreteblock structure is possible in the framework of geometrically regular operators,while the construction of a Kreiss symmetrizer also requires the the fulfillmentof the UKLC.

(iv)Once the case of zero initial data is clarified (it reduces more or less to verifyingthe UKLC), the remaining part of the theory consists in incorporating arbitraryinitial data and proving semigroup estimates. This does not seem possible with-out any further assumption on the numerical schemes that we consider. In these


notes, we have presented a general argument that works for many one time stepschemes.

In the case of zero initial data, the stability theory presented here seems to becomplete since we do not know of any stable discretization for the one-dimensionalCauchy problem that violates the geometric regularity condition. The situation be-comes far less clear in several space dimensions. In that case, even simple examplesshow that geometric regularity can be lost and further arguments need to be de-veloped. This can be seen for instance on the Lax-Friedrichs scheme for the two-dimensional wave equation (this scheme is known to be strongly stable but geometricregularity seems to fail as long as the matrices of the hyperbolic system do not com-mute). The so-called dissipative14 schemes were considered in [Mic83] and we hopeto push the analysis beyond this class in a near future. From a practical point ofview, it would also be very interesting to develop powerful computational tools toverify the UKLC in some situations where it cannot be done analytically.

Incorporating nonzero initial data for one time step schemes works the same inone or several space dimensions with the argument presented here, see [Cou11b] forthe details. Hence the main open problem is to consider numerical schemes withseveral time steps (even in one space dimension).

A Other examples of discretizations for the Cauchyproblem

In this appendix, we give two other possible discretizations for the Cauchy problem(2.5). The goal is to convince the reader that there exist some numerical schemesfor which the eigenvalues of the amplification matrix can have a highly singularbehavior as they approach the unit circle.

A.1 The Runge-Kutta schemes or how to produce singular pointsof even order

In this paragraph we follow [GKO95, chapter 6] and introduce a class of high ordernumerical schemes based on the Runge-Kutta approximation for ordinary differentialequations. The general method is the following: we start from (2.5) and first intro-duce a discretization of the space variable (this is usually called semi-discretization).This amounts to introducing a space step ∆x > 0 and approximating the solutionu(t, x) to (2.5) by a sequence of function (vj(t))j∈Z where for all j ∈ Z, vj(t) rep-resents an approximation of u(t, j ∆x). One example is obtained by observing thatfor all sufficiently smooth function f , there holds

2

3 ε(f(ε)− f(−ε))− 1

12 ε(f(2 ε)− f(−2 ε)) = f ′(0) +O(ε4) .

Then the Cauchy problem (2.5) can be approximated by the semi-discrete problem15

14 Here dissipativity refers to dissipativity in Kreiss sense, see [GKO95, chapter 5].15 Here we use the rather standard “dot” notation for the time derivative in an

ordinary differential equation.

2 Stability of finite difference schemes for boundary value problems 215vj = − 2

3∆xA (vj+1 − vj−1) +

1

12∆xA (vj+2 − vj−2) , j ∈ Z , t ≥ 0 ,

vj(0) = f(j ∆x) , j ∈ Z .

The latter problem is a linear (infinite) system of ordinary differential equations forwhich we can apply the fourth order Runge-Kutta integration rule16 with time step∆t = λ∆x (recall that the CFL number λ is a fixed constant). The following obser-vation follows from a straightforward computation: for a linear ordinary differentialequation

X = LX , X(0) = X0 ,

the fourth order Runge-Kutta method reads

Xn+1 =

4∑k=0

(∆tL)k

k!Xn .

Applying this rule to the above linear system for the vj ’s, we obtain the followingfully discrete approximation for (2.5):U

n+1j =

4∑k=0

(λA Q

)kk!

Unj , j ∈ Z , n ∈ N ,

U0j = fj , j ∈ Z ,

(2.140)

with

Q := −2

3(T−T−1) +

1

12(T2 −T−2) .

The scheme (2.140) can be written under the form (2.8), (2.9) with p = r = 8.Our goal is now to determine the values of the CFL number λ for which the scheme(2.140) is stable. Applying Lemma 2, we already know that it is sufficient to verify thevon Neumann condition. Once again, we let µ1, . . . , µN denote the (real) eigenvaluesof A, and we compute the eigenvalues of the corresponding amplification matrix Aby diagonalizing A. The eigenvalues z1(η), . . . , zN (η) of A (ei η) are given by

∀ j = 1, . . . , N , zj(η) =

4∑`=0

(λµj q(η)

)``!

, q(η) := −i sin η

3(4− cos η) .

The modulus of zj(η) is computed by using the fact that q(η) is purely imaginary,and we obtain

|zj(η)|2 = 1− (λµj)6

52488h(η)6

(1− (λµj)

2

72h(η)2

), h(η) := sin η (4− cos η) .

(2.141)It follows from (2.141) that the scheme (2.140) satisfies the von Neumann conditionif and only if λ ρ(A) maxR |h| ≤ 6

√2. The maximum of |h| on R can be explicitely

computed (!) by studying the variations of h and we obtain

maxR|h| =

(3 +

√6

2

) √√6− 3

2.

16 We refer to [Sch02] for an introduction to the discretization of ordinary differentialequations.


The maximum value for λ ρ(A) that ensures stability equals approximately 2.06.The reader can check that |h| attains its maximum for η ± η0 ∈ Z 2π where η0 isuniquely determined by η0 ∈ ]π/2, π[ and cos η0 = 1−

√3/2.

We now wish to analyze the behavior of the parametrized curve zj(η) , η ∈ Raccording to the possible values of λµj . For simplicity again, we assume that 0 doesnot belong to sp(A). Let us first consider the case where λ |µj | maxR |h| < 6

√2. Then

it follows from (2.141) that zj(η) belongs to S1 if and only if η ∈ Zπ. Moreover,there holds zj(0) = zj(π) = 1, z′j(0) = −i λ µj 6= 0 and z′j(π) = 5 i λ µj/3 6= 0.Consequently the curve zj(η) , η ∈ R has one regular contact point with the unitcircle (this point is attained in two different ways but each time it corresponds to aregular point). An example of such a curve is depicted in red in the left picture ofFigure 2.6. The unit circle is depicted in black.

Let us now consider the more interesting case where λ |µj | maxR |h| = 6√

2, andlet us even further assume µj > 0, the case µj < 0 being entirely similar. The formula(2.141) shows that zj(η) ∈ S1 if and only if η ∈ Zπ or η ± η0 ∈ Z 2π. As above,we compute zj(0) = zj(π) = 1, z′j(0) = −i λ µj 6= 0 and z′j(π) = 5 i λ µj/3 6= 0. Wealso compute z′j(±η0) = 0 since h′(±η0) = 0. An elementary calculation yields therelations

zj(η0) = zj(−η0) = −1

3+ i

2√

2

3,

z′′j (η0) = z′′j (−η0) = λµj h′′(η0)

(2√

2

9+ i

), h′′(η0) < 0 .

The points zj(±η0) are singular points of order 2 on the curve zj(η) , η ∈ R.Moreover, there exists a constant c > 0 such that for all η close to η0, there holds

|zj(η)| = 1− c (η − η0)2 + o((η − η0)2) ,

and there is a similar behavior in the neighborhood of −η0. The curve parametrizedby zj is depicted in blue in the left picture of Figure 2.6.

The scheme (2.140) gives an example for an eigenvalue zj of the amplificationmatrix such that the curve zj(η) , η ∈ R has a singular contact point of order2 with S1 and this curve is not included in S1 (as was the case with the leap-frogscheme). As a matter of fact, it is now not so difficult to generalize the example(2.140) in order to give an example of a stable scheme which produces some eigen-values whose corresponding parametrized curves have a singular contact point withS1 of arbitrarily large even order. Moreover these parametrized curves will not beincluded in S1. Let us detail how this generalization can be performed.

Let us consider an integer J ∈ N that is fixed once and for all. Then we definethe numbers

∀ j = 0, . . . , J , qj :=CJ−j2 J+1

22 J+1 (2 j + 1), (2.142)

where Ckn denotes the binomial coefficient. Using these numbers, we define the fol-lowing finite difference operator (we feel free to use similar notation as above)

Q :=J∑j=0

qj(T1+2 j −T−1−2 j) .


Fig. 2.6. Left : parametrized curves of eigenvalues for the Runge-Kutta scheme(2.140) (the unit circle in black, the eigenvalue curve for λ |µj | maxR |h| = 6

√2 ×

0.8 in red, and the eigenvalue curve for λ |µj | maxR |h| = 6√

2 in blue). Right :parametrized curves of eigenvalues for the Runge-Kutta scheme (2.143) (the unitcircle in black, the eigenvalue curve for λ |µj |MJ = 3

√3/4 in red and the eigenvalue

curve for λ |µj |MJ =√

3 in blue).

This operator is constructed as an approximation of the space derivative ∂x. In-deed, the properties of the binomial coefficients show that for all sufficiently smoothfunction f , there holds

J∑j=0

qj(f((1 + 2 j) ε)− f(−(1 + 2 j) ε)

)= ε f ′(0) +O(ε3) .

We now consider the Runge-Kutta integration rule of order 3 for the linear system ofordinary differential equations obtained after semi-discretizing the space derivative∂x by means of the operator Q/∆x (we recall that λ still denotes the CFL number∆t/∆x)17. This procedure gives the fully discretized schemeU

n+1j =

3∑k=0

(− λA Q

)kk!

Unj , j ∈ Z , n ∈ N ,

U0j = fj , j ∈ Z .

(2.143)

For the scheme (2.143), we have p = r = 3 (1 + 2 J), and applying Lemma 2 again,stability is equivalent to the von Neumann condition. The latter condition is verifiedby diagonalizing the matrix A. The eigenvalues zj(η) of the amplification matrixA (ei η) are given by

zj(η) = 1− (λµj)2

2h(η)2 − i λ µj h(η)

(1− (λµj)

2

6h(η)2

),

h(η) :=

J∑j=0

2 qj sin((2 j + 1) η) . (2.144)

17 We could have used again the Runge-Kutta integration rule of order 4 as in thepreceeding example, but we propose this new example to convince the reader thatthere is a very wide choice of approximation procedures.


We compute

|zj(η)|2 = 1− (λµj)4

12h(η)4

(1− (λµj)

2

3h(η)2

),

so stability of (2.143) is equivalent to the condition λ ρ(A) maxR |h| ≤√

3. The mainproperties of the function h are summarized in Lemma 29 below.

Lemma 29. Let the numbers qj be defined by (2.142) and let h be defined in (2.144).Then h is odd and satisfies

∀ η ∈ R , h′(η) = cos2 J+1 η .

The function h vanishes exactly for η ∈ Zπ. The maximum of h on R, that wedenote MJ , is positive and is attained when η− π/2 ∈ Z 2π. The minimum of h onR equals −MJ , and is attained when η + π/2 ∈ Z 2π.

Proof (Proof of Lemma 29). It is clear that h is odd, and we now differentiate husing the expression (2.142) of the qj ’s, obtaining

h′(η) =1

22 J

J∑j=0

CJ−j2 J+1 cos((2 j + 1) η)

=1

22 J

J∑j=0

Cj2 J+1 cos((2 J + 1− 2 j) η)

=1

22 J+1

2 J+1∑j=0

Cj2 J+1 cos((2 J + 1− 2 j) η)

= Re

(ei η + e−i η

2

)2 J+1

= cos2 J+1 η ,

where we have first changed j for J−j, and then used the symmetry of the binomialcoefficients.

It follows that h behaves exactly as the sine function: h vanishes at 0, is increasingon [0, π/2], attains its maximum at π/2, is decreasing on [π/2, 3π/2] and vanishesat π, attains its minimum at 3π/2, and so on.

Remark 12. The value of MJ in Lemma 29 coincides with the Wallis integral∫ π/20

cos2 J+1 η dη, that is 22 J (J !)2/(2 J + 1)!. Since MJ tends to 0 as J tends to

+∞, we see that the range of stability λ ρ(A) ∈ [0;√

3/MJ ] for the scheme (2.143)is getting larger and larger with J going to +∞ (meaning that for large J , the CFLnumber λ can be chosen large).

We now analyze the behavior of the curve zj(η) η ∈ R, dealing first withthe easier case λ |µj |MJ <

√3. We also assume that 0 does not belong to sp(A) for

simplicity. Then zj(η) ∈ S1 if and only if η ∈ Zπ, and we compute zj(0) = zj(π) = 1,z′j(0) = z′j(π) = −i λ µj 6= 0. The contact point with the unit circle is a regular point,as can be seen in the right picture of Figure 2.6 (red curve).

Let us now assume that the CFL number is chosen such that λµjMJ =√

3(we consider the case µj > 0). Then Lemma 29 shows that zj(η) ∈ S1 if and only if


η ∈ Zπ/2. We still have zj(0) = zj(π) = 1, z′j(0) = z′j(π) 6= 0, and we focus from now

on on the behavior of zj near η = π/2. We first compute zj(π/2) = −1/2− i√

3/2.Using Lemma 29, we also have

h′(π/2) = · · · = h(2 J+1)(π/2) = 0 , h(2 J+2)(π/2) = −(2 J + 1)! .

Performing a Taylor expansion in (2.144), we obtain

zj(η) = −1

2− i√

3

2+

λµj2 J + 2

(√3− i

2

)(η − π/2)2 J+2 +O((η − π/2)2 J+3) .

In particular, zj(π/2) is a singular point of order 2 J + 2 and we have

|zj(η)| = 1− 3

8MJ (J + 1)(η − π/2)2 J+2 + o((η − π/2)2 J+2) .

The behavior of the curve parametrized by zj near η = −π/2 is similar (it is justobtained by a complex conjugation). We refer to the right picture in Figure 2.6 fora representation of this curve with two singular points of high order18.

A.2 Multisteps schemes or how to produce singular points of oddorder

In this paragraph, we are going to construct an example of a scheme of the form(2.14) with s = 1, r = 3, p = 4, that is stable as long as λ ρ(A) ≤ 1, that isgeometrically regular and such that in the case λ ρ(A) = 1, one of the parametrizedcurves associated with eigenvalues of the amplification matrix has a singular contactpoint of order 3 with S1. This example is mainly constructed in order to convincethe reader that singular contact points of odd order do exist ! However the readershould keep in mind that the scheme defined below is probably useless for practicalapplications, as was the case for the scheme (2.143). Its interest is purely theoretical.As it will appear below, it is not so straightforward to construct such an example,or at least we have not found - despite repeated efforts - an easier construction.

We start from (2.5), semi-discretize the space variable by means of a finite dif-ference operator, leading to the system of ordinary differential equations

vj =1

∆xAQ] vj , j ∈ Z .

Then we apply the Adams-Bashforth quadrature rule of order 2. The numericalscheme thus readsUn+1

j = Unj + λ

(3

2AQ] U

nj −

1

2AQ] U

n−1j

), j ∈ Z , n ≥ 1 ,

U0j = f0

j , U1j = f1

j , j ∈ Z .(2.145)

We choose the finite difference operator Q] of the form

18 Of course, when one only considers the curve and not its parametrization, it isimpossible to distinguish between a singular point of order 2 and a singular pointof order 2 J+2. The two pictures in Figure 2.6 look similar even though the rightpicture represents a more degenerate situation.


Q] :=

4∑`=−3

q` T` ,

where the real numbers q−3, . . . , q4 are defined as the unique solution to the linearsystem

1 1 1 1 1 1 1 1−3 −2 −1 0 1 2 3 49 4 1 0 1 4 9 16−1 1 −1 1 −1 1 −1 13 −2 1 0 −1 2 −3 4−9 4 −1 0 −1 4 −9 1627 −8 1 0 −1 8 −27 64−81 16 −1 0 −1 16 −81 256

q−3

q−2

q−1

q0q1q2q3q4

=

0−11−100−11

. (2.146)

The first two rows of the linear system (2.146) ensure that for all smooth functionf , there holds

4∑`=−3

q` f(` ε) = −f ′(0) ε+ o(ε) ,

so (2.145) is really an approximation of (2.5). The reader can easily check eitherby hand made calculations or on a computer that the matrix of the above linearsystem is invertible so the scheme (2.145) is well-defined. The amplification matrixof (2.145) is given by the formula (2.16). Diagonalizing A and permuting rows andcolumns, there exists an invertible matrix T such that for all η ∈ R, there holds

A (ei η) = diag

((1 +

3λµj2

q(η) −λµj2

q(η)

1 0

))j=1,...,N

,

q(η) :=

4∑`=−3

q` ei ` η .

The function q satisfies

q(0) = 0 , q′(0) = −i , q′′(0) = −1 ,

q(π) = −1 , q′(π) = q′′(π) = 0 , q(3)(π) = i , q(4)(π) = 1 , (2.147)

as can be checked by using (2.146).We now wish to determine the CFL numbers λ for which the scheme (2.145) is

stable. More precisely, we are going to show that if all eigenvalues of A are nonneg-ative and if λ ρ(A) ≤ 1, then the operators in (2.145) are geometrically regular andthe amplification matrix of (2.145) satisfies the von Neumann condition. This willenable us to apply Proposition 3 and deduce stability for (2.145). We shall need thefollowing preliminary result.

Lemma 30. The mapping

κ ∈ S1 7−→ 2κ (κ− 1)

3κ− 1,

is injective and thus defines a closed simple curve C ⊂ C ' R2. The interior I ofC is a strictly convex region that contains the segment ]− 1, 0[. Moreover, 1 belongsto the exterior of C .


Proof (Proof of Lemma 30). We consider the mapping

θ ∈ [−π, π] 7−→2 ei θ

(ei θ − 1

)3 ei θ − 1

= x(θ) + i y(θ) .

Direct computations yield y(0) = y(±π) = 0, and ±y(θ) > 0 if ±θ ∈ ]0, π[. Further-more, x is increasing on [−π, 0] and decreasing on [0, π]. These properties imply thatC is a simple closed curve (see Figure 2.7 for a representation of C ). The reader canalso check that (x′)2 + (y′)2 does not vanish so every point of C is regular.

The interior of C is well-defined thanks to Jordan’s Theorem. It is strictly convexprovided that the curvature of C is nonnegative and vanishes at finitely many points.This amounts to proving that x′ y′′ − x′′ y′ is nonnegative and vanishes at finitelymany points. We compute

x′(θ) y′′(θ)− x′′(θ) y′(θ) =6 (1−X) (3X2 − 3X + 4)

(5− 3X)3

∣∣∣X=cos θ

≥ 0 ,

so I is strictly convex.

Fig. 2.7. The curve C (black line), and the curve λµj q(η) , η ∈ R for λµj = 1/4,λµj = 1/2 and λµj = 1 (red dots). The crosses represent the points −2/9±i 4

√2/9.

The following Lemma explains the link between the curve C and stability of thescheme (2.145).

Lemma 31. Let us assume that for all η 6∈ Zπ, q(η) ∈ I , where the region I isdefined in Lemma 30. If all eigenvalues of A are nonnegative and if furthermoreλ ρ(A) ≤ 1, then the scheme (2.145) is stable.

Proof (Proof of Lemma 31). • Let us start with the following simple observations.The matrix


M(α) :=

(1 + 3α/2 −α/2

1 0

)has at least one eigenvalue in S1 if and only if α ∈ C . By a connectedness argument,this means that for α ∈ I , M(α) has two eigenvalues in D (just look at the caseα = −1/2). If α belongs to the exterior of C , then M(α) has one eigenvalue in Dand one eigenvalue in U (look at the case α = 1). Moreover, M(α) has a doubleeigenvalue if and only if α = −2/9 ± i 4

√2/9, and in that case the double root

belongs to D. If α ∈ C , then M(α) can not have two distinct eigenvalues on S1 (useLemma 30) so M(α) has exactly one eigenvalue in D and one eigenvalue on S1. Ifwe summarize, the eigenvalues of M(α) belong to the closed unit disk provided thatα belongs to I ∪ C .• According to the reduction of the amplification matrix, the von Neumann con-

dition will be satisfied if for all eigenvalue µj of A and for all η ∈ R, the eigenvaluesof M(λµj q(η)) belong to the closed unit disk. We compute q(0) = 0 ∈ C andq(π) = −1 ∈ C , so for all η ∈ R, there holds q(η) ∈ I ∪C thanks to the assumptionof Lemma 31. The convexity of I shows that under the CFL condition λ ρ(A) ≤ 1,there holds λµj q(η) ∈ I ∪C . (Here we have used the fact that eigenvalues of A arenonnegative.) Using the above observations, we conclude that the eigenvalues of thematrix M(λµj q(η)) belong to the closed unit disk. Consequently the von Neumanncondition is satisfied.• It remains to show that the amplification matrix satisfies the geometric regu-

larity condition stated in Definition 3 and we shall be able to apply Proposition 3 toconclude. Using the diagonalization of A (ei η) in blocks of the form M(λµj q(η)),we already see that it is sufficient to prove a geometric regularity condition on each2× 2 block. Moreover, the exponential is locally a biholomorphic diffeomorphism soworking in a complex neighborhood of some κ = ei η ∈ S1 is equivalent to workingin a complex neighborhood of η ∈ R.

Let us first consider the case λµj < 1. The strict convexity of I shows thatλµj q(η) ∈ C if and only if q(η) ∈ Z 2π. For η = 0, the eigenvalues of M(0) are 0and 1, so 1 is a simple hence geometrically regular eigenvalue of M(λµj q(η)). If weconsider the case λµj = 1, we have λµj q(η) ∈ C if and only if q(η) ∈ Zπ. For η = π,the eigenvalues of M(−1) are −1 and 1/2 so −1 is also a simple hence geometricallyregular eigenvalue of M(λµj q(η)). The proof of Lemma 31 is complete.

Figure 2.7 gives some numerical evidence that the curve q(η) , η ∈ R remainswithin the interior of C . However, we must confess that we have not been able (ornot brave enough) to find a complete proof of this fact. As such, stability of (2.145)under the appropriate CFL condition remains an “if result”.

Let us focus on the behavior of the eigenvalues of the block M(q(η)), assumingthat λµj = 1. As we have seen in the proof of Lemma 31, M(q(η)) has an eigenvalueon S1 if and only if η ∈ Zπ. If η = 0, 1 is a simple eigenvalue whose Taylor expansionnear η = 0 reads (use the relations (2.147))

z(η) = 1− i η − η2 + o(η2) , |z(η)| = 1− 1

2η2 + o(η2) .

If η = π, −1 is a simple eigenvalue whose Taylor expansion near η = π reads (usethe relations (2.147) again)


z(η) = −1 +2 i

9(η − π)3 +

1

18(η − π)4 + o((η − π)4) ,

|z(η)| = 1− 1

18(η − π)4 + o((η − π)4) .

In particular, the above Taylor expansions show that for all η 6= 0 sufficiently smalland for all η 6= π sufficiently close to π, the eigenvalues of M(q(η)) belong to D.Furthermore, the eigenvalue curve passing through −1 has a singular contact pointof order 3. We refer to Figure 2.8 for a representation of the spectrum of M(q(η)),that is for the case λµj = 1.

Fig. 2.8. The eigenvalues of M(q(η)) in red and the unit circle in black.

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[BG07] Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Dif-ferential Equations. First-Order Systems and Applications. Oxford Uni-versity Press (2007)

[Cha82] Chazarain, J., Piriou, A.: Introduction to the Theory of Linear PartialDifferential Equations. North-Holland (1982)

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[Cou11a] Coulombel, J.-F.: Stability of finite difference schemes for hyperbolicinitial boundary value problems II. Ann. Sc. Norm. Super. Pisa Cl. Sci.,X(1), 37–98 (2011)


[Cou11b] Coulombel, J.-F., Gloria, A.: Semigroup stability of finite differenceschemes for multidimensional hyperbolic initial boundary value problems.Math. Comp., 80(273), 165–203 (2011)

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3

Boundary control and boundary inverse theoryfor non-homogeneous second-order hyperbolicequations: A common Carleman estimatesapproach

Shitao Liu1 and Roberto Triggiani2

1 Department of Mathematical Sciences, Clemson University, Clemson, SC 29634,USA and Department of Mathematics and Statistics, University of Helsinki,Helsinki, 00014, Finland. [email protected]

2 Department of Mathematical Sciences, University of Memphis, Memphis, TN38152, USA. [email protected]

Summary. In these lecture notes, we present some recent results on boundarycontrol and boundary inverse problems for non-homogeneous, second-order hyper-bolic equations. The focus will be on the application of sharp Carleman estimatesfor second-order hyperbolic equations, which serve as a common umbrella to ap-proach both control theory issues as well as inverse problems. We especially studythe inverse problem of determining the interior damping and/or the pair of dampingand potential coefficients of a mixed, second-order hyperbolic equation with non-homogeneous Neumann or Dirichlet boundary datum on a general multidimensionalbounded domain. It employs a single additional measurement: the Dirichlet-traceor Neumann-trace, respectively, of the solution over a suitable, explicit sub-portionof the boundary of the domain, and over a computable and sharp/close to optimaltime interval [0, T ]. Weak (optimal) regularity requirements on the data are imposed.Two canonical results in inverse problems are established: (i) global uniqueness and(ii) Lipschitz stability estimates of the linear as well as of the nonlinear inverseproblems. In addition, similar uniqueness and stability issues for coupled hyperbolicequations are also studied.

In the control theory part, Carleman estimates yield continuous observabilityinequalities (hence, exact controllability results by direct duality). In our treatment,continuous observability inequality (COI) is also employed in the study of inverseproblems (e.g. to boost the original regularity properties of the PDE-solutions).Another technical tool (at least in our treatment) in common between control theoryanalysis and inverse theory analysis is the approach by ‘compactness–uniqueness’ toabsorb lower order terms in the preliminary estimates in both cases. It appears that,in control theory, the first use of compactness–uniqueness was made by W. Littman[Lit87], who quotes Hormander. It has been widely used subsequently in controltheory since the late 80’s, see e.g., Lions [Lio88], W. Littman [Lit87], Lasiecka-Triggiani [LT89], Triggiani [Tri88], not only for hyperbolic equations, but also forplate equations, e.g., Lasiecka-Triggiani [LT00b], and for Schrodinger equations, e.g.,

228 Shitao Liu and Roberto Triggiani

Lasiecka-Triggiani [LT92b]; and not only in the linear case but also in the nonlinearsetting Lasiecka-Tataru [LT93], Lasiecka-Triggiani [LT06]. Apparently, the first useof a compactness-uniqueness argument in the context of inverse problems occurs inYamamoto [Yam99].

An additional important ingredient in our uniqueness proof of the linear inverseproblems is that it takes advantage of a recent, convenient tactical shortcut “post-Carleman estimates” to be found in Isakov [Isa06, Theorem 8.2.2, p. 231] which wasnot available in [Isa98], where a different route “post-Carleman estimates” was fol-lowed (see Acknowledgement). The present treatment of the two classical issues ininverse problems – ‘uniqueness’ and ‘stability’ – shows a sort of parallelism withthe control-theoretic issues of “observability” and “continuous observability,” re-spectively as pursued e.g., in Lasiecka-Triggiani [LT89], Triggiani [Tri88], Lasiecka-Triggiani-Zhang [LTZ00], or Gulliver-Lasiecka-Littman-Triggiani [GLL03].

The use of Carleman estimates to prove uniqueness of multidimensional inverseproblems with a single boundary observation was originated by Bukhgeim-Klibanov[BK81]. Since then, many papers concerning inverse hyperbolic problems with singleboundary measurement by using Carleman estimates have been published. Whilewe refer to the “Notes and Literature” at the end of each main section for a detailedcomparison between the content of this article and the literature prior to the presentauthors’ work (i.e. prior to 2011), we wish to provide here, at the outset, a broadorientation and insight.

The results collected in this article are either outright new or else a definiteimprovement of those of the literature prior to 2011.

Preliminary technical reasons responsible for this progress include (but are notlimited to) the following: (i) use of sharp Carleman estimates with explicit bound-ary terms defined in the entire lateral boundary Σ (from Lasiecka-Triggiani-Zhang[LTZ00]); (ii) use of sharp/optimal interior and boundary regularity theory for non-homogeneous mixed Dirichlet or Neumann direct problems (from Lasiecka-Lions-Triggiani [LLT86], Lasiecka-Triggiani [LT81], [LT90], [LT91], [LT94]) respectively;(iii) sharp control of tangential traces (from Lasiecka-Triggiani [LT92a]). In thisstated form, these critical ingredients were not used in prior literature in inverseproblems.

Cumulatively, advances or refinements over prior literature include the follow-ing areas (all with just one boundary measurement): (a) recovery (‘uniqueness’ and‘stability’) of both damping and source coefficients in one shot; (b) treatment ofboundary non-homogeneous Dirichlet or Neumann problems; (c) weaker geomet-rical conditions; that is, a smaller sub-portion of the boundary needed for theobservation/measurement to achieve recovery; (d) topologically stronger stabilityestimates; (e) sharp minimal assumptions on the data of the problem rather thanexcessive (untested) regularity properties of the solutions in the stability of linearand nonlinear inverse problems with non-homogeneous boundary data.

We focus here only on second order hyperbolic equations defined on an Euclideandomain Ω ⊂ Rn, with the Laplacian ∆ as principle part of the dynamic operator. Itis the intent of the authors to provide counterpart inverse theory results also whereΩ is a bounded set with boundary of an n-dimensional Riemannian manifold. Thisincludes the case of the Euclidean domain with ∆ replaced by an elliptic differentialoperator with variable coefficients in space. To this end, the Carleman estimates inthe Riemannian setting from Triggiani-Yao [TY02] will be needed: they have alreadyproduced control theory results in this reference. Numerous references for inverse

3 Boundary control and boundary inverse theory 229

problems for PDEs are given in the bibliography which however is not meant to beexhaustive.

As a potential direction of future research, we also include a recent referenceCannarsa-Tort-Yamamoto [CTY10] on inverse problems for a degenerate equation,which is however parabolic.

3.1 Preparatory material: Carleman estimates, interiorand boundary regularity of mixed problems

In this first part, we recall from [LTZ00] the Carleman estimates which will becritically used throughout these lecture notes as a common supporting pillar forboth control and inverse theory. In addition, we also collect here from [LLT86],[LT81], [LT83], [LT90], [LT91], [LT94], [Tat96], some sharp interior and boundaryregularity results for mixed second-order hyperbolic equations which are needed inthe proofs of the results in Sections 3.4-3.7.

3.1.1 Carleman estimates for H1 solutions of second-orderhyperbolic equations with explicit boundary terms: Dirichlet andNeumann cases

In this section, we recall from [LTZ00] a Carleman estimate at the H1×L2-level forsecond-order hyperbolic equations with explicit boundary terms. Such estimate willplay a key role in our presented control and inverse theory results.

Throughout these notes, we let Ω ⊂ Rn, n ≥ 2, be an open bounded domainwith boundary Γ = ∂Ω of class C2, consisting of the closure of two disjoint parts:Γ0 (uncontrolled or unobserved part) and Γ1 (controlled or observed part), bothrelatively open in Γ . Namely, Γ = ∂Ω = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅. Let ν = [ν1, · · · , νn]be the unit outward normal vector on Γ , and let ∂

∂ν= ∇·ν denote the corresponding

normal derivative. We also use Hs(Ω) to denote the standard Sobolev space [LM72].Consider the following second-order hyperbolic equation in the unknown w(x, t):

wtt(x, t)−∆w(x, t) = F (w) + f(x, t), (x, t) ∈ Q = Ω × [0, T ], (3.1a)

where the forcing termf(x, t) ∈ L2(Q), (3.1b)

and F (w) is given by

F (w) = q1(x, t)w + q2(x, t)wt + q3(x, t) · ∇w, (3.1c)

subject to the following standing assumption on the coefficients: q1, q2, |q3| ∈ L∞(Q),so that the following pointwise estimate holds true:

|F (w)| ≤ CT [w2 + w2t + |∇w|2], (x, t) ∈ Q. (3.1d)

Remark 1. At this point, we do not impose any boundary conditions (B.C.) or initialconditions (I.C.) for the equation (3.1a). In the subsequent sections we shall applyeither Dirichlet or Neumann B.C., as well as different I.C. to (3.1a) depending onthe problem under consideration.


Main geometrical assumptions. In addition to the standing assumptions (3.1b),(3.1c), on the forcing term and the first-order operator F , the following assumptionis postulated throughout these notes:

(A.1) Given the triple Ω,Γ0, Γ1, ∂Ω = Γ0 ∪ Γ1, there exists a strictly convex(real-valued) non-negative function d : Ω → R+, of class C3(Ω), such that, if weintroduce the (conservative) vector field h(x) = [h1(x), . . . , hn(x)] ≡ ∇d(x), x ∈ Ω,then the following two properties hold true:

(i) Dirichlet case:

∂d

∂ν

∣∣∣∣Γ0

= ∇d · ν = h · ν ≤ 0 on Γ0, h ≡ ∇d; (3.2a)

or (i’) Neumann case:

∂d

∂ν

∣∣∣∣Γ0

= ∇d · ν = h · ν = 0 on Γ0, h ≡ ∇d. (3.2b)

(ii) the (symmetric) Hessian matrix Hd of d(x) [i.e., the Jacobian matrix Jh of h(x)]is strictly positive definite on Ω: there exists a constant ρ > 0 such that for allx ∈ Ω:

Hd(x) = Jh(x) =

dx1x1 · · · dx1xn......

dxnx1 · · · dxnxn

=

∂h1

∂x1· · · ∂h1

∂xn...

...∂hn∂x1

· · · ∂hn∂xn

≥ ρI. (3.3)

(A.2) d(x) has no critical point on Ω:

infx∈Ω|h(x)| = inf

x∈Ω|∇d(x)| = s > 0. (3.4)

Remark 2. Assumption (A.1) responds to the expectation that it is natural to makegeometric assumptions only on the uncontrolled or unobserved portion of the bound-ary Γ0. The more demanding condition (i’) is because of the Neumann B.C. of thehyperbolic problem to follow. It was introduced in [Tri88, Section 5], and we alsofollow [LT89], [LTZ00] here. Assumption (A.2) is needed for the validity of the point-wise Carleman estimate in Theorem 1 below (it will imply that the constant β bepositive, β > 0, in estimate (3.11)–(3.12) below). Actually, as noted in [LTZ00, Re-mark 1.1.3, p. 229], Assumption (A.2) = (3.4) is needed to hold true only with theinfimum computed for x ∈ Γ0 (uncontrolled or unobserved part of the boundary Γ ).Moreover, (A.2) can, in effect, be entirely dispensed with, by use of two vector fields[LTZ00, Section 10]. For sake of keeping the exposition simpler, we shall not exploitthis substantial generalization. Assumptions (A.1) and (A.2) hold true for largeclasses of triples Ω,Γ0, Γ1 (even for the more demanding Neumann B.C. case):One canonical illustration where (A.1) = (3.1b) holds true in the Neumann caseis that Γ0 be flat: here then we can take d(x) = |x − x0|2, with x0 collocated onthe hyperplane containing Γ0 and outside Ω. Then h(x) = ∇d(x) = 2(x − x0) isradial. Another case where (A.1) = (3.1b) holds true is where Γ0 is either convexor concave and subtended by a common point; more precisely see [LTZ00, TheoremA.4.1, p. 301]; in which case, the corresponding required d( · ) can also be explicitly


constructed. For more examples, see illustrative configurations in the Neumann casein the Appendix. We also refer to the Appendix of [LTZ00] for other classes andmore details.

Pseudo-convex function. [LTZ00, p. 230] Having chosen, on the strength of as-sumption (A.1), a strictly convex potential function d(x) satisfying the preliminaryscaling condition minx∈Ω d(x) = m > 0, we next introduce the pseudo-convex func-tion ϕ(x, t) : Ω × R→ R of class C3 by setting

ϕ(x, t) = d(x)− c(t− T

2

)2

; x ∈ Ω, t ∈ [0, T ], (3.5a)

where T > T0, where T0 is defined by

T 20 ≡ 4 max

x∈Ωd(x). (3.5b)

Moreover, 0 < c < 1 is selected as follows: By (3.5b), there exists δ > 0 such that

T 2 > 4 maxx∈Ω

d(x) + 4δ, (3.5c)

and for this δ > 0, there exists a constant c, 0 < c < 1, such that

cT 2 > 4 maxx∈Ω

d(x) + 4δ. (3.5d)

Henceforth, let ϕ(x, t) be defined by (3.5a) with T and c chosen as describedabove, unless otherwise explicitly noted. It is easy to check that such function ϕ(x, t)has the following properties:

(a) For the constant δ > 0 fixed in (3.5c), we have via (3.5d),

ϕ(x, 0) ≡ ϕ(x, T ) = d(x)− c T2

4≤ max

x∈Ωd(x)− c T

2

4≤ −δ uniformly in x ∈ Ω;

(3.6a)

ϕ(x, t) ≤ ϕ(x,T

2

), for any t > 0 and any x ∈ Ω. (3.6b)

(b) There are t0 and t1, with 0 < t0 <T2< t1 < T , say, chosen symmetrically

about T2

, such that

minx∈Ω,t∈[t0,t1]

ϕ(x, t) ≥ σ, where 0 < σ < m = minx∈Ω

d(x), (3.7)

since ϕ(x, T

2

)= d(x) ≥ m > 0, under present choice. Moreover, let Q(σ) be the

subset of Ω × [0, T ] ≡ Q defined by

Q(σ) = (x, t) : ϕ(x, t) ≥ σ > 0, x ∈ Ω, 0 ≤ t ≤ T, (3.8)

-

6

x

tT

Q(σ)

T2

••t0

t1


We recall the following important property of Q(σ) [LTZ00, Eqn. (1.1.20), p. 232]which will be needed later

[t0, t1]×Ω ⊂ Q(σ) ⊂ [0, T ]×Ω. (3.9)

Carleman estimate for second-order hyperbolic equation (3.1a) at theH1 × L2-level. We next return to the main equation (3.1a), with the forcing termf and first order operator F satisfies (3.1b), (3.1c), (3.1d), at first without theimposition of boundary conditions. We shall consider initially solutions w(x, t) of(3.1a) in the class

w ∈ H2,2(Q) ≡ L2(0, T ;H2(Ω)) ∩H2(0, T ;L2(Ω)). (3.10)

For these solutions the following Carleman estimate was established in [LTZ00,Theorem 5.1]. To this end, the property σ > −δ, δ > 0, here achieved from (3.7), iscritically used [LTZ00, p. 261, in going from (6.5) to (6.6)].

Theorem 1 ([LTZ00, p. 255]). Assume the geometrical assumptions (A.1) and(A.2). Let ϕ(x, t) be defined by (3.5a). Assume (3.1b), (3.1c). Let w be a solutionof equation (3.1a) in the class (3.10). Then, the following one-parameter family ofestimates hold true, with ρ > 0 as in (3.3), β > 0 a suitable constant (β is positiveby virtue of (A.2) = (3.4)), for all τ > 0 sufficiently large and ε > 0 small:

BT |Σ + 2

∫ T

0

∫Ω

e2τϕ|f |2dQ+ C1,T e2τσ

∫ T

0

∫Ω

w2dQ

≥ C1,τ

∫Q

e2τϕ[w2t + |∇w|2]dQ

+ C2,τ

∫Q(σ)

e2τϕw2dxdt− cT τ3e−2τδ[Ew(0) + Ew(T )]. (3.11)

C1,τ = τερ− 2CT , C2,τ = 2τ3β + O(τ2)− 2CT . (3.12)

Here δ > 0, σ > 0 and σ > −δ are the constants in (3.6), (3.7), CT , cT and C1,T

are positive constants depending on T , as well as d (but not on τ). In particular,CT is the constant in (3.1d) depending on the L∞(Ω)-norm of the coefficients. Inaddition, the boundary terms BT |Σ are given explicitly by


BT |Σ = 2τ

∫ T

0

∫Γ

e2τϕ (w2t − |∇w|2

)h · νdΓdt

+8cτ

∫ T

0

∫Γ

e2τϕ

(t− T

2

)wt

∂w

∂νdΓdt

+4τ

∫ T

0

∫Γ

e2τϕ (h · ∇w)∂w

∂νdΓdt

+4τ2

∫ T

0

∫Γ

e2τϕ

[|h|2 − 4c2

(t− T

2

)2

+α

2τ

]w∂w

∂νdΓdt

+2τ

∫ T

0

∫Γ

e2τϕ

[2τ2

(|h|2 − 4c2

(t− T

2

)2)

+ τ(α−∆d− 2c)

]w2h · ν.dΓdt (3.13)

where h(x) = ∇d(x), α(x) = ∆d(x) − 2c − 1 + k for 0 < k < 1 a constant. Inwriting (3.13) we have deliberately not taken advantage of assumption (3.1b) in theNeumann case. This will be done in the sequel. Moreover, Q(σ) is as in (3.8). Theenergy function Ew(t) is defined as

Ew(t) =

∫Ω

[w2(x, t) + w2t (x, t) + |∇w(x, t)|2]dΩ. (3.14)

For what follows, it is important to recall also the following extension of theCarleman estimate (3.11) to finite energy solutions. To this end, we introduce thefollowing class of solutions for equation (3.1a), subject to assumptions (3.1b), (3.1c),see [LTZ00, Eqn. (8.1), p. 264]:

w ∈ H1,1(Q) = L2(0, T ;H1(Ω)) ∩H1(0, T ;L2(Ω));

wt,∂w

∂ν∈ L2(Σ) ≡ L2(0, T ;L2(Γ )).

(3.15a)

(3.15b)

As noted in [LTZ00, p. 264], the main difficulty in carrying out the extensionof Carleman estimate (3.11) to the class (3.15a-b) in the Neumann case is the factthat finite energy solutions subject to Neumann B.C. do not produce (in dimension≥ 2) H1-traces on the boundary [LT90], [LT91], [LT00b].

Theorem 2 ([LTZ00, Theorem 8.2, p. 266]). Assume the geometrical assump-tions (A.1), (A.2) and, moreover, (3.1b), (3.1c). Let w ∈ H2,2(Q) be a solutionof equation (3.1a) for which inequality (3.11) holds true, at least as guaranteed byTheorem 1. Let w be a solution of equation (3.1a) in the class defined by (3.15a-b).Then, the Carleman estimate (3.11) is satisfied by such solution w as well.

Remark 3. Theorems 1 and 2 have a perfect counterpart on the bounded set Ω ofa finite-dimensional Riemannian manifold with boundary ∂Ω [TY02]. This settingincludes, in particular, the case where the Laplacian (−∆) in (3.1a) is replacedby a uniformly strongly elliptic operator A(x, ∂) of the second order, with space-dependent coefficients of class C2. Here, because of space constraints, we must re-strict Eqn. (3.1a).


3.1.2 Some sharp interior and boundary regularity results formixed second-order hyperbolic problems of Dirichlet andNeumann type

In the following sections we recall from [LLT86], [LT90], [LT91] some sharp regularitytheory results for mixed type second-order hyperbolic problems. These results arecrucial in our efforts to achieve the minimal/optimal assumptions on the data forthe control and inverse problems in this notes.

3.1.3 Sharp regularity theory for second-order hyperbolicequations of Dirichlet type

Consider the second-order hyperbolic equation (3.1a) with non-homogeneous Dirich-let B.C g and initial data w0, w1:

wtt(x, t)−∆w(x, t) = F (w) + f(x, t) in Q = Ω × [0, T ];

w ( · , 0) = w0(x); wt ( · , 0) = w1(x) in Ω;

w(x, t)|Σ = g(x, t) in Σ = Γ × [0, T ].

(3.16a)

(3.16b)

(3.16c)

Here f(x, t) satisfies (3.1b) and F (w) satisfies (3.1c), (3.1d). For this initial boundaryvalue problem, the following interior and boundary regularity results hold true:

Theorem 3 ([LLT86, Theorem 2.1, p. 151]). Suppose we have the data

f ∈ L1(0, T ;L2(Ω)); w0 ∈ H1(Ω), w1 ∈ L2(Ω); g ∈ H1(Σ), (3.17)

with the compatibility condition g|t=0 = w0|Γ . Then the uniqueness solution of theproblem (3.16) satisfies

w,wt ∈ C([0, T ];H1(Ω)× L2(Ω));∂w

∂ν∈ L2(Σ). (3.18)

In addition, by increasing the regularity of the data and interpolation, we also havethe following more general result.

Theorem 4 ([LLT86, Remark 2.10, p. 167]). Suppose now the data satisfy (withm not necessarily an integer)

f ∈ L1(0, T ;Hm(Ω)), f (m) ∈ L1(0, T ;L2(Ω)); (3.19a)

w0 ∈ Hm+1(Ω), w1 ∈ Hm(Ω); g ∈ Hm+1(Σ), (3.19b)

with all compatibility conditions (trace coincidence) which make sense are satisfied.Then, we have the following regularity results

w ∈ C([0, T ];Hm+1(Ω)), w(m+1) ∈ C([0, T ];L2(Ω));∂w

∂ν∈ Hm(Σ). (3.20)


3.1.4 Sharp regularity theory for second-order hyperbolicequations of Neumann type

Next, we consider again the second-order hyperbolic equation (3.1a), however, thistime with non-homogeneous Neumann B.C g and I.C. w0, w1:

wtt(x, t)−∆w(x, t) = F (w) + f(x, t) in Q = Ω × [0, T ];w ( · , 0) = w0(x); wt ( · , 0) = w1(x) in Ω;∂w∂ν

(x, t)|Σ = g(x, t) in Σ = Γ × [0, T ].

(3.21a)(3.21b)(3.21c)

For this problem, we first define the parameters α and β to be the followingvalues:

α =3

5− ε, β =

3

5: for a general smooth, bounded domain Ω;

α = β =2

3: for a sphere domain Ω;

α = β =3

4− ε : for a parallelepiped domain Ω

(3.22a)(3.22b)(3.22c)

where ε > 0 is arbitrary. Then we have the following sharp regularity results:

Theorem 5 ([LT90, Theorem 1.2 (ii), (iii), 1.3, p.290]). With reference to thecorresponding mixed problem (3.21), the following regularity results hold true, withα and β defined above in (3.22).

I [LT91, Theorem 2.0, (2.6), (2.9), p.123; Theorem A, p.117; Theorem 2.1,p.124]. Suppose we have f = 0, w0, w1 ∈ H1(Ω) × L2(Ω) and g ∈ L2(Σ). Thenwe have the unique solution of (3.21) satisfies

w ∈ Hα(Q) = C([0, T ];Hα(Ω)) ∩Hα(0, T ;L2(Ω)); w|Σ ∈ H2α−1(Σ). (3.23)

II [LT91, Theorem 5.1, (5.4), (5.5), p.149; Theorem C, p.118; Theorem 7.1,p.158]. Suppose now f ∈ L2(Q), w0, w1 ∈ H1(Ω) × L2(Ω) and g = 0. Then wehave

w ∈ C([0, T ];H1(Ω)), wt ∈ C([0, T ];L2(Ω)); w|Σ ∈ Hβ(Σ). (3.24)

For smoother data, we have the following results, which are needed in the latersections.

Theorem 6 ([LT91, Theorem 3.1, p. 129]). With reference to the correspondingmixed problem (3.21), the following regularity results hold true, with α defined in(3.22).

I. Suppose f = 0, w0 = w1 = 0 and g ∈ H1(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12 (Γ ));

g(0) = 0. Then we have the unique solution of (3.21) satisfies

w ∈ C([0, T ];Hα+1(Ω)). (3.25)

II. If g ∈ H1(0, T ;L2(Γ )); g(0) = 0. Then

w ∈ C([0, T ];H32 (Ω)), wt ∈ C([0, T ];Hα(Ω)), wtt ∈ C([0, T ];Hα−1(Ω)). (3.26)


Theorem 7 ([LT91, Theorem 3.2, p. 132]). With reference to the correspondingmixed problem (3.21), the following regularity results hold true, with α defined in(3.22).

I. Suppose f = 0, w0 = w1 = 0 and g ∈ H2(0, T ;L2(Γ )) ∩ C([0, T ];Hα+ 12 (Γ ));

g(0) = g(0) = 0. Then we have the unique solution of (3.21) satisfies

w ∈ C([0, T ];Hα+2(Ω)). (3.27)

II. If g ∈ H2(0, T ;L2(Γ )), g ∈ C([0, T ];Hα− 12 (Γ )); g(0) = g(0) = 0. Then

wt ∈ C([0, T ];Hα+1(Ω)). (3.28)

III. If g ∈ H2(0, T ;L2(Γ )); g(0) = g(0) = 0. Then

wtt ∈ L2(0, T ;Hα(Ω)). (3.29)

Remark 4. Regarding the parameter α defined in (3.22), the result has been laterrefined in [Tat96] by obtains α = 2

3for a general domain, except for α = 3

4for a

parallelepiped ([LT90, Counterexample, p. 294] showed that α = 34

+ε is impossible).

Remark 5. In the models (3.16) and (3.21), the presence of the term F (w), whichsatisfies (3.1c), (3.1d), does not affect the invoked results here from [LLT86], [LT90],[LT91] as bounded perturbations.

3.1.5 Notes on Carleman estimates

The idea of introducing suitable exponential weights in estimates for solutions ofPDEs goes back to Carleman [Car39] in 1939, who used these estimates to obtainthe uniqueness in the Cauthy Problem in two variables. Such idea “dominated alllater work in the field” according to Hormander [Hor69, p.61]. We also refer to Taylor[Tay81, p.158]. Extended to a more general case, Carleman estimates of the form∑

|α|<m

τ2(m−|α|)−1

∫|Dαue2τϕ|2dx ≤ K

∫|Pue2τϕ|2dx (3.30)

were subsequently used by Hormander to prove the Unique Continuation Property(UCP): Suppose u solves the PDE P (x,D)u = 0 on a bounded domain Ω ⊂ Rn andsuppose u = 0 for ϕ(x) > 0, where ϕ : Ω → R is a smooth function with ∇ϕ 6= 0on ϕ = 0 (so that ϕ defines a smooth hyper surface in Ω). Does it follow then thatu = 0 on a neighborhood of ϕ = 0? More recently Isakov [Isa00] obtained similarresults as Hormander’s for more complicated operators with anisotropic principalsymbols. Examples include Schrodinger equations and Euler–Bernoulli models forplates, for which Hormander’s uniqueness result is ineffective.

All of the above results refer to solutions which are compactly supported so thatCarleman estimates involve no boundary terms. In areas such as control (continuousobservability inequalities, stabilization inequalities) as well as inverse theory forPDEs, a critical role is played precisely by the traces (restrictions) of the solutionson the boundary. The simple procedure of homogenizing the Cauchy data gives inthe estimates a RHS term which involves norms of boundary traces which are 1

2

derivative higher than the LHS-norm of u. Thus classical Carleman–type estimates,


which are a strong tool in proving unique continuation, do not give good resultswhen applied directly to boundary value problems. Control theory and inverse theoryrequire sharp Carleman–type estimates to boundary value problems, with optimalnorms for the RHS boundary traces.

From the point of view of the senior co–author of the present notes, two sourceswere instrumental in obtaining a first form of sharp Carleman–type inequalities forsolutions of boundary value problems: D. Tataru in his Ph.D. thesis [Tat92] (May1992) at the University of Virginia (from which some of the above informationis taken) as well as Lavrentev–Romanov–Shishataskii [LRS86] of the Novosibirskischool. These two contributions developed independently of each other. Tataru’swork published in [Tat94]–[Tat96] referred to a general evolution problem in pseudo-differential form and the resulting Carleman–type estimates included lower orderterms. It inspired the case-by-case treatment via differential multipliers to obtainexplicit Carleman estimates for second order hyperbolic equations in Lasiecka–Triggiani [LT97] and Schrodinger equations in Triggiani [Tri96] which were geo-metrically more refined but also contained lower order terms. Their counterpartsin the context of a Riemannian manifold, which includes the Euclidean domaincase where the Laplacian ∆ is replaced by an elliptic operator with variable (inspace) coefficients of modest regularity is given in Lasiecka–Triggiani–Yao [LTY97],[LTY99a] and Triggiani–Yao [TY99], respectively. The work of Lavrentev–Romanov–Shishataskii [LRS86] focused instead on second order hyperbolic equations andyielded first pointwise Carleman–type estimates, whose resulting integral form wasthen localized in the interior set Q(σ) in (3.8) and applied to H2,2(Q(σ))-solutions.Motivated by these two sources were the subsequent works of Lasiecka–Triggiani–Zhang [LTZ00] yielding the Carleman–type estimates first in point wise form andthen in the form of Theorem 1 in the case of general second order hyperbolic equa-tions, for H1,1(Q)-solutions, with explicit boundary terms on the lateral surfaces Σof the cylinder as in (3.13) and focus on the Neumann B.C. case. CorrespondingCarleman–type estimates for Schodinger equations were given by the same authorsin [LTZ04a] at the H1-level and in [LTZ04b] at the L2-level. The latter requireda pseudo-differential lift of topologies, inspired by [Tat92]. The corresponding Rie-mannian versions are then given in Triggiani–Yao [TY02] in the case of second orderhyperbolic equations on a Riemannian manifold and in Triggiani–Xu [TX07] in thecase of a Schrodinger equation on a Riemannian manifold, at the H1-level. All theseworks were motivated by, and directed to, control theory problems. The unique-ness of multidimensional inverse problems with a single boundary observation waspioneered by Bukhgeim and Klibanov [BK81] by means of Carleman–type estimates.

Carleman estimates (with lower order terms) for a general plate-model in aRiemannian manifold with challenging free boundary conditions (2nd and 3rd order)are given in [LTY99b]. The bibliography includes numerous other references usingCarleman–type inequalities for inverse problems.


3.2 Control theory results

3.2.1 Continuous observability and stabilization inequalities:Dirichlet and Neumann cases

In this section we present the following continuous observability inequality and reg-ularity estimates for second-order hyperbolic equation (3.1a). Such inequalities arenot only important for control theory (they imply the exact controllability of (3.1a)with Dirichlet or Neumann boundary control [GLL03], [LT89], [LT97], [LT00a]), butalso needed in our inverse problems presented in Section 3. In the more demandingNeumann case, the observability inequality is derived in [LTZ00, Section 9] as a con-sequence of the Carleman estimate [LTZ00, Theorem. 5.1, p. 255], here reproducedas Theorem 1 above. Proceeding analogously, we can also get the observability forthe Dirichlet case (see Theorem 8 below). We also refer, in the present context, to[Ho86], [Lio88], [Tri88]. The regularity estimates are standard and can be found, forexamples, in [LLT86, Theorem 2.1, p. 151], [LT81], [LT83].

3.2.2 Dirichlet case

We consider here again the Dirichlet problem (3.16), however, with the homogeneousDirichlet B.C. (g ≡ 0). Namely,

wtt(x, t)−∆w(x, t) = F (w) + f(x, t) in Q = Ω × [0, T ];w ( · , 0) = w0(x); wt ( · , 0) = w1(x) in Ω;w(x, t)|Σ = 0 in Σ = Γ × [0, T ].

(3.31a)(3.31b)(3.31c)

where f(x, t) satisfies (3.1b) and F (w) satisfies (3.1c), (3.1d). Moreover, let theinitial data w0, w1 satisfy

w0 ∈ H10 (Ω), w1 ∈ L2(Ω). (3.32)

Then, the unique solution w of (3.31) satisfies

w∈C([0, T ];H10 (Ω)) ∩ C1([0, T ];L2(Ω)), a-fortiori w∈H1,1(Q), continuously.

(3.33)

Theorem 8 (Counterpart of [LTZ00, Theorem. 9.2, p. 269], [LLT86], [LT81],[LT83]). Assume hypotheses (3.2a), (3.3), (3.4). For problem (3.31) with data asassumed in (3.1b), (3.1c), (3.1d) and (3.32), the following continuous observabil-ity/regularity inequalities hold true:

CT(‖w0‖2H1

0 (Ω) + ‖w1‖2L2(Ω)

)≤∫ T

0

∫Γ1

(∂w

∂ν

)2

dΓ1 dt+‖f‖2L2(Q), T > T0; (3.34)

∫ T

0

∫Γ1

(∂w

∂ν

)2

dΓ1 dt ≤ cT(‖w0‖2H1

0 (Ω) + ‖w1‖2L2(Ω)

)+ ‖f‖2L2(Q)), ∀T > 0. (3.35)

Here, (3.35) is a restatement of Theorem 3, Eqn (3.18), T0 is defined by (3.5b) forthe first inequality (3.34) (the second inequality (3.35) holds for all T > 0); Γ1 is thecontrolled or observed portion of the boundary, and cT , CT are positive constantsdepending on T .


Remark 6. The COI (3.34) may be interpreted also as follows: if problem (3.31a),(3.31c) has non-homogeneous forcing term f ∈ L2(Q) and Neumann boundary trace∂w∂ν|Σ1 ∈ L2(Σ1), then necessarily the I.C. w0, w1 must lie in H1

0 (Ω)×L2(Ω). Thiswill be used, for example, in connection with the (utt)-over-determined problem(3.219) in Section 3.5.

3.2.3 Neumann case

Now we consider the Neumann problem (3.21), however, with homogeneous Neu-mann B.C. (g ≡ 0). Namely,

wtt(x, t)−∆w(x, t) = F (w) + f(x, t) in Q = Ω × [0, T ];w ( · , 0) = w0(x); wt ( · , 0) = w1(x) in Ω;∂w

∂ν(x, t)|Σ = 0 in Σ = Γ × [0, T ],

(3.36a)(3.36b)(3.36c)

where again f(x, t) satisfies (3.1b), F (w) satisfies (3.1c), (3.1d) and the initial dataw0, w1 satisfy

w0 ∈ H1(Ω), w1 ∈ L2(Ω). (3.37)

Then, the unique solution w of (3.36) satisfies

w ∈ C([0, T ];H1(Ω)) ∩ C1([0, T ];L2(Ω)), a-fortiori w ∈ H1,1(Q), continuously.(3.38)

Theorem 9 ([LTZ00, Theorem 9.2, p. 269]). Assume hypotheses (3.2b), (3.3),(3.4). For problem (3.36) with data as assumed in (3.1b), (3.1c), (3.1d) and (3.37),the following continuous observability inequality holds true:

CT(‖w0‖2H1(Ω) + ‖w1‖2L2(Ω)

)≤∫ T

0

∫Γ1

[w2 +w2t ]dΓ1 dt+ ‖f‖2L2(Q), T > T0, (3.39)

whenever the right-hand side is finite. Here, T > T0, with T0 defined by (3.5b); Γ1,is the controlled or observed portion of the boundary, with Γ0 = Γ\Γ1 satisfying(3.2b), and CT > 0 is a positive constant depending on T .

Remark 7 (Counterpart of Remark 6). The COI (3.39) may be interpreted also asfollows: If problem (3.36a-c) has non-homogeneous forcing term f ∈ L2(Q), andDirichlet boundary traces w,wt ∈ L2(Σ1), then necessarily the I.C. w0, w1 mustlie in H1(Ω)×L2(Ω). This will be used, for example, in connection with the utt-over-determined problem (3.73) in Section 3.4.

As noted in [LTZ00, p. 269], the above Theorem 9 is critically based on [LT92a,Section 7.2] for a sharp trace theory result that expresses the tangential derivativein terms of the normal derivative and the boundary velocity, modulo interior lower-order terms. Its proof is by microlocal analysis. A counterpart with an energy levelterm (rather than lower-order terms) is given in [LTZ00, Lemma 8.1, p. 265].

Next we present a COI at the H1 × L2-level for a coupled hyperbolic equationswhich is need in Section 6. It is again a consequence of the Carleman estimate inTheorem 1 and can be seen as a generalized result of the above Theorem 9. Considerthe following coupled initial boundary value problem


ytt = ∆y + F1(ψ) + f1; ψtt = ∆ψ + F2(y) + f2 in Q; (3.40a)

y

(· , T

2

)= y0(x), yt

(· , T

2

)= y1(x) in Ω; (3.40b)

ψ

(· , T

2

)= ψ0(x), ψt

(· , T

2

)= ψ1(x) in Ω; (3.40c)

∂y

∂ν

∣∣∣∣Σ

= 0;∂ψ

∂ν

∣∣∣∣Σ

= 0 in Σ, (3.40d)

where f1, f2 ∈ L2(Q), the lower-order operators F1, F2 are defined similarly as F in(3.1c) and satisfy (3.1d) and the initial conditions satisfy

y0, y1, ψ0, ψ1 ∈ H1(Ω)× L2(Ω). (3.41)

Then, its solution satisfies

y, yt, ψ, ψt ∈ C([0, T ];H1(Ω)× L2(Ω)×H1(Ω)× L2(Ω)), (3.42)

a-fortiori y, ψ ∈ H1,1(Q)×H1,1(Q), continuously.

Theorem 10. Assume hypotheses (3.2b), (3.3), (3.4). For problem (3.9) with dataas assumed in (3.41), the following COI holds true:

CT(‖y0‖2H1(Ω) + ‖y1‖2L2(Ω) + ‖ψ0‖2H1(Ω) + ‖ψ1‖2L2(Ω)

)≤∫ T

0

∫Γ1

[y2 + y2t + ψ2 + ψ2

t ]dΓ1 dt+ ‖f1‖2L2(Q) + ‖f2‖2L2(Q), (3.43)

whenever the right-hand side is finite.

Remark 8. As in Remark 6, the COI (3.39) and (3.43) may be interpreted also asfollows: if problems (3.36), (3.40) have non-homogeneous forcing terms f , f1, f2 ∈L2(Q) and Dirichlet boundary traces w|Σ1 , y|Σ1 , ψ|Σ1 ∈ L2(Σ1), then necessarilythe I.C. w0, w1, y0, y1, ψ0, ψ1 must lie in H1(Ω)× L2(Ω). This will be used,for example, in connection with the over-determined problem (3.370) in Section 3.6.

3.3 Inverse theory results

This part constitutes the core of the present lecture notes. It is based on recentresults by the authors [LT11b] for Section 3.4 (single hyperbolic equation with non-homogeneous Neumann B.C.); [LT12] for Section 3.5 (single hyperbolic equationwith non-homogeneous Dirichlet B.C.); [LT11d] for Section 3.6 (a coupled systemof two hyperbolic equation with non-homogeneous Neumann B.C.); and [LT13] forSection 3.7 (recover two coefficients in one shot for single hyperbolic equation withnon-homogeneous Dirichlet B.C.). The common goal is to recover the damping coef-ficient by means of a single boundary measurement (observation). ‘Recovery’ meansboth (i) global uniqueness and (ii) Lipschitz stability, the two canonical issues ininverse problems. More precisely, in Section 3.4, we seek to determine the damp-ing coefficient of a single, second-order hyperbolic equation with non-homogeneous


Neumann B.C. by means of a single Dirichlet boundary observation on the (con-trolled or observed) part of the boundary Σ1; then in Section 3.5 we investigate asimilar problem of determining the damping coefficient by interchanging the roleof the B.C. and of the boundary observation: that is, non-homogeneous DirichletB.C. coupled with a Neumann boundary observation on Σ1. This change, however,will bring about distinct technical challenges in the problem of stability (see Section3.5.1). Then, in Section 3.6 we generalize the inverse problem to a coupled hyperbolicsystem with Neumann B.C. and consider the problem of determining two dampingcoefficients by a single Dirichlet boundary measurement. Finally, in Section 3.7 weshow one can actually recover both damping and potential coefficients in one shotwith a single boundary measurement (with more technicalities). We remark thatdue to space constraints, although all the inverse problems we consider in this partare mainly of the determination of the damping coefficients, the ideas in the proofwork equally well with the determination of potential coefficients, or damping andpotential coefficients together for coupled hyperbolic equations [LT11a], [LT11e].

A recent contribution by the authors is [LT], which offers several novel features.It seeks to recover (both uniqueness and stability) a space variable coefficient ofa third order (in time) physically significant PDE by means of just one boundarymeasurement. Such equation arises in high-intensity ultrasound, where the Fourierlaw is replaced by the Cattaneo law. The original third order equation, say in thesolution u, is first converted into a second order hyperbolic equation, say in the solu-tion z, by means of a mathematically natural change of variable. However, the newz-equation offers some novelties: the original variable u (depending on the parameterto be recovered) is present in both the initial conditions as well as in the internalforcing term to the z-equation. This makes the map from the parameter to be re-covered to the z-equation non-linear, unlike a traditional “linear inverse problem”,such as those discussed in these notes. The aforementioned pathology is a source ofadditional difficulties.

3.4 Inverse problems for second-order hyperbolicequations with non-homogeneous Neumann boundarydata: Global uniqueness and Lipschitz stability

3.4.1 Problem formulation I: The original hyperbolic problemsubject to an unknown damping coefficient q(x)

We consider the following second-order hyperbolic equation with damping

wtt(x, t) = ∆w(x, t) + q(x)wt(x, t) in Q = Ω × [0, T ];

w

(· , T

2

)= w0(x); wt

(· , T

2

)= w1(x) in Ω;

∂w(x, t)

∂ν

∣∣∣∣Σ

= g(x, t) in Σ = Γ × [0, T ].

(3.44a)

(3.44b)

(3.44c)

Given data: The initial data w0, w1, as well as the Neumann boundary termg are given in appropriate function spaces.


Unknown term: Instead, the time-independent damping coefficient q(x) ∈ L∞(Ω)is assumed to be unknown

The following two remarks apply also to all the formulations in the sequel andwill not be repeated again.

Remark 9. We could, in effect, add the terms r1(x, t) ·∇w(x, t), r2(x, t)w(x,t) on theRHS of (3.44a), with known coefficient ri(x, t), i = 1, 2 satisfying |r1|, r2 ∈ L∞(Q).The proofs remain essentially unchanged as the Carleman estimate (3.11) is derivedfor the more general second-order hyperbolic equation (3.1a).

Remark 10. In model (3.44) we regard t = T2

as initial time instead of t = 0. This isnot essential because the change of independent variable t→ t− T

2transforms t = T

2

to t = 0. However, this present choice is convenient in order to invoke the Carle-man estimates recalled in Section 3.1, which use the pseudo-convex function ϕ(x, t)in (3.5a), whose time-dependent term is centered around T

2. It is the consequent

property (3.6b) of such ϕ(x, t) that is later invoked in obtaining (3.85).

We shall denote by w(q) the solution of problem (3.44) due to the unknowndamping coefficient q (and the fixed data w0, w1, g). Notice that the map q → w(q)is nonlinear. Thus, this setting generates the following two nonlinear inverse problemissues.

I(1): Uniqueness in the nonlinear inverse problem for the w-system(3.44). In the above setting, let w = w(q) be a solution to (3.44). Does measurement(knowledge) of the Dirichlet boundary trace w(q)|Γ1×[0,T ] over the observed part Γ1

of the boundary and over a sufficiently long time T determine q uniquely, undersuitable geometrical conditions on the complementary unobserved part Γ0 = Γ\Γ1

of the boundary Γ = ∂Ω? In other words, if w(q) and w(p) denote the solutions ofproblem (3.44) due to the damping coefficients q(·) and p(·) (in L∞(Ω)) respectively,and common data w0, w1, h,

does w(q)|Γ1×[0,T ] = w(p)|Γ1×[0,T ] imply =⇒ q(x) ≡ p(x) a.e. in Ω? (3.45)

Remark 11. As in the exact controllability and uniform stabilization theories [GLL03,LT89, Tri88, Tri89], one expects that geometrical conditions be needed only in thecomplementary part Γ0 of that part Γ1 where measurement takes place.

Assuming that the answer to the aforementioned uniqueness question (3.45) is inthe affirmative, one then asks the following more demanding, quantitative stabilityquestion.

I(2): Stability in the nonlinear inverse problem for the w-system(3.44). In the above setting, let w(q), w(p) be solutions to (3.44) due to corre-sponding damping coefficients q(·) and p(·) (in L∞(Ω)) and fixed common dataw0, w1, g. Under geometric conditions on the complementary unobserved part ofthe boundary Γ0 = Γ\Γ1, is it possible to estimate the norm ‖q−p‖L2(Ω) of the dif-ference of the two damping coefficients by a suitable norm of the difference of theircorresponding Dirichlet boundary traces (measurements) (w(q)− w(p))|Γ1×[0,T ]?

II: The corresponding homogeneous problem. Next, we shall turn theabove inverse problems for the original w-system (3.44) into corresponding inverse


problems for a related (auxiliary) problem. This is a standard step. Define, in theabove setting

f(x) = q(x)− p(x); u(x, t) = w(q)(x, t)− w(p)(x, t); R(x, t) = wt(p)(x, t) (3.46)

Then, u(x, t) is readily seen to satisfy the following (homogeneous) mixed problem

utt(x, t)−∆u(x, t)− q(x)ut(x, t) = f(x)R(x, t) in Q = Ω × [0, T ];

u

(· , T

2

)= 0; ut

(· , T

2

)= 0 in Ω;

∂u(x, t)

∂ν

∣∣∣∣Σ

= 0 in Σ = Γ × [0, T ].

(3.47a)

(3.47b)

(3.47c)

Thus, in this setting (3.47) for the u-problem, we have that: the coefficient f(x) ∈L∞(Ω) in (3.47a) is assumed to be unknown while the function R is “suitable.”

The above serves only as a motivation. Henceforth, we shall consider the u-problem (3.47), under the following setting:

Given data: The coefficient q ∈ L∞(Ω) and the term R( · , · ) are given, subjectto appropriate assumptions.

Unknown term: The term f( · ) ∈ L∞(Ω) is assumed to be unknown.The u-problem (3.47) has three advantages over the w-problem (3.44): it has

homogeneous I.C. and B.C., and above all, the map from f → corresponding solutionu(f) is linear (with q, R fixed data). We then introduce the corresponding uniquenessand stability problems for the multidimensional hyperbolic u-system (3.47).

II(1): Uniqueness in the linear inverse problem for the u-system (3.47).In the above setting, let u = u(f) be a solution to (3.47). Does measurement (knowl-edge) of the Dirichlet boundary trace u(f)|Γ1×[0,T ] over the observed part Γ1 of theboundary and over a sufficiently long time T determine f uniquely under suitablegeometrical conditions on the unobserved part Γ0 = Γ\Γ1 of the boundary Γ = ∂Ω?In other words, in view of linearity,

does u(f)|Γ1×[0,T ] = 0 imply =⇒ f(x) = 0 a.e. on Ω? (3.48)

Assuming that the answer to the aforementioned uniqueness question (3.48) is inthe affirmative, one then asks the following more demanding, quantitative stabilityquestion.

II(2): Stability in the linear inverse problem for the u-system (3.47). Inthe above setting, let u = u(f) be a solution to (3.47). Under geometrical conditionson the unobserved portion of the boundary Γ0 = Γ\Γ1, is it possible to estimate thenorm ‖f‖L2(Ω) by a suitable norm of the corresponding Dirichlet boundary trace(measurement) u(f)|Γ1×[0,T ]?

3.4.2 Main results

Next we state the results of the above questions in the natural order of which they areproved. We begin with a uniqueness result for the linear inverse problem involvingthe u-system (3.47).


Theorem 11 (Uniqueness of linear inverse problem). Assume the preliminarygeometric assumptions (A.1): (3.2b), (3.3) and (A.2) = (3.4). Let

T > T0 ≡ 2√

maxx∈Ω

d(x). (3.49)

With reference to the u-problem (3.47), assume further the following regularity prop-erties on the fixed data q( · ) and R( · , · ) and unknown term f( · ):

R,Rt, Rtt ∈ L∞(Q), Rxi

(x,T

2

)∈ L∞(Ω), i = 1, · · · , n;

q ∈ L∞(Ω); f ∈ L2(Ω), (3.50)

as well as the following positivity property at the initial time T2

:∣∣∣∣R(x, T2)∣∣∣∣ ≥ r0 > 0, for some constant r0 and x ∈ Ω. (3.51)

If, moreover, the solution to problem (3.47) satisfies the additional homogeneousDirichlet boundary trace condition

u(f)(x, t) = 0, x ∈ Γ1, t ∈ [0, T ] (3.52)

over the observed part Γ1 of the boundary Γ and over the time interval T as in(3.49), then, in fact,

f(x) ≡ 0, a.e. x ∈ Ω. (3.53)

Next, we provide the stability result for the linear inverse problem involving theu-system (3.47), and the determination of the term f( · ) in (3.47a). We shall seek fin L2(Ω).

Theorem 12 (Lipschitz stability of linear inverse problem). Assume the pre-liminary geometric assumptions (A.1): (3.2b), (3.3), and (A.2) = (3.4). Considerproblem (3.47) on [0, T ] with T > T0, as in (3.49) and data as in (3.50), where,moreover, R satisfies the positivity condition (3.51) at the initial time t = T

2. Then

there exists a constant C = C(Ω, T, Γ1, ϕ, q, R) > 0, i.e., depending on the data ofproblem (3.47), but not on the unknown coefficient f , such that

‖f‖L2(Ω) ≤ C(‖ut(f)‖L2(Γ1×[0,T ]) + ‖utt(f)‖L2(Γ1×[0,T ])

), (3.54)

for all f ∈ L2(Ω). (More precisely, C depends on the L∞(Ω)-norm of q.)

Now we are ready to state the corresponding uniqueness and stability resultsfor the original nonlinear inverse problem with reference to the original w-problem(3.44).

Theorem 13 (Uniqueness of nonlinear inverse problem). Assume the pre-liminary geometric assumptions (A.1): (3.2b), (3.3), and (A.2) = (3.4). Let T > T0

as in (3.49). With reference to the w-problem (3.44), assume the following a-prioriregularity of two damping coefficients

q, p ∈Wm,∞(Ω). (3.55)


plus boundary Compatibility Conditions depending on dimΩ (corresponding to thoseof Proposition 6, Section 3.5.3, in the Dirichlet case). Let w(q) and w(p) denote thecorresponding solutions of problem (3.44). Assume further the following regularityproperties on the data:

(i)

w0, w1 ∈ H`+1(Ω)×H`(Ω), ` >dimΩ

2+ 2; |∇w1| ∈ L∞(Ω) (3.56)

(ii) g ∈ Hm(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12

+(m−1)(Γ )),

α = 23

for a general domain; α = 34

for a parallelepiped,(3.57)

with Compatibility Relations (C.R.) (trace coincidence which make sense) for

m >dimΩ

2+ 3− α. (3.58)

In addition, assume the following positivity condition on the initial velocity w1 in(3.44b) to match condition (3.51) on R(·, T

2):

|w1(x)| ≥ v1 > 0, for some constant v1 > 0 and x ∈ Ω. (3.59)

Finally, if w(q) and w(p) have the same Dirichlet boundary traces on Σ1:

w(q)(x, t) = w(p)(x, t), x ∈ Γ1, t ∈ [0, T ] (3.60)

over the observed part Γ1 of the boundary Γ and over the time interval T , as in(3.49), then, in fact, the two damping coefficients coincide

q(x) ≡ p(x), a.e. x ∈ Ω. (3.61)

See Remark 15 for appropriate comments, tailored to the Dirichlet case.

Theorem 14 (Lipschitz stability of nonlinear inverse problem). Assumepreliminary geometric assumptions (A.1): (3.2b), (3.3), and (A.2) = (3.4). Con-sider problem (3.44) on [0, T ], with T > T0 as in (3.49), and let w(q), w(p)denote the corresponding solutions, with damping coefficients q, p in Wm,∞(Ω),respectively, subject to the assumptions of Theorem 13. Assume further the regu-larity and positivity properties (3.56), (3.57), (3.59) on the data. Then, the fol-lowing stability result holds true for the w-problem (3.44): there exists a constantC = C(Ω, T, Γ1, ϕ, q, w0, w1, g) > 0, i.e., depending on the data of problem (3.44),and on the L∞(Ω)-norm of the coefficient q, such that

‖q − p‖L2(Ω) ≤ C(‖wt(q)− wt(p)‖L2(Γ1×[0,T ]) + ‖wtt(q)− wtt(p)‖L2(Γ1×[0,T ])

)(3.62)

for all such coefficients q, p (a consequence of the independence of the constant C in(3.54) on f , and of f = q−p by (3.46)). In particular, the constant C in (3.62) maybe thought of as dependent only on the radius M (arbitrarily large) of an L∞(Ω)-ball,for all coefficients q in such a ball, independently of the coefficients p ∈ L∞(Ω).


3.4.3 Proofs

In the present section,we provide the proofs for the above results Theorem 11–14.We begin with the proof of Theorem 11.

Uniqueness of linear inverse problem for the u-problem (3.47):Proof of Theorem 11

Orientation. Returning to the u-problem (3.47), we have at the outset utt(x,T2

) =f(x)R(x, T

2), x ∈ Ω, with R(x, T

2) satisfying assumption (3.51). Thus, to show that

f ≡ 0, one has to prove that utt(x,T2

) ≡ 0. This is the goal of the proof. It will beaccomplished in Eqn. (3.97) below.

Step 1. We return to the u-mixed problem (3.47) under the assumptions (3.50),supplemented by the required Dirichlet boundary observation u|Σ1 = 0 in (3.52).We thus obtain

utt(x, t)−∆u(x, t)− q(x)ut(x, t) = f(x)R(x, t) in Q; (3.63a)

u

(· , T

2

)= 0; ut

(· , T

2

)= 0 in Ω; (3.63b)

∂u

∂ν

∣∣∣∣Σ

= 0 in Σ; (3.63c)

u|Σ1 = 0 in Σ1. (3.63d)

We think of the solution u as dependent on the unknown term f : u = u(f).A-fortiori from assumptions (3.50) we obtain the standard result

u ∈ H1,1(Q) = L2(0, T ;H1(Ω)) ∩H1(0, T ;L2(Ω)). (3.64)

In view of (3.64) and because of the over-determined B.C. (3.63c-d), combinedwith the assumed property (3.2b) h ·ν = 0 on Γ0 (unobserved portion of the bound-ary), we readily see that Theorem 2 implies that such solution in (3.64) satisfies theCarleman estimate (3.11), with boundary terms BT |Σ = BTu|Σ in (3.13) that nowvanish:

BTu|Σ = 0. (3.65)

Thus, the solution u = u(f) of the over-determined problem (3.63) satisfies thefollowing special version of (3.11):

C1,τ

∫ T

0

∫Ω

e2τϕ[u2t + |∇u|2]dQ+ C2,τ

∫Q(σ)

e2τϕu2dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fR|2dQ+ CT,u e2τσ + cT τ

3e−2τδ[Eu(0) + Eu(T )]. (3.66)

where we have set via (3.11) for fixed u = u(f):

CT,u = C1,T

∫ T

0

∫Ω

u2dQ. (3.67)


Step 2. In this step, we differentiate in time the u-mixed problem (3.63), thusobtaining

(ut)tt(x, t)−∆(ut)(x, t)− q(x)(ut)t(x, t) = f(x)Rt(x, t) in Q;

(ut)

(· , T

2

)= 0; (ut)t

(· , T

2

)= f(x)R

(x,T

2

)in Ω;

∂(ut)

∂ν

∣∣∣∣Σ

= 0 in Σ;

(ut)|Σ1 = 0 in Σ1,

(3.68a)

(3.68b)

(3.68c)

(3.68d)

Here, by virtue of (3.50), (3.63a-b), one obtains

f(x)Rt(x, t) ∈ L∞(0, T ;L2(Ω)); utt

(x,T

2

)= f(x)R

(x,T

2

)∈ L2(Ω). (3.69)

Therefore, a-fortiori from (3.69), we have the following regularity for problem(3.68):

ut ∈ H1,1(Q) = L2(0, T ;H1(Ω)) ∩H1(0, T ;L2(Ω)). (3.70)

In view this time of (3.70), and because of the over-determined B.C. (3.68c-d),combined with the assumed property (3.2b) h · ν = 0 on Γ0 (unobserved portion ofthe boundary), we readily see again that Theorem 2 implies that such solution utin (3.70) satisfies the Carleman estimate (3.11) again with zero boundary terms

BTut |Σ = 0. (3.71)

Thus, such solution ut = ut(f) of the over-determined problem (3.68), satisfies thecounterpart of inequality (3.66), with u there replaced by ut now, thus yielding forall τ sufficiently large:

C1,τ

∫ T

0

∫Ω

e2τϕ[u2tt + |∇ut|2] dQ+ C2,τ

∫Q(σ)

e2τϕu2t dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fRt|2dQ+ CT,ut e2τσ + cT τ

3e−2τδ[Eut(0) + Eut(T )]. (3.72)

Step 3. In this step, we differentiate in time, once more, namely, the ut-mixedproblem (3.68), thus obtaining

(utt)tt(x, t)−∆(utt)(x, t)− q(x)(utt)t(x, t) = f(x)Rtt(x, t)in Q; (3.73a)

(utt)

(· , T

2

)=f(x)R

(x,T

2

)∈ L2(Ω)in Ω; (3.73b)

(utt)t

(· , T

2

)=f(x)

[q(x)R

(x,T

2

)+Rt

(x,T

2

)]∈ L2(Ω)in Ω; (3.73c)

∂(utt)

∂ν= 0in Σ; (3.73d)

(utt)|Σ1 = 0in Σ1. (3.73e)

Here, by virtue of (3.68a-b) and (3.50), one obtains


uttt

(x,T

2

)= ∆ut

*

(x,T

2

)+ q(x)utt

(x,T

2

)+ f(x)Rt

(x,T

2

)= q(x)f(x)R

(x,T

2

)+ f(x)Rt

(x,T

2

)∈ L2(Ω),

f(x)Rtt(x, t) ∈ L∞(0, T ;L2(Ω))

(3.74a)

(3.74b)

Similarly, again with f a-priori only in L2(Ω), and R(x, T2

) ∈ L∞(Ω), oneobtains–at a first glance at least–that utt(·, T2 ) = f(x)R(·, T

2) ∈ L2(Ω) only. If

this was so, one would only have utt ∈ C([0, T ];L2(Ω)), while application of theCarleman estimates in Theorem 2 on the (utt)-problem (3.73) requires H1,1(Q)-solutions. This would prevent us from justifying a further application of Carlemanestimates (3.11) to the (utt)-problem (3.73). However, as we shall see in the nextLemma 1, the special structure of the (utt)-over-determined problem will allow usto boost the regularity of f from the a-priori level f ∈ L2(Ω) to the enhanced level[f(x)R(x, T

2)] ∈ H1(Ω) first, and f ∈ H1(Ω) next.

Lemma 1. With reference to the (utt)-problem (3.73), with a-priori f ∈ L2(Ω),Rtt ∈ L∞(Q), q ∈ L∞(Ω), R(x, T

2), Rxi(x,

T2

) ∈ L∞(Ω) and under (3.51), we have

1

C2RT

2,r0

‖f(·)‖2H1(Ω) ≤∥∥∥∥f(·)R

(·, T

2

)∥∥∥∥2

H1(Ω)

≤ CT ‖fRtt‖2L2(Q) <∞. (3.75)

Proof. The inequality on the RHS of (3.75) is the COI (3.39), as applied to the over-determined problem (3.73), under present assumptions (after dropping the L2(Ω)-velocity term utt( · , T2 )), see Remark 7. We now show the inequality of the LHS of(3.75) (this was already noted in [IY01a, implication (6.21) → (6.30), p. 221]). Thisamounts to saying that R(·, T

2) is a multiplier H1 → H1. As a sufficient condition,

let∣∣R (x, T

2

)∣∣ ≥ r0 > 0, x ∈ Ω and Rxi(x,T2

) ∈ L∞(Ω) by assumption. Taking the

space derivative ∂xi of the identity f(x) =f(x)R(x,T

2)

R(x,T2

), we readily obtain

|∂xif(x)| ≤ CRT2

∣∣∣∣f(x)R

(x,T

2

)∣∣∣∣+

∣∣∣∣∂xi [f(x)R

(x,T

2

)]∣∣∣∣ ,from which the LHS of (3.75) follows.

Remark 12. The reason behind the process of differentiating the original u-problem(3.47) twice to obtain the (utt)-problem (3.73) is explained and justified in Remarkbelow. At any rate, while preserving the L∞(0, T ;L2(Ω))-regularity of the successiveright-hand side ‘forcing terms’ fR, fRt, fRtt (under present assumptions), passingfrom the u-problem to the (utt)-problem shifts the unknown term f from the RHSof (3.47a) to the initial conditions in (3.73b-c). This creates a convergence of aimswith the COI, which is then natural to invoke.

Henceforth, we proceed with the proof, having at our disposal the enhancedregularity from (3.73b), (3.75):

utt

(x,T

2

)∈ H1(Ω), f(x) ∈ H1(Ω), with Rxi

(x,T

2

)∈ L∞(Ω). (3.76)


We wish to establish that, indeed, f ≡ 0. Thus, under the gained regularity(3.76) and by virtue of (3.73b-c), we have the following regularity from problem(3.73): utt ∈ H1,1(Q). In view of this regularity and because of the over-determinedB.C. (3.73c-d), combined with the assumed property (3.2b) h · ν = 0 on Γ0, wereadily see again that Theorem 2 implies that such solution utt ∈ H1,1(Q) of (3.73)satisfies the Carleman estimate (3.11) again with boundary terms BTutt |Σ = 0.

BTutt |Σ = 0. (3.77)

Thus, such solution utt = utt(f) of the over-determined problem (3.73) satisfiesthe counterpart of inequality (3.66) for u, and (3.72) for ut, thus yielding for all τsufficiently large:

C1,τ

∫ T

0

∫Ω

e2τϕ[u2ttt + |∇utt|2]dQ+ C2,τ

∫Q(σ)

e2τϕu2tt dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fRtt|2dQ+ CT,utte2τσ + cT τ

3e−2τδ[Eutt(0) + Eutt(T )]. (3.78)

Step 4. Adding up (3.66), (3.72), and (3.78) together yields the combined in-equality

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt

]dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ

[|fR|2 + |fRt|2 + |fRtt|2

]dQ+ [CT,u + CT,ut + CT,utt ] e

2τσ

+ cT τ3e−2τδ

[Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )

].

(3.79)

Next, we invoke the property (3.50) for R,Rt, Rtt ∈ L∞(Q) (already used to claimproperties (3.66) for u, (3.72) for ut, (3.78) for utt):

|f(x)R(x, t)| ≤ CR|f(x)|; |f(x)Rt(x, t)| ≤ CRt |f(x)|;|f(x)Rtt(x, t)| ≤ CRtt |f(x)|, (3.80)

a.e. in Q, with CR = ‖R(x, t)‖L∞(Q), etc. Using (3.80) into the RHS of (3.79) yieldsfinally


C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt

]dx dt

≤ CR,T,u

∫Q

e2τϕ|f |2dQ+ e2τσ + τ3e−2τδ[Eu(0) + Eut(0)

+ Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

, (3.81)

where CR,T,u is a positive constant depending on R, T , and u (it combines CR, CRt ,CRtt as well as CT,u, CT,ut , CT,utt and cT ).

Step 5. In this step, we follow the strategy proposed in [Isa06, Theorem 8.2.2,p. 231]. We evaluate equation (3.47a) for u at the initial time t = T

2, use the vanishing

I.C. (3.47b), to obtain via hypothesis (3.51) on R(x, T

2

):∣∣∣∣utt(x, T2

)−∆u

*

(x,T

2

)− q(x)ut

*

(x,T

2

)∣∣∣∣ = |f(x)|∣∣∣∣R(x, T2

)∣∣∣∣ ≥ r0|f(x)|,

(3.82)with r0 > 0, x ∈ Ω. Hence, (3.82) allows one to obtain the pointwise inversion

|f(x)| ≤ 1

r0

∣∣∣∣utt(x, T2)∣∣∣∣ , x ∈ Ω. (3.83)

Claim: Using (3.83) in the first integral term∫Qe2τϕ|f |2dQ on the RHS of (3.81)

yields, as we shall see below, the following estimate for it:∫Q

e2τϕ|f |2dQ =

∫ T

0

∫Ω

e2τϕ|f |2dΩ dt ≤ T

r20

∫Ω

|utt(x, 0)|2dΩ

+T

r20

(2cτT + 1)

∫Ω

∫ T/2

0

e2τϕ(x,s)|utt(x, s)|2ds dΩ

+T

r20

∫Ω

∫ T/2

0

e2τϕ(x,s)|uttt(x, s)|2ds dΩ. (3.84)

Proof (Proof of (3.84)). By (3.83) and (3.6b), we preliminarily compute∫ T

0

∫Ω

e2τϕ|f |2dΩ dt

≤ 1

r20

∫ T

0

∫Ω

e2τϕ(x,t)

∣∣∣∣utt(x, T2)∣∣∣∣2 dΩ dt ≤ T

r20

∫Ω

e2τϕ(x,T2 )∣∣∣∣utt(x, T2

)∣∣∣∣2 dΩ=

T

r20

(∫Ω

∫ T2

0

d

ds

(e2τϕ(x,s)|utt(x, s)|2

)ds dΩ +

∫Ω

e2τϕ(x,0)|utt(x, 0)|2dΩ

).

(3.85)


We next perform the derivative in s of the integrand of (3.85), use ddsϕ(x, s) =

−2c(s− T

2

)from (3.5a) and obtain

∫Q

e2τϕ|f |2dQ ≤ T

r20

4cτ

∫Ω

∫ T2

0

(T

2− s)e2τϕ(x,s)|utt(x, s)|2ds dΩ

+ 2

∫Ω

∫ T2

0

e2τϕ(x,s)|utt(x, s)| |uttt(x, s)|ds dΩ +

∫Ω

e2τϕ(x,0)|utt(x, 0)|2dΩ.

(3.86)

Next, we recall that ϕ(x, 0) ≤ −δ by (3.6a) so that exp(2τϕ(x, 0)) ≤ 1 in thethird integral of (3.86), while obvious majorizations on the other two integrals of(3.86) yield∫

Q

e2τϕ|f |2dQ ≤(T

r0

)2

2cτ

∫Ω

∫ T2

0


+T

r20

∫Ω

∫ T2

0

e2τϕ(x,s) [|utt(x, s)|2 + |uttt(x, s)|2]ds dΩ

+T

r20

∫Ω

|utt(x, 0)|2dΩ, (3.87)

which is (3.84), as desired.

Step 6. We substitute (3.84) for the first integral term on the RHS of (3.81)and obtain, after obvious majorizations,

C1,τ

∫Q

e2τϕ [u2t+u2

tt+u2ttt+|∇u|2+|∇ut|2+|∇utt|2

]dQ

+C2,τ

∫Q(σ)

e2τϕ [u2+u2t+u2

tt

]dx dt

≤ CR,T,u

T

r20

[(2Tcτ+1)

∫Q

e2τϕ|utt|2dQ+

∫Q

e2τϕ|uttt|2dQ+

∫Ω

|utt(x, 0)|2dΩ]

+ e2τσ+τ3e−2τδ[Eu(0)+Eut(0)+Eutt(0)+Eu(T )+Eut(T )+Eutt(T )]

.

(3.88)

Remark 13. It is the term uttt on the RHS of estimate (3.87), which then occurs alsoon the RHS of estimate (3.88)—the price to pay in (3.73) to eliminate the unknownterm f in terms of the solution, from the RHS of (3.84)—that requires the need todifferentiate the original u-problem (3.47) to obtain the (utt)-problem (3.73). Thus,the Carleman estimate on the (utt)-problem also produces an uttt-term on the LHSof estimate (3.88) which eventually will absorb the uttt-term on the RHS of (3.88)and the resulting term will be dropped in Step 8 below.

Next, we recall that e2τϕ < e2τσ on Q\Q(σ) by (3.8), so that the followingestimate holds:

252 Shitao Liu and Roberto Triggiani∫Q

e2τϕ|utt|2dQ =

∫Q(σ)

e2τϕ|utt|2dt dx+

∫Q\Q(σ)

e2τϕ|utt|2dx dt

≤∫Q(σ)

e2τϕ|utt|2dt dx+ e2τσ

∫Q\Q(σ)

|utt|2dx dt (3.89)

Substituting (3.89) for the first integral term on the RHS of (3.88), we rewrite(3.88) as

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt

]dx dt

≤ CR,T,u,r0(2Tcτ + 1)

[∫Q(σ)

e2τϕu2ttdt dx+ e2τσku

]+ CR,T,uτ

3e−2τδ [Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

+ CR,T,ue2τσ + CR,T,u,r0

∫Q

e2τϕu2tttdQ+ CR,T,u,r0 ku, (3.90)

where we have set

CR,T,u,r0 = CR,T,u

(T

r20

); ku =

∫Q\Q(σ)

u2ttdx dt; ku =

∫Ω

|utt(x, 0)|2dΩ. (3.91)

Step 7. (Analysis of (3.90)) In (3.90) we shall use the following facts:

(a) the right-hand side term

CR,T,u,r0

∫Q

e2τϕu2tttdQ

can be absorbed by left-hand side term

C1,τ

∫Q

e2τϕ[u2t + u2

tt + u2ttt]dQ,

when τ is large enough, since C1,τ = τερ− 2CT by (3.12);

(b) the right-hand side term

CR,T,u,r0(2Tcτ + 1)

∫Q(σ)

e2τϕu2ttdt dx

can be absorbed by the left-hand term

C2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt,

when τ is large enough, since C2,τ = 2τ3β + O(τ2) − 2CT by (3.12), β > 0 as aconsequence of (A.2) = (3.4). Therefore, (3.90) becomes


C′1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C′2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ CR,T,u

e2τσ + τ3e−2τδ[Eu(0) + Eut(0) + Eutt(0) + Eu(T )

+Eut(T ) + Eutt(T )]

+ CR,T,u,r0

[ku(2Tcτ + 1)e2τσ + ku

], (3.92)

where

C′1,τ = C1,τ − CR,T,u,r0 = τερ− 2CT − CR,T,u,r0 (3.93a)

C′2,τ = C2,τ − CR,T,u,r0(2Tcτ + 1)

= 2τ3β + O(τ2)− 2CT − CR,T,u,r0(2Tcτ + 1), (3.93b)

and C′1,τ > 0, C′2,τ > 0 for all τ sufficiently large. Likewise, as limτ→∞

τ3e−2τδ = 0, we

can take τ sufficiently large, say ∀τ > some τ0, such that the quantity

CR,T,uτ3e−2τδ[Eu(0)+Eut(0)+Eutt(0)+Eu(T )+Eut(T )+Eutt(T )] ≤ CR,T,u,δ,τ0

is bounded by some constant which is independent of τ .

Step 8. We then return to inequality (3.92), drop here the first positive termand obtain for all τ > τ0:

C′2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ CR,T,u,r0

[ku(2Tcτ + 1)]e2τσ + ku

+ CR,T,ue

2τσ + CR,T,u,r0,δ,τ0

≤ Cdataτe2τσ, for all τ large enough > τ0, (3.94)

where Cdata is a constant depending on the data R, T, u, r0, δ, τ0, but not on τ .We note again that from the definition of Q(σ) in (3.8), we have e2τϕ ≥ e2τσ on

Q(σ). Thus, (3.94) implies

C′2,τe2τσ

∫Q(σ)

[u2 + u2t + u2

tt]dx dt ≤ Cdataτe2τσ. (3.95)

Dividing by τe2τσ on both sides of (3.95) yields

C′2,ττ

∫Q(σ)

[u2 + u2t + u2

tt]dx dt ≤ Cdata. (3.96)

But the expression of C′2,τ ∼ τ3 in (3.93b) shows that

C′2,ττ→∞ as τ →∞, hence (3.96) implies ⇒ u = ut = utt = 0 on Q(σ). (3.97)

Step 9. We now return to equation (3.47a), and use here (3.97) on Q(σ) toobtain


f(x)R(x, t) = utt(x, t)−∆u(x, t)− q(x)ut(x, t) ≡ 0, (x, t) ∈ Q(σ). (3.98)

We next invoke property (3.9) that [t0, t1]×Ω ⊂ Q(σ) ⊂ Q, and that T2∈ [t0, t1].

Thus, (3.98) in particular yields

f(x)R

(x,T

2

)≡ 0, for all x ∈ Ω, (3.99)

Finally, we recall from (3.51) that∣∣R (x, T

2

)∣∣ ≥ r0 > 0, x ∈ Ω, and thus obtain from(3.99)

f(x) ≡ 0 a.e. in Ω, (3.100)

as desired. Thus, with this step, the proof of Theorem 11 is complete.

Stability of linear inverse problem for the u-problem: Proof ofTheorem 12

Step 1. Let u(f) be the solution of problem (3.47), with data as in (3.50) and (3.51),viewed as a function of the unknown term f ∈ L2(Ω). We set

v = v(f) = ut(f). (3.101)

Then, v satisfies problem (3.68a-c), which we rewrite here for convenience:

vtt(x, t) = ∆v(x, t) + q(x)vt(x, t) + f(x)Rt(x, t) in Q; (3.102a)

v

(· , T

2

)= 0; vt

(· , T

2

)= f(x)R

(x,T

2

)in Ω; (3.102b)

∂v

∂ν(x, t)

∣∣∣∣Σ

= 0 in Σ, (3.102c)

so that f(x)R(x, T

2

)∈ L2(Ω). Accordingly, by linearity, we split v into two compo-

nents:v = ψ + z, (3.103)

where ψ satisfies the same problem as v, however, with homogeneous forcing term

ψtt(x, t) = ∆ψ(x, t) + q(x)ψt(x, t) in Q; (3.104a)

ψ

(· , T

2

)= 0; ψt

(· , T

2

)= f(x)R

(x,T

2

)in Ω; (3.104b)

∂ψ

∂ν(x, t)

∣∣∣∣Σ

= 0 in Σ, (3.104c)

while z satisfies the same problem as v, however, with homogeneous initial condi-tions:

ztt(x, t) = ∆z(x, t) + q(x)zt(x, t) + f(x)Rt(x, t) in Q; (3.105a)

z

(· , T

2

)= 0; zt

(· , T

2

)= 0 in Ω; (3.105b)

∂z

∂ν(x, t)

∣∣∣∣Σ

= 0 in Σ. (3.105c)


Step 2. Here we apply the COI (3.39), Theorem 9, to the ψ-problem (3.104),as all the assumptions in Theorem 9 are satisfied. Accordingly, there is a constantCT > 0 depending on T (and on the datum q) but not on f , such that∥∥∥∥f( · )R

(· , T

2

)∥∥∥∥2

L2(Ω)

≤ C2T

∫ T

0

∫Γ1

[ψ2 + ψ2t ]dΓ1 dt, (3.106)

whenever the RHS is finite, where T > T0 (see (3.49)), as assumed. Since∣∣R (x, T

2

)∣∣ ≥r0 > 0, x ∈ Ω by assumption (3.51), we then obtain from (3.106) by use of (3.103),the triangle inequality and (3.101), with CT,r0 = CT

r0:

‖f‖L2(Ω) ≤ CT,r0(‖ψ‖L2(Γ1×[0,T ]) + ‖ψt‖L2(Γ1×[0,T ])

)≤ CT,r0

(‖v − z‖L2(Γ1×[0,T ]) + ‖vt − zt‖L2(Γ1×[0,T ])

)≤ CT,r0

(‖v‖L2(Γ1×[0,T ]) + ‖vt‖L2(Γ1×[0,T ])

+ ‖z‖L2(Γ1×[0,T ]) + ‖zt‖L2(Γ1×[0,T ])

)≤ CT,r0

(‖ut‖L2(Γ1×[0,T ]) + ‖utt‖L2(Γ1×[0,T ])

+ ‖z‖L2(Γ1×[0,T ]) + ‖zt‖L2(Γ1×[0,T ])

). (3.107)

Inequality (3.107) is the desired, sought-after estimate (3.54) of Theorem 12, mod-ulo (polluted by) the z- and zt-terms. Such terms will be next absorbed by acompactness-uniqueness argument (for which we refer to the overview at the be-ginning of the notes for a historical view in control theory and in inverse problems).To carry this through, we need the following lemma.

Step 3. Lemma 2 Consider the z-system (3.105), with data

q ∈ L∞(Ω), f ∈ L2(Ω), Rt, Rtt ∈ L∞(Q); (3.108)

Define the following operators K and K1:

(Kf)(x, t) = z(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]); (3.109)

(K1f)(x, t) = zt(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]), (3.110)

where z is the unique solution of problem (3.105). Then,

both K and K1 are compact operators. (3.111)

Proof. Preliminaries. We shall invoke sharp (Dirichlet) trace theory results Theo-rem 5 II, Eqn. (3.24) for the Neumann hyperbolic problem (3.105). More precisely,regarding the z-problem (3.105), the following Dirichlet trace results hold true:

(a) Assumptions f(x) ∈ L2(Ω),Rt ∈ L∞(Q) as in (3.108) imply that f(x)Rt(x, t)∈ L2(Q), and then

f(x)Rt(x, t) ∈ L2(Q)⇒ z|Σ ∈ Hβ(Σ) continuously; (3.112)


(b) Similarly, assumptions f(x) ∈ L2(Ω) and Rt, Rtt ∈ L∞(Q) as in (3.108)imply f(x)Rtt(x, t) ∈ L2(Q), f(x)Rt(x, t) ∈ H1(0, T ;L2(Ω)), and then

f(x)Rt(x, t) ∈ H1(0, T ;L2(Ω))⇒ D1t z|Σ = zt|Σ ∈ Hβ(Σ), (3.113)

continuously with β the following constant (see (3.37)):

β=3

5, for a general Ω; β=

2

3, if Ω is a sphere; β=

3

4− ε, if Ω is parallelepiped.

(3.114)Then implication (3.113) is an immediate consequence of Theorem 5 II, Eqn.

(3.24) (as in implication (3.112)) for problem (3.105), as then the assumption in(b) implies f(x)Rtt(x, t) ∈ L2(Q), zt( · , T2 ) = 0, (zt)t( · , T2 ) ∈ L2(Ω), and then oneapplies the regularity (3.112) to zt, solution of problem (3.105), differentiated intime once.

After these preliminaries, we can now draw the desired conclusions on the com-pactness of the operators K and K1 defined in (3.109) and (3.110).

(c) Compactness of K. According to (3.112), it suffices to have Rt ∈ L∞(Q) inorder to have that the map

f ∈ L2(Ω)→ Kf |Σ = z|Σ ∈ Hβ−ε(Σ) is compact, (3.115)

∀ ε > 0 sufficiently small, for then f(x)Rt(t, x) ∈ L2(Q) as required, by (3.112).

Compactness of K1. According to (3.113), it suffices to have Rtt ∈ L∞(Q) inorder to have that the map

f ∈ L2(Ω)→ K1f |Σ = zt|Σ ∈ Hβ−ε(Σ) is compact, (3.116)

∀ ε > 0, sufficiently small, for then f(x)Rtt(t, x) ∈ L2(Q) as required by (3.113).Lemma 2 is proved.

Remark 14 (A more refined analysis). By interpolation between (3.112) and (3.113),one obtains, for 0 ≤ θ ≤ 1, still under the hypotheses (3.108) on f,Rt, Rtt:

f(x)Rt(x, t) ∈ Hθ(0, T ;L2(Ω))⇒ Dθt z∣∣∣Σ∈ Hβ(Σ), (3.117)

continuously. In particular, for θ = 1− β, (3.117) yields

f(x)Rt(x, t) ∈ H1−β(0, T ;L2(Ω))⇒ zt|Σ ∈ L2(Σ), (3.118)

continuously. Moreover, for θ = 1− β + ε, one obtains

f(x)Rt(x, t) ∈ H1−β+ε(0, T ;L2(Ω))⇒ zt|Σ ∈ Hε(Σ) (3.119)

continuously, for any ε > 0. In fact, one half (time version) of (3.119):

D1−β+εt z|Σ ∈ Hβ(0, T ;L2(Γ )); or Dtz|Σ = zt|Σ ∈ Hε(0, T ;L2(Γ )) (3.120)

follows from (3.117). Moreover, interpolating between (3.113) and (3.118) yields for0 ≤ θ ≤ 1:

f(x)Rt(x, t) ∈ H(1−θ)+(1−β)θ(0, T ;L2(Γ ))

⇒ zt ∈ L2(

0, T ;Hβ(1−θ)(Γ )). (3.121)


But, for (1− θ) + (1− β)θ = 1− β + ε, that is, for θ = (β − ε)/β, then β(1− θ) = ε,and thus (3.121) specializes to

f(x)Rt(x, t) ∈ H1−β+ε(0, T ;L2(Γ ))⇒ zt|Σ ∈ L2(0, T ;Hε(Γ )), (3.122)

which is the second half (space version) of (3.119).

Step 4. Lemma 2 will allow us to absorb the terms

‖Kf = z‖L2(Γ1×[0,T ]) and ‖K1f = zt‖L2(Γ1×[0,T ]) (3.123)

on the RHS of estimate (3.107), by a compactness-uniqueness argument.

Proposition 1. Consider the u-problem (3.47) with T > T0 as in (3.49) underassumption (3.50) which contains (3.108)) for its data q( · ), f( · ) and R( · , · ), withR satisfying also (3.51), so that both estimate (3.107), as well as Lemma 2 hold true.Then, the terms Kf = z|Σ1 and K1f = zt|Σ1 measured in the L2(Γ1 × [0, T ])-normcan be omitted from the RHS of inequality (3.107) (for a suitable constant CT,r0,...independent of the solution u), so that the desired conclusion, equation (3.54), ofTheorem 12 holds true:

‖f‖2L2(Ω) ≤ CT,data

∫ T

0

∫Γ1

[u2t + u2

tt]dΓ1 dt

, (3.124)

for all f ∈ L2(Ω), with CT,data independent of f .

Proof. Step (i). Suppose, by contradiction, that inequality (3.124) is false. Then,there exists a sequence fn∞n=1, fn ∈ L2(Ω), such that

(i) ‖fn‖L2(Ω) ≡ 1, n = 1, 2, . . . ;

(ii) limn→∞

(‖ut(fn)‖L2(Γ1×[0,T ]) + ‖utt(fn)‖L2(Γ1×[0,T ])

)= 0,

(3.125a)

(3.125b)

where u(fn) solves problem (3.47) with f = fn:

(u(fn))tt = ∆u(fn) + q(x)(u(fn))t + fn(x)R(x, t) in Q;

u(fn)

(· , T

2

)= 0; (u(fn))t

(· , T

2

)= 0 in Ω;

∂u(fn)(x, t)

∂ν

∣∣∣∣Σ

= 0 in Σ

(3.126a)

(3.126b)

(3.126c)

In view of (3.125a), there exists a subsequence, still denoted by fn, such that:

fn converges weakly in L2(Ω) to some f0 ∈ L2(Ω). (3.127)

Moreover, since the operators K and K1 are both compact by Lemma 2, it thenfollows by (3.127) that we have strong convergence [Bal76, Theorem 3.2.3, p. 86]:

limm,n→+∞

‖Kfn −Kfm‖L2(Γ1×[0,T ]) = 0; (3.128a)

limm,n→+∞

‖K1fn −K1fm‖L2(Γ1×[0,T ]) = 0. (3.128b)


Step (ii). On the other hand, since the map f → u(f) is linear, and recalling thedefinition of the operators K and K1 in (3.109), (3.110), it follows from estimate(3.107) that

‖fn − fm‖L2(Ω) ≤ CT,r0(‖ut(fn)− ut(fm)‖+ ‖utt(fn)− utt(fm)‖L2(Σ1))

+ CT,r0‖Kfn −Kfm‖+ CT,r0‖K1fn −K1fm‖L2(Σ1)

≤ CT,r0‖ut(fn)‖L2(Σ1) + CT,r0‖ut(fm)‖L2(Σ1)

+ CT,r0‖utt(fn)‖L2(Σ1) + CT,r0‖utt(fm)‖L2(Σ1)

+ CT,r0‖Kfn −Kfm‖L2(Σ1) + CT,r0‖K1fn −K1fm‖L2(Σ1)

(3.129)

CT,r0 = CTr0

. It then follows from (3.125b) and (3.128) as applied to the RHS of(3.129) that

limm,n→+∞

‖fn − fm‖L2(Ω) = 0. (3.130)

Therefore, fn is a Cauchy sequence in L2(Ω). By uniqueness of the limit, recall(3.127), it then follows that

limn→∞

‖fn − f0‖L2(Ω) = 0. (3.131)

Thus, in view of (3.125a), then (3.131) implies

‖f0‖L2(Ω) = 1. (3.132)

Step (iii). We now apply to the the u-problem (3.47) the same trace theory re-sults Theorem 5 II, Eqn. (3.24), that we have invoked in Lemma 2 for the z-problem(3.105) (If we replace f(x)Rt(x, t) by f(x)R(x, t), then the z-problem (3.105) be-comes the u-problem (3.47))); that is, as f ∈ L2(Ω), R,Rt ∈ L∞(Q) by assumption:

f(x)R(x, t) ∈ L2(Q) ⇒ u|Σ ∈ Hβ(Σ); (3.133)

f(x)R(x, t) ∈ H1(0, T ;L2(Ω)) ⇒ ut|Σ ∈ Hβ(Σ), (3.134)

continuously, hence by interpolation (counterpart of (3.118))

f(x)R(x, t) ∈ H1−β(0, T ;L2(Ω))⇒ ut|Σ ∈ L2(Σ). (3.135)

Here β is defined in (3.114) (see also (3.37)).

Step (iv). Thus, since R ∈ L∞(Q), we deduce from (3.133) thatf(x) ∈ L2(Ω)→ u(f)|Σ ∈ Hβ(Σ)

continuously, i.e., ‖u(f)|Σ‖Hβ(Σ) ≤ CR‖f‖L2(Ω),

(3.136a)

(3.136b)

with CR = ‖R‖L∞(Q), for then f(x)R(x, t) ∈ L2(Q), as required by (3.133).As the map f → u(f)|Σ is linear, it then follows in particular from (3.136b),

since fn, f0 ∈ L2(Ω)

‖|u(fn)|Σ1 − u(f0)|Σ1‖Hβ(Σ1) ≤ CR‖fn − f0‖L2(Ω). (3.137)


Recalling (3.131) on the RHS of (3.137), we conclude first that

limn→∞

‖u(fn)|Σ1 − u(f0)|Σ1‖Hβ(Σ1) = 0, (3.138)

next thatlimn→∞

‖u(fn)|Σ1 − u(f0)|Σ1‖C([0,T ];L2(Γ1)) = 0, (3.139)

since β > 12, so that Hβ(0, T ) embeds in C[0, T ].

Step (v). Similarly, from (3.134), where in addition, Rt ∈ L∞(Q), we deduce inaddition that

f(x) ∈ L2(Ω)⇒ ut|Σ ∈ Hβ(Σ) (3.140)

continuously, that is, in particular,

‖ut(f)‖Hβ(Σ1) ≤ C‖Rtf‖L2(Q) ≤ CR‖f‖L2(Ω) (3.141)

As the map f → u(f) is linear, it then follows from (3.141) that

‖ut(fn)|Σ1 − ut(f0)|Σ1‖Hβ(Σ1) ≤ CR‖fn − f0‖L2(Ω). (3.142)

Recalling (3.131) on the RHS of (3.142), we conclude first that

limn→∞

‖ut(fn)|Σ1 − ut(f0)|Σ1‖Hβ(Σ1) = 0, (3.143)

next thatlimn→∞

‖ut(fn)|Σ1 − ut(f0)|Σ1‖C([0,T ];L2(Γ1)) = 0, (3.144)

since β > 12, as in (3.139). Then, by virtue of (3.125b), combined with (3.144), we

obtain that

ut(f0)|Σ1 ≡ 0; or u(f0)|Σ1 ≡ function of x ∈ Γ1, constant in t ∈ [0, T ]. (3.145)

Step (vi). We return to problem (3.126): with fn ∈ L2(Ω) and data q ∈ L∞(Ω),R ∈ L∞(Q). We have again from Theorem 5 the following regularity results, con-tinuously:

u(fn), (u(fn))t ∈ C([0, T ];H1(Ω)× L2(Ω)); (3.146)

u(fn)|Σ ∈ Hβ(Σ), (3.147)

where the sharp trace regularity (3.147) is the same result noted in (3.112) and in(3.133), with β the constant in (3.114). As a consequence of (3.131), we also havevia (3.146), (3.147):

u(fn), (u(fn))t → u(f0), (u(f0))t in C([0, T ];H1(Ω)× L2(Ω)); (3.148)

u(fn)|Σ → u(f0)|Σ in Hβ(Σ). (3.149)

On the other hand, recalling the initial condition (3.126b), we have

u(fn)

(x,T

2

)≡ 0, x ∈ Ω, hence u(fn)

(x,T

2

)≡ 0, x ∈ Γ1, (3.150)

in the sense of trace in H12 (Γ1). Then (3.150) combined with (3.138), (3.139) yields

a-fortiori


u(f0)

(x,T

2

)≡ 0, x ∈ Γ1, (3.151)

and next, by virtue of (3.145), the desired conclusion,

u(f0)|Σ1 ≡ 0. (3.152)

Here, u(f0) satisfies weakly the limit problem, (3.131), (4.105), (4.106) appliedto (3.126):

utt(f0)−∆u(f0)− q(x)ut(f0) = f0(x)R(x, t) in Q;

u(f0)

(· , T

2

)= 0; ut(f0)

(· , T

2

)= 0 in Ω;

∂u(f0)

∂ν

∣∣∣∣Σ

= 0 and u(f0)|Σ1 = 0 in Σ, Σ1,

(3.153a)

(3.153b)

(3.153c)

via also (3.152), where

f0 ∈ L2(Ω); q ∈ L∞(Ω); R,Rt, Rtt ∈ L∞(Q); (3.154)

Rxi

(x,T

2

)∈ L∞(Ω), i = 1, · · · , n, (3.155)

by assumption (3.50). Moreover, (3.51) holds. Thus, the uniqueness Theorem 11applies and yields

f0(x) ≡ 0, a.e. x ∈ Ω. (3.156)

Then (3.156) contradicts (3.132). Thus, assumption (3.125) is false and inequal-ity (3.124) holds true. Proposition 1, as well as Theorem 12 are then established.

Uniqueness and stability for nonlinear inverse problem ofrecovering the damping coefficient: Proof of Theorems 13 and 14

Step 1. We return to the non-homogeneous w-problem (3.44) and split it, for con-venience, into two components:

w = w + w (3.157)

wtt(x, t) = ∆w(x, t) + q(x)wt(x, t) in Q; (3.158a)w

(· , T

2

)= w0(x); wt

(· , T

2

)= w1(x) in Ω; (3.158b)

∂w(x, t)

∂ν

∣∣∣∣Σ

= 0 in Σ. (3.158c)

wtt(x, t) = ∆w(x, t) + q(x)wt(x, t) in Q; (3.159a)

w

(· , T

2

)= 0; wt

(· , T

2

)= 0 in Ω; (3.159b)

∂w(x, t)

∂ν

∣∣∣∣Σ

= g(x, t) in Σ. (3.159c)


Step 2. We rewrite the w-problem (3.158) abstractly as

wtt = −AN w + q( · )wt; w

(T

2

)= w0, wt

(T

2

)= w1; (3.160)

d

dt

[w

wt

]=

[0 I

−AN q( · )

][w

wt

]= A

[w

wt

], (3.161)

where

ANh ≡ −∆h, D(AN ) =

h ∈ H2(Ω) :

∂h

∂ν

∣∣∣∣Γ

= 0

; (3.162)

A =

[0 I

−AN q( · )

]: E ⊃ D(A ) ≡ D(AN )×D(A

12N )→ E; (3.163)

E ≡ D(A12N )× L2(Ω); D(A

12N ) ≡ H1(Ω). (3.164)

For q ∈ L∞(Ω) as assumed, A is the generator of a s.c. group eA t on theenergy space E, as the perturbation [ 0

00

q( · ) ] is bounded on E, while A0 = [ 0−AN

I0],

D(A0) = D(A ), is the generator of a unitary group on E. In particular, as aconsequence, following regularity results hold true:

w0, w1 ∈ E ⇒ w, wt = eA t[w0, w1] ∈ C([0, T ];E); (3.165)

w0, w1 ∈ D(A ) ⇒ w, wt = eA t[w0, w1] ∈ C([0, T ]; D(A )). (3.166)

continuously. Moreover, we have, in general, for m = 0, 1, 2, . . . ,

D(A m) = D(A m0 ) = D

(Am+1

2N

)×D

(Am2N

)⊂ Hm+1(Ω)×Hm(Ω); (3.167)

w0, w1 ∈ D(A m) ⇒ w, wt = eA t[w0, w1] ∈ C([0, T ]; D(A m)); (3.168)

⇒ wt, wtt = A eA t[w0, w1] ∈ C([0, T ]; D(A m−1)); (3.169)

⇒ wtt, wttt=A 2eA t[w0, w1]∈C([0, T ]; D(A m−2)) (3.170)

continuously. As a matter of fact, the above relationships (3.167)–(4.170), hold truealso for m real positive. Henceforth, accordingly, m may be taken to be a real positivenumber, in order to get sharp/optimal results.

Orientation. Let w(q) = w(q) + w(q), w(p) = w(p) + w(p) be solutions ofproblem (3.44) [respectively, (3.157) and (3.158)] due to the damping coefficientsq(·) and p(·), respectively. By the change of variable as in (3.46),

f(x) ≡ q(x)− p(x); (3.171a)

u(x, t) = w(q)(x, t)− w(p)(x, t)

= [w(q)(x, t) + w(q)(x, t)]− [w(p)(x, t) + w(p)(x, t)]; (3.171b)

R(x, t) = wt(p)(x, t) = wt(p)(x, t) + wt(p)(x, t), (3.171c)

then the variable u satisfies problem (3.47), for which Theorem 11 and Theorem 12provide the corresponding uniqueness and stability results. We here seek to reduce


the nonlinear uniqueness and stability results for the original w-problem (3.44) tothe linear uniqueness and stability results for the u-problem (3.47). To this end, weneed to verify for the term R(x, t) = wt(p)(x, t) in (3.171c) the assumptions requiredin (3.50), (3.51) of Theorem 11 and Theorem 12. For this, it is expedient to splitw as in (3.157) and check them for wt(p) and wt(p) separately. Since q, p ∈ L∞(Ω)by assumption, we have f = q − p ∈ L∞(Ω) ⊂ L2(Ω) as required in (3.50). Thus,in order to be able invoke the uniqueness and stability results, Theorem 11 andTheorem 12, to the variable u = w(q) − w(p) in (3.171b), solving problem (3.47),what is left is to verify the regularity properties (3.50) on R defined by (3.171c),i.e., the following regularity properties:

wt(p), wtt(p), wttt(p) ∈ L∞(Q); (3.172)

wt(p), wtt(p), wttt(p) ∈ L∞(Q), (3.173)

as a consequence of suitably smooth I.C. w0, w1 in the w-problem (3.157) [withq(·) replaced by p(·)] and, respectively, of suitably smooth Neumann boundary termg in the w-problem (3.158) [again, with q(·) replaced by p(·)] along with Rxi(x,

T2

) =w1xi(x) ∈ L∞(Ω), i = 1, · · · , n, which is true by hypothesis, see (3.56). This programwill be accomplished in the following two steps.

Step 3. Proposition 2 With reference to the w-problem (3.157), with q ∈ L∞(Ω),let (with m non-necessarily integer):

w0, w1 ∈ D(A m), m >dimΩ

2+ 2. (3.174)

Then, the corresponding solution w(q) satisfies (a-fortiori) the following regular-ity properties:

wt(q), wtt(q), wttt(q) ∈ C([0, T ];Hm(Ω)×Hm−1(Ω)×Hm−2(Ω)), (3.175)

continuously, where, moreover, the following embedding holds:

Hm−2(Ω) → C(Ω) ⊂ L∞(Ω). (3.176)

A-fortiori, properties (3.172) are fulfilled (for q, rather than p):

wt(q), wtt(q), wttt(q) ∈ L∞(Q). (3.177)

Proof. Regularity (3.175) follows a-fortiori from (3.168), (3.167). Then, assumption(3.174) implies the embedding (3.176) [LM72, Corollary 9.1, p. 46], which along with(3.175) yields (3.177).

Step 4. Proposition 3 We return to the w-problem (3.158).(a) Under the following assumptions on the data:

q( · ) ∈ L∞(Ω); (3.178)


g ∈ Hm(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12

+(m−1)(Γ )),

α =2

3for a general domain; α =

3

4for a parallelepiped,

with Compatibility Relations (C.R.)

g

(T

2

)= g

(T

2

)= · · · = g(m−1)

(T

2

)= 0

(3.179)

(the regularity in (3.177) is a-fortiori implied via [LM72, Theorem 3.1, p.19] by

g ∈ Hm(2α−1),m(Σ) = L2(0, T ;Hm(2α−1)(Γ )) ∩Hm(0, T ;L2(Γ )), (3.180)

then the solution w = w(q) of problem (3.158) satisfies the following regularityproperty

w, wt, wtt, wttt∈ C([0, T ];Hα+m(Ω)×Hα+(m−1)(Ω)×Hα+(m−2)(Ω)×Hα+(m−3)(Ω).

(3.181)

continuously.(b) If, moreover,

m >dimΩ

2+ 3− α (3.182)

then a-fortiori, properties (3.173) are fulfilled (for q rather than p)

w(q), wt(q), wtt(q), wttt(q) ∈ L∞(Q). (3.183)

Proof. (a) The result in (a) relies critically on sharp regularity results Theorem 11,in terms of a parameter α, which was specified in (3.37): α = 2

5− ε for a general

domain, α = 23

for a sphere and certain other domains; α = 34

for a parallelepiped.More precisely,

Case m = 1. Let

g ∈ H1(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12 (Γ )), with C.R. g

(T

2

)= 0. (3.184)

Then Theorem 6 implies that

w, wt, wtt ∈ C([0, T ];Hα+1(Ω)×Hα(Ω)×Hα−1(Ω)), (3.185)

continuously. Equation (3.185) is result (a), (3.181), for m = 1, except for wttt.

Case m = 2. Let now

g ∈ H2(0, T ;L2(Γ )) ∩ C([0, T ];Hα+ 12 (Γ )), with C.R. g

(T

2

)= g

(T

2

)= 0,

(3.186)then Theorem 11 implies that

w, wt, wtt ∈ C([0, T ];Hα+2(Ω)×Hα+1(Ω)×Hα(Ω)

), (3.187)


continuously. Equation (3.186) is result (a), (3.181), for m = 2, except for wttt.

General case m. As noted in Theorem 11, the general case is similar and yieldsg as in (3.178)⇒

w, wt, wtt ∈ C(

[0, T ];Hα+m(Ω)×Hα+(m−1)(Ω)×Hα+(m−2)(Ω)

),

(3.188)

continuously, to which we add

wttt ∈ C([0, T ];Hα+(m−3)(Ω)), (3.189)

as the above theorems for the map g → w, wt, wtt (with zero I.C.) can be appliednow to the map gt → wt, wtt, wttt (still with zero I.C.), as q(·) is time-independent.Thus (3.181) is proved.

(b) If α+ (m− 3) > dimΩ2

, then from [LM72, Corollary 9.1, p. 96] the followingembedding holds

Hα+(m−3)(Ω) → C(Ω) ⊂ L∞(Ω), (3.190)

which, along with properties (3.181), yields (3.183) under (3.182).Step 5. Having verified properties (3.172): wt(p), wtt(p), wttt(p) ∈ L∞(Q) in

Proposition 2 and properties (3.173)): wt(p), wtt(p), wttt(p) ∈ L∞(Q) in Proposition3, it follows via (3.171c) that we have verified the properties

R,Rt, Rtt ∈ L∞(Q) (3.191)

for the u-problem (3.47), with u defined by (3.171b). Thus, Theorem 11 and Theorem12 apply and we then obtain the nonlinear uniqueness Theorem 13 and for f(x) =q(x) − p(x) and C depending on the data, in particular, on the L∞(Ω) norm of q,but not on f :

‖q − p‖L2(Ω) ≤ C(‖wt(q)− wt(p)‖L2(Γ1×[0,T ]) + ‖wtt(q)− wtt(p)‖L2(Γ1×[0,T ])

)(3.192)

for all p ∈ L∞(Ω). This is the desired conclusion of the nonlinear stability Theorem14.

3.4.4 Notes and literature

Section 3.4 is a streamlined improvement of [LT11b] specifically in the use of Lemma1 (which is nothing but the COI (3.39) as applied to the over-determined problem(3.73)) to obtain the enhanced regularity (3.76) for the utt-problem (3.73a-e). Whatfollows is a comparison between the present Section 3.4 and prior literature, takenfrom [LT11b].

Each of the results of the present section is either new or a definite improvement,just per se, over past literature. Cumulatively, the advances over the literature in-clude the following three areas:

(i) Treatment of a more general problem characterized in turn by three main fea-tures: (i1) recovery of damping coefficient, in one shot, corresponding to the energylevel term wt (rather than just potential coefficient, corresponding to the lower-order


term w): we refer to the enlightening introduction of [BCI01] to appreciate the seri-ous additional difficulties that arise while seeking to recover the damping coefficientas opposed to the potential coefficient, even in the study of just the uniqueness issue,and in the case of homogeneous Dirichlet boundary condition (B.C.), let alone in thecase of both uniqueness and stability with non-homogeneous Neumann B.C. of thepresent notes; (i2) presence of non-homogeneous B.C. g 6= 0 (as opposed to homo-geneous B.C. g = 0); (i3) of Neumann-type, which violates the uniform Lopatinskicondition (rather than Dirichlet type, an easier problem from the point of view ofthe regularity of the mixed problem).

(ii) Weaker geometrical conditions: that is, a smaller portion Γ1 of the boundaryis needed for the Dirichlet boundary measurement. A case in point (see Appendix):if Ω = 2-d disk, our results require the observed portion Γ1 of the boundary to justexceed 1

2(circumference)= 1

2(∂Ω)—in line with controllability/stabilization theories

[LT90], [Tri88]—while available literature (just in the homogeneous boundary caseh = 0) requires at least 3

4(circumference) [IY00], [IY03] or the entire boundary

[IY01b].(iii) Stronger analytical estimates: indeed, our stability estimates show a penal-

ization in H1 only in time, while available literature with h = 0 gives a penalizationin H1 both in time and space. The main result in the present notes, with g 6= 0, isin contrast, altogether new.

Our overall proofs rely on three main ingredients (see Section 3.1):(a) Sharp Carleman estimates at the H1 ×L2-level for second-order hyperbolic

equations as obtained in [LTZ00]; sharpness is at least on three fronts: (a1) reducedgeometrical requirements of Ω,Γ0, Γ1; (a2) reduced regularity assumptions to aminimal level of the solutions claimed to satisfy the Carleman estimates: this issueof reduced regularity is the key issue tacked in paper [BCI01] for uniqueness of thedamping coefficient in the Dirichlet case; (a3) Carleman estimates on the entirecylinder Ω × [0, T ], not only on Q(σ), as in [LRS86], [BCI01], etc, and with explicitboundary terms on the lateral surface ∂Ω × [0, T ] of this cylinder.

(b) A correspondingly implied continuous observability inequality at the sameenergy level H1 × L2, which penalizes the boundary trace of the solution in H1

only in time, but avoids H1-penalization in space, by critical use of the sharp tracetheory result in [LT92a, Section 7.2].

(c) Sharp interior and boundary regularity theory of mixed problems for second-order hyperbolic equations in a general bounded domain with Neumann boundarydata [LT90], [LT91], [LT94], [LT00a, Ch. 8, Section 8A, p. 755], [Tat96].

An additional important ingredient is that, in the linear uniqueness proof, ittakes advantage of a new, convenient tactical shortcut “post-Carleman estimates”to be found in [Isa06, Theorem 8.2.2, p. 231] which was not available in [Isa98], wherea different route “post-Carleman estimates” was followed. The present treatment ofthe two issues—“uniqueness” and “stability”—of, say, the linear inverse problemsshow a sort of parallelism with the control-theoretic issues of “observability” and“continuous observability,” respectively as pursued e.g., in [LT89], [Tri88], [LTZ00],or [GLL03]. The latter continuous observability inequality is critically used in bothour uniqueness and stability proofs. Another shared technical difficulty which arisesin this context between the stability estimates in inverse problems and the continu-ous observability inequality in control theory is the need to employ a compactness-uniqueness argument, for absorbing lower-order terms, though the case of the sta-bility estimate presents issues of its own.


Compactness-uniqueness arguments in control theory have been employed quitefrequently since the late 80’s, see e.g., [Lit87], [LT89], not only for hyperbolic equa-tions, but also for plate equations, e.g., [LT00b], and for Schrodinger equations, e.g.,[LT92b]; and not only in the linear case but also in the nonlinear setting [LT93],[LT06]. Apparently, the first use of a compactness-uniqueness argument in the con-text of inverse problems occurs in [Yam99].

3.5 Inverse problems for second-order hyperbolicequations with non-homogeneous Dirichlet boundarydata: Global uniqueness and Lipschitz stability

3.5.1 Problem formulation

I: The original hyperbolic problem subject to an unknown damping co-efficient q(x). On Ω we consider the following second-order hyperbolic equation

wtt(x, t) = ∆w(x, t) + q(x)wt(x, t) in Q = Ω × [0, T ];

w

(· , T

2

)= w0(x); wt

(· , T

2

)= w1(x) in Ω;

w(x, t)|Σ = g(x, t) in Σ = Γ × [0, T ].

(3.193a)

(3.193b)

(3.193c)

Given data: The initial conditions w0, w1, as well as the Dirichlet boundaryterm g are given in appropriate function spaces.

Unknown term: Instead, the damping coefficient q(x) ∈ L∞(Ω) is assumed tobe unknown.

We shall denote by w(q) the solution of problem (3.193) due to the dampingcoefficient q (and the fixed data w0, w1, g). The map q → w(q) is nonlinear. Thus,this setting generates the following two nonlinear inverse problem issues.

I(1): Uniqueness in the nonlinear inverse problem for the w-system(3.193). In the above setting, let w = w(q) be a solution to (3.193). Does mea-

surement (knowledge) of the Neumann boundary trace ∂w(q)∂ν|Γ1×[0,T ] over the ob-

served part Γ1 of the boundary and over a sufficiently long time T determine quniquely, under suitable geometrical conditions on the complementary unobservedpart Γ0 = Γ\Γ1 of the boundary Γ = ∂Ω? In other words, if w(q) and w(p) denotethe solutions of problem (3.193) due to the damping coefficients q(·) and p(·) (inL∞(Ω)) respectively, and common data w0, w1, g

does∂w(q)

∂ν

∣∣∣∣Γ1×[0,T ]

=∂w(p)

∂ν

∣∣∣∣Γ1×[0,T ]

imply =⇒ q(x) ≡ p(x) a.e. in Ω? (3.194)

Assuming that the answer to the uniqueness question (3.194) is in the affirmative,one then asks the following more demanding, quantitative stability question.

I(2): Stability in the nonlinear inverse problem for the w-system(3.193). In the above setting, let w(q), w(p) be solutions to (3.193) due to cor-responding damping coefficients q(·) and p(·) in (3.193a) (in L∞(Ω)) and fixed


common data w0, w1, g. Under geometric conditions on the complementary un-observed part of the boundary Γ0 = Γ\Γ1, is it possible to estimate a suitablenorm ‖q − p‖ of the difference of the two damping coefficients by a suitable normof the difference of their corresponding Neumann boundary traces (measurements)

( ∂w(q)∂ν− ∂w(p)

∂ν)|Γ1×[0,T ]?

II: The corresponding homogeneous problem. Next, as in Section 3.4,we shall turn the above inverse problems for the original w-system (5.193) intocorresponding inverse problems for a related (auxiliary) problem. Define, in theabove setting, as in (3.46)

f(x) = q(x)− p(x); u(x, t) = w(q)(x, t)− w(p)(x, t); R(x, t) = wt(p)(x, t) (3.195)

Then, u(x, t) is readily seen to satisfy the following (homogeneous) mixed prob-lem

utt(x, t)−∆u(x, t)− q(x)ut(x, t) = f(x)R(x, t) in Q = Ω × [0, T ];

u

(· , T

2

)= 0; ut

(· , T

2

)= 0 in Ω;

u(x, t)|Σ = 0 in Σ = Γ × [0, T ].

(3.196a)

(3.196b)

(3.196c)

Thus, in this setting (3.196) for the u-problem, we have that: the coefficientf(x) ∈ L∞(Ω) in (3.196a) is assumed to be unknown while the function R is “suit-able.” The above serves again as a motivation. Henceforth, we shall consider theu-problem (3.196), under the following setting:

Given data: The coefficient q ∈ L∞(Ω) and the term R( · , · ) are given, subjectto appropriate assumptions.

Unknown term: The term f( · ) ∈ L∞(Ω) is assumed to be unknown.The u-problem (3.196) has three advantages over the w-problem (3.193): it has

homogeneous I.C. and B.C., above all, the map from f → corresponding solutionu(f) is linear (with q, R fixed data). We then introduce the corresponding uniquenessand stability problems for the multidimensional hyperbolic u-system (3.196).

II(1): Uniqueness in the linear inverse problem for the u-system(3.196). In the above setting, let u = u(f) be a solution to (3.196). Does measure-

ment (knowledge) of the Neumann boundary trace ∂u(f)∂ν|Γ1×[0,T ] over the observed

part Γ1 of the boundary and over a sufficiently long time T determine f uniquelyunder suitable geometrical conditions on the unobserved part Γ0 = Γ\Γ1 of theboundary Γ = ∂Ω? In other words, in view of linearity,

does∂u(f)

∂ν

∣∣∣∣Γ1×[0,T ]

= 0 imply =⇒ f(x) = 0 a.e. on Ω? (3.197)

Assuming that the answer to the uniqueness question (3.197) is in the affirmative,one then asks the following more demanding, quantitative stability question.

II(2): Stability in the linear inverse problem for the u-system (3.196).In the above setting, let u = u(f) be a solution to (3.196). Under geometricalconditions on the unobserved portion of the boundary Γ0 = Γ\Γ1, is it possibleto estimate suitable norm ‖f‖ by a suitable norm of the corresponding Neumann

boundary trace (measurement) ∂u(f)∂ν|Γ1×[0,T ]?


3.5.2 Main results

Next we give affirmative and quantitative answers to the above uniqueness andstability questions for the w-problem (3.193), as well as the u-problem (3.196) inthe following theorems in the order in which they are proved below.

Uniqueness results. We begin with the uniqueness result for the linear inverseproblem involving the u-system (3.196).

Theorem 15 (Uniqueness of linear inverse problem). Assume the preliminarygeometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Let T > T0 be asin (3.49) With reference to the u-problem (3.196), assume further the followingregularity properties on the fixed data q( · ) and R( · , · ) and unknown term f( · ):

q ∈ L∞(Ω); R,Rt, Rtt ∈ L∞(Q), R

(x,T

2

)∈W 1,∞(Ω), f ∈ L2(Ω), (3.198)

as well as the following positivity property at the initial time T2

:∣∣∣∣R(x, T2)∣∣∣∣ ≥ r0 > 0, for some constant r0 and x ∈ Ω (3.199)

[same as in (3.40), (3.51)]. If the solution to problem (3.196) satisfies the additionalhomogeneous Neumann boundary trace condition

∂u(f)

∂ν= 0, x ∈ Γ1, t ∈ [0, T ], (3.200)


f(x) ≡ 0, a.e. x ∈ Ω. (3.201)

Stability results. Next, we provide the stability result for the linear inverse prob-lem involving the u-system (3.196), and the determination of the term f( · ) in(3.196a). We shall seek f in Hθ

0 (Ω) for 0 < θ ≤ 1, θ 6= 12.

Theorem 16 (Lipschitz stability of linear inverse problem). Assume the pre-liminary geometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Considerproblem (3.196) on [0, T ] with T > T0, as in (3.49), f ∈ Hθ

0 (Ω) for some fixedbut otherwise arbitrary θ, 0 < θ ≤ 1, θ 6= 1

2where of course Hθ

0 (Ω) = Hθ(Ω),0 < θ < 1

2[LM72, p. 55] and data

q ∈ L∞(Ω); R,Rt, Rtt ∈ L∞(Q);

Rt ∈ H2θ(0, T ;W θ,∞(Ω)), R

(x,T

2

)∈W 1,∞(Ω), (3.202)

where, moreover, R satisfies the positivity condition (3.199) at the initial time t = T2

.Then there exist constants C = C(Ω, T, Γ1, ϕ, q, R, θ) > 0 and likewise constantc > 0, i.e., depending on the data of problem (3.196), but not on the unknowncoefficient f , such that

c

∥∥∥∥∂ut(f)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

≤ ‖f‖Hθ0 (Ω) ≤ C∥∥∥∥∂ut(f)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

, (3.203)

for all f ∈ Hθ0 (Ω), 0 < θ ≤ 1, θ 6= 1

2fixed (More precisely, C, c depend on the

L∞(Ω)-norm of q). [See Remark 18 why θ = 0 is excluded.]


We next give the corresponding uniqueness result to the nonlinear inverse prob-lem involving the determination of the damping coefficient q( · ) in the w-problem(3.193).

Theorem 17 (Uniqueness of nonlinear inverse problem). Assume the pre-liminary geometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Let T > T0

as in (3.49). With reference to the w-problem (3.193), assume the following a-prioriregularity of two damping coefficients (in (3.193a))

q, p ∈Wm,∞(Ω). (3.204)

plus boundary Compatibility Conditions depending on dimΩ. For dimΩ = 2, 3, theseare identified in the proof of Proposition 6 in Section 3.5.3. Let w(q) and w(p)denote the corresponding solutions of problem (3.193). Assume further the followingregularity properties on the initial and boundary data

w0, w1 ∈ Hm+1(Ω)×Hm(Ω), m >dimΩ

2+ 2, |∇w1| ∈ L∞(Ω), (3.205)

g ∈ Hm+1(Σ), (3.206)

along with all Compatibility Conditions (trace coincidence) which make sense. Letthe initial velocity w1 in (3.193b) satisfy the following positivity condition

|w1(x)| ≥ v1 > 0, for some constant v1 > 0 and x ∈ Ω. (3.207)

Finally, if w(q) and w(p) have the same Neumann boundary traces on Σ1:

∂w(q)

∂ν(x, t) =

∂w(p)

∂ν(x, t), x ∈ Γ1, t ∈ [0, T ], (3.208)

over the observed part Γ1 of the boundary Γ and over the time interval T , as in(3.49), then, in fact, the two damping coefficients coincide

q(x) ≡ p(x), a.e. x ∈ Ω. (3.209)

Finally, we state the stability result for the nonlinear inverse problem involvingthe w-problem (3.193) with damping coefficient q( · ).

Theorem 18 (Lipschitz stability of nonlinear inverse problem). Assume pre-liminary geometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Consider prob-lem (3.193) on [0, T ], with T > T0 as in (3.49), one time with damping coefficientq ∈Wm,∞(Ω), and one time with damping coefficient p ∈Wm,∞(Ω), subject to theassumptions of Theorem 17, and let w(q), w(p) denote the corresponding solutions.Assume further the properties (3.205) and (3.206)) on the data and on m. Then, thefollowing stability result holds true for the w-problem (3.193): there exists constantC = C(Ω, T, Γ1, ϕ, q, w0, w1, g) > 0 and likewise c > 0 depending on the data of theproblem (3.193) and on the L∞(Ω)-norm of the coefficient q such that

c

∥∥∥∥∂wt(q)∂ν− ∂wt(p)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

≤ ‖q − p‖Hθ(Ω)

≤ C∥∥∥∥∂wt(q)∂ν

− ∂wt(p)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

, (3.210)


for all such coefficients q, p with 0 < θ < min

12,m−1− dimΩ

23

. In particular, the

constant C in (3.210) may be thought of as dependent only on the radius M (arbi-trarily large) of an L∞(Ω)-ball, for all coefficients q in such a ball, independently ofthe coefficients p ∈ L∞(Ω).

Remark 15. The sufficient condition R(x, T2

) ∈W 1,∞(Ω) in (3.198) of Theorem 15–as well as in (3.202) of Theorem 16–could be replaced by the weaker assumption:“R(x, T

2)

is a multiplier H1(Ω)→ H1(Ω),” or that R(x, T2

) ∈M(H1(Ω)→ H1(Ω)), of which[MS85, Theorem 1, m = l = 1, p=2, p.243] provides a characterization. Similarly,in (2.205) of Theorem 17, the sufficient condition |∇w1| ∈ L∞(Ω) could be replacedby the weaker w1 ∈ M(H1(Ω) → H1(Ω)). In addition, Theorem 17 also requiresadditional conditions on the coefficient q, which can be precise and explicit. They in-volve two aspects: (i) suitable regularity assumptions on q, which can be expressed interms of the coefficient itself as belonging to appropriate multipliers spaces [MS85],such as q belongs to M(Hm(Ω)→ Hm(Ω)); as well as (ii) suitable boundary com-patibility conditions, depending on dim Ω. A checkable sufficient condition on theregularity requirements (i) is: q belong to Wm,∞(Ω). A most direct condition onthe boundary compatibility requirement (ii) for any dimension of Ω is that q hascompact support on Ω. More specifically, for dim Ω = 2, the only C.C. is ∂q

∂ν= 0

on Γ . For dim Ω = 3, these conditions have to be supplemented by ∂(∆q)∂ν

= 0 onΓ ; ∇(qxi) = 0. Details are provided in Proposition 6 in Section 3.5.3 (as well as inProposition 20 in Section 3.7.5) after Proposition 6.1 of [LT13]. All these conditionsare needed to guarantee that the solution w, wt, wtt, wttt have the same regularityas R, Rt, Rtt, Rttt as assumed in Theorem 15 and 16.

3.5.3 Proofs

Uniqueness of linear inverse problem for the u-dynamics (3.196):Proof of Theorem 15

Orientation. Returning to the u-problem (3.196), we have at the outset thatutt(x,

T2

) = f(x)R(x, T2

), x ∈ Ω, with R(x, T2

) satisfying assumption (3.199). Thus,as in Theorem 11, to show that f ≡ 0, one has to prove that utt(x,

T2

) ≡ 0. This isthe goal of the proof. It will be accomplished in Eqn. (3.239) below.

Step 1. We return to the u-mixed problem (3.196a-c) under the assumptionsdata q∈L∞(Ω), R,Rt, Rtt∈L∞(Q) and R

(x,T

2

)∈W 1,∞(Ω)

unknown term f(·) ∈ L2(Ω),

(3.211a)

(3.211b)

as per assumption (3.198), supplemented by the required Neumann boundary ob-servation ∂u

∂ν|Σ1 = 0 in (3.200). We thus obtain

utt(x, t)−∆u(x, t)− q(x)ut(x, t) = f(x)R(x, t) in Q;

u

(· , T

2

)= 0; ut

(· , T

2

)= 0 in Ω;

u|Σ = 0,∂u

∂ν

∣∣Σ1

= 0 in Σ, Σ1.

(3.212a)

(3.212b)

(3.212c)


We think of the solution u as dependent on the unknown term f : u = u(f).A-fortiori from assumptions (3.211) we obtain the regularity (Theorem 3)

u ∈ H1,1(Q) = L2(0, T ;H1(Ω)) ∩H1(0, T ;L2(Ω));∂u

∂ν∈ L2(Σ). (3.213)

In view of (3.213), we can apply Theorem 2 to problem (3.212), so that it satisfiesthe Carleman estimate (3.11). In evaluating the boundary term BTu|Σ in (3.13), wesee that u|Σ = 0 in (3.212c), hence ut|Σ = 0, has two implications: (i) it forces the2nd, 4th, 5th integral terms of BTu|Σ in (3.13) to vanish along with the contributiondue to ut|Σ = 0 in the first integral term; (ii) on Σ, it yields: |∇u|2 = | ∂u

∂ν|2;

h ·∇u = (h ·ν) ∂u∂ν

. As a result, we obtain, also recalling in the last step the propertyh · ν ≤ 0 on Γ0 from assumption (3.2a) and ∂u

∂ν|Σ1 = 0 by (3.212c):

BTu|Σ = −2τ

∫ T

0

∫Γ0

e2τϕ

∣∣∣∣∂u∂ν∣∣∣∣2 h · νdΓdt+ 4τ

∫ T

0

∫Γ0

e2τϕ (h · ∇u)∂u

∂νdΓdt

= 2τ

∫ T

0

∫Γ0

e2τϕ

∣∣∣∣∂u∂ν∣∣∣∣2 h · νdΓdt ≤ 0. (3.214)

Thus, the solution u = u(f) of the over-determined problem (3.212) satisfies thefollowing specialized version of (3.11), where we may drop BTu|Σ by (3.214):

C1,τ

∫ T

0

∫Ω

e2τϕ[u2t + |∇u|2]dQ+ C2,τ

∫Q(σ)

e2τϕu2dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fR|2dQ+ CT,u e2τσ + cT τ

3e−2τδ[Eu(0) + Eu(T )], (3.215)

where we have set via (3.11) for fixed u = u(f): CT,u = C1,T

∫ T0

∫Ωu2dQ.

Step 2. In this step, we differentiate in time the u-mixed problem (3.212), thusobtaining

(ut)tt(x, t)−∆(ut)(x, t)− q(x)(ut)t(x, t) = f(x)Rt(x, t) in Q;

(ut)

(· , T

2

)= 0; (ut)t

(· , T

2

)= f(x)R

(x,T

2

)in Ω;

(ut)|Σ = 0,∂(ut)

∂ν

∣∣∣∣Σ1

= 0

(3.216a)

(3.216b)

(3.216c)

Here, by virtue of (3.211), (3.212a-b), one obtains

f(x)Rt(x, t) ∈ L∞(0, T ;L2(Ω)); utt

(x,T

2

)= f(x)R

(x,T

2

)∈ L2(Ω). (3.217)

Moreover, a-fortiori from (3.217), we have the following regularity from problem(3.216), see Theorem 3, ut ∈ H1,1(Q). In view of this regularity, and because of theover-determined B.C. (3.216c), combined with the assumed property (3.2a) h ·ν ≤ 0on Γ0, we readily see again that Theorem 2 implies that such solution ut ∈ H1,1(Q) of(3.216) satisfies the Carleman estimate (3.11) again with boundary terms: BTut |Σ ≤0, counterpart of (3.214). Thus, such solution ut = ut(f) of the over-determined


problem (3.216), satisfies the counterpart of inequality (3.215), with u there replacedby ut now, thus yielding for all τ sufficiently large:

C1,τ

∫ T

0

∫Ω

e2τϕ[u2tt + |∇ut|2] dQ+ C2,τ

∫Q(σ)

e2τϕu2t dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fRt|2dQ+ CT,ut e2τσ + cT τ

3e−2τδ[Eut(0) + Eut(T )]. (3.218)

Step 3. In this step, we differentiate in time, once more, namely, the ut-mixedproblem (3.216), thus obtaining

(utt)tt(x, t)−∆(utt)(x, t)− q(x)(utt)t(x, t) = f(x)Rtt(x, t) in Q;

(utt)

(·, T

2

)=f(x)R

(x,T

2

)in Ω;

(utt)t

(· , T

2

)=f(x)

[q(x)R

(x,T

2

)+Rt

(x,T

2

)]∈ L2(Ω) in Ω;

(utt)|Σ = 0,∂(utt)

∂ν

∣∣∣∣Σ1

= 0

(3.219a)

(3.219b)

(3.219c)

(3.219d)

Here, by virtue of (3.216a-b) and (3.211), one obtains, with f ∈ L2(Ω),uttt

(x,T

2

)= q(x)f(x)R

(x,T

2

)+ f(x)Rt

(x,T

2

)∈ L2(Ω),

f(x)Rtt(x, t) ∈ L∞(0, T ;L2(Ω)).

(3.220a)

(3.220b)

Similarly, again with f a-priori only in L2(Ω), and R(x, T2

) ∈ L∞(Ω), oneobtains–at a first glance at least–that utt(·, T2 ) = f(x)R(·, T

2) ∈ L2(Ω) only. If

this was so, one would only have utt ∈ C([0, T ];L2(Ω)), while application of theCarleman estimates in Theorem 2 on the (utt)-problem (3.219) requires H1,1(Q)-solutions. This would prevent us from justifying a further application of Carlemanestimates (3.11)–(3.13) to the (utt)-problem (3.219). However, as we shall see in thenext lemma (see also Lemma 1), the special structure of the (utt)-over-determinedproblem will allow us to boost the regularity of f from the a-priori level f ∈ L2(Ω)to the enhanced level [f(x)R(x, T

2)] ∈ H1

0 (Ω) first, and f ∈ H10 (Ω) next.

Lemma 3. With reference to the (utt)-problem (3.219), with a-priori f ∈ L2(Ω),Rtt ∈ L∞(Q), q ∈ L∞(Ω), R(x, T

2), Rt(x,

T2

) ∈ L∞(Ω) and under (3.199), we have

1

C2RT

2,r0

‖f(·)‖2H10 (Ω) ≤

∥∥∥∥f(·)R(·, T

2

)∥∥∥∥2

H10 (Ω)

≤ CT ‖fRtt‖2L2(Q) <∞. (3.221)

Proof. The inequality on the RHS of (3.221) is the COI (3.34), as applied to theover-determined problem (3.219), under present assumptions (after dropping theL2(Ω)-velocity term utt( · , T2 ). The LHS of (3.221) is already shown in Lemma 1.Moreover, we note that, in addition, we have now[

f(x)R

(x,T

2

)] ∣∣∣∣Γ

= 0 and (3.199) imply f |Γ = 0, (3.222)

since f(x)R(x, T2

) ∈ H10 (Ω), where (3.222) is needed on the LHS of (3.221).


Remark 16. The reason behind the process of differentiating the original u-problem(3.212) twice to obtain the (utt)-problem (3.219) is explained and justified in Re-mark 17 below. At any rate, while preserving the L∞(0, T ;L2(Ω))-regularity of thesuccessive right-hand side ‘forcing terms’ fR, fRt, fRtt (under present assump-tions), passing from the u-problem to the (utt)-problem shifts the unknown termf from the RHS of (3.212a) to the initial conditions in (3.219b-c). This creates aconvergence of aims with the COI.

Henceforth, we proceed with the proof, having at our disposal the enhancedregularity from (3.219b), (3.221):

utt

(x,T

2

)∈ H1

0 (Ω), f(x) ∈ H10 (Ω), with R

(x,T

2

)∈W 1,∞(Ω). (3.223)

We seek to establish that, indeed, f ≡ 0. Thus, under the gained regularity(3.223), we have the following regularity from problem (3.219): utt ∈ H1,1(Q). Inview of this regularity and because of the over-determined B.C. (3.219d), combinedwith the assumed property (3.2a) h ·ν ≤ 0 on Γ0, we readily see again that Theorem2 implies that such solution utt ∈ H1,1(Q) of (3.216) satisfies the Carleman estimate(3.11) again with boundary terms BTutt |Σ ≤ 0. Thus, such solution utt = utt(f) ofthe over-determined problem (3.219) satisfies the counterpart of inequality (3.215)for u, and (3.218) for ut, thus yielding for all τ sufficiently large:

C1,τ

∫ T

0

∫Ω

e2τϕ[u2ttt + |∇utt|2]dQ+ C2,τ

∫Q(σ)

e2τϕu2tt dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ|fRtt|2dQ+ CT,utte2τσ + cT τ

3e−2τδ[Eutt(0) + Eutt(T )]. (3.224)

Step 4. Adding up (3.215), (3.218), and (3.224) together yields the combinedinequality

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ 2

∫ T

0

∫Ω

e2τϕ

[|fR|2 + |fRt|2 + |fRtt|2

]dQ+ [CT,u + CT,ut + CT,utt ] e

2τσ

+ cT τ3e−2τδ

[Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )

].

(3.225)

[This is the counterpart of (3.79).]Next, we invoke the properties (3.198) = (3.111) for R,Rt, Rtt ∈ L∞(Q) (already

used to claim properties (3.213) for u, and similar properties for ut and utt):

|f(x)R(x, t)| ≤ CR|f(x)|; |f(x)Rt(x, t)| ≤ CRt |f(x)|;|f(x)Rtt(x, t)| ≤ CRtt |f(x)|, (3.226)


with CR = ‖R(x, t)‖L∞(Q), etc. Using (3.226) into the RHS of (3.225) yields finally

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt

]dx dt ≤ CR,T,u

∫Q

e2τϕ|f |2dQ+ e2τσ

+ τ3e−2τδ[Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

, (3.227)

where CR,T,u is a positive constant depending on R, T , and u (it combines CR, CRt ,CRtt as well as CT,u, CT,ut , CT,utt defined below (3.215) and cT ).

Step 5. In this step we get a same claim as in Section 3.4.1, Step 5, a strategyproposed in [Isa06, Theorem. 8.2.2, p. 231]. From (3.212a) evaluated at t = T

2via

(3.212b) and hypothesis (3.199) on R(x, T2

), we get∣∣∣∣utt(x, T2)∣∣∣∣ = |f(x)|

∣∣∣∣R(x, T2)∣∣∣∣ ≥ r0|f(x)|, (3.228)

with r0 > 0. Hence, (3.228) allows one to obtain the pointwise inversion

|f(x)| ≤ 1

r0

∣∣∣∣utt(x, T2)∣∣∣∣ , x ∈ Ω. (3.229)

Claim: Using (3.229) in the first integral term∫Qe2τϕ|f |2dQ on the RHS of (3.227)

yields, the following estimate was proved in (3.84):∫Q

e2τϕ|f |2dQ ≤ T

r20

[(2cτT + 1)

∫Ω

∫ T/2

0


+

∫Ω

∫ T/2

0

e2τϕ(x,s)|uttt(x, s)|2dsdΩ +

∫Ω

|utt(x, 0)|2dΩ

]. (3.230)

Step 6. We substitute (3.230) for the first integral term on the RHS of (3.227)and obtain, after obvious majorizations,

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt

]dx dt

≤ CR,T,u

(T

r20

)[(2Tcτ + 1)

∫Q

e2τϕ|utt|2dQ+

∫Q

e2τϕ|uttt|2dQ

+

∫Ω

|utt(x, 0)|2dΩ]

+ e2τσ + τ3e−2τδ[Eu(0) + Eut(0)

+ Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

. (3.231)

[This is the counterpart of (3.88).]


Remark 17. As noted in Remark 13, it is the term uttt on the RHS of estimate(3.230), which then occurs also on the RHS of estimate (3.231)—the price to payin (3.229) to eliminate the unknown term f in terms of the solution, from the RHSof (3.227)—that requires the need to differentiate the original u-problem (3.212) toobtain the (utt)-problem (3.219). Thus, the Carleman estimate on the (utt)-problemalso produces an uttt-term on the LHS of estimate (3.231) which eventually willabsorb the uttt-term on the RHS of (3.231) and the resulting term will be droppedin Step 8 below.

We proceed as in (3.89)–(3.97). Next, we recall that e2τϕ < e2τσ on Q\Q(σ) by(3.8), so that the following estimate holds:∫

Q

e2τϕ|utt|2dQ =

∫Q(σ)

e2τϕ|utt|2dt dx+

∫Q\Q(σ)

e2τϕ|utt|2dx dt

≤∫Q(σ)

e2τϕ|utt|2dt dx+ e2τσ

∫Q\Q(σ)

|utt|2dx dt. (3.232)

Substituting (3.232) for the first integral term on the RHS of (3.231), we rewrite(3.231) as

C1,τ

∫Q

e2τϕ[u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2]dQ

+ C2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ CR,T,u,r0(2Tcτ + 1)

[ ∫Q(σ)

e2τϕu2ttdt dx+ e2τσku

]+ CR,T,u,r0

∫Q

e2τϕu2tttdQ+ CR,T,u,r0 ku + CR,T,ue

2τσ

+ CR,T,uτ3e−2τδ[Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )],

(3.233)

where we have set

CR,T,u,r0 = CR,T,u

(T

r20

); ku =

∫Q\Q(σ)

u2ttdx dt; ku =

∫Ω

|utt(x, 0)|2dΩ.

(3.234)Step 7. In (3.233) we shall use the following facts: Two terms on the right-hand

side

CR,T,u,r0

∫Q

e2τϕu2tttdQ,

and

CR,T,u,r0(2Tcτ + 1)

∫Q(σ)

e2τϕu2ttdt dx,

can be absorbed by left-hand side terms

C1,τ

∫Q

e2τϕ[u2t + u2

tt + u2ttt]dQ,

and


C2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt,

when τ is large enough, since C1,τ = τερ− 2CT and C2,τ = 2τ3β + O(τ2)− 2CT by(3.12), β > 0 as a consequence of (3.4). Therefore, (3.233) becomes

C′1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + |∇u|2 + |∇ut|2 + |∇utt|2

]dQ

+ C′2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ CR,T,u,r0

[ku(2Tcτ + 1)e2τσ + ku

]+ CR,T,u

e2τσ + τ3e−2τδ[Eu(0)

+ Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

; (3.235)

C′1,τ = C1,τ − CR,T,u,r0 = τερ− 2CT − CR,T,u,r0 ; (3.236a)

C′2,τ = 2τ3β + O(τ2)− 2CT − CR,T,u,r0(2Tcτ + 1), (3.236b)

and C′1,τ > 0, C′2,τ > 0 for all τ sufficiently large. Likewise, as limτ→∞

τ3e−2τδ = 0, we

can take τ sufficiently large, say ∀τ > some τ0, such that the quantity

CR,T,uτ3e−2τδ[Eu(0) + Eut(0) + Eutt(0) + Eu(T ) + Eut(T ) + Eutt(T )]

≤ constR,T,u,r0,δ,τ0

is bounded by some constant which is independent of τ .

Step 8. We then return to inequality (3.235), drop here the first positive termC′1,τ

∫QdQ, and obtain for all τ > τ0:

C′2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt]dx dt

≤ CR,T,u,r0

[ku(2Tcτ + 1)]e2τσ + ku

+ CR,T,ue

2τσ + constR,T,u,r0,δ,τ0

≤ Cdata,uτe2τσ, for all τ large enough > τ0, (3.237)

where Cdata,u is a constant depending on the data R, T, u, r0, δ, τ0, but not on τ .We note again that from the definition of Q(σ) in (3.8), we have e2τϕ ≥ e2τσ on

Q(σ). Thus, (3.237) implies

C′2,τe2τσ

∫Q(σ)

[u2 + u2t + u2

tt]dx dt ≤ Cdata,uτe2τσ. (3.238)

Dividing by τe2τσ on both sides of (3.238) and recalling the expression of C′2,τ ∼τ3 in (3.236b) yield

C′2,ττ→∞ as τ →∞, hence (3.238) implies ⇒ u = ut ≡ utt = 0 on Q(σ). (3.239)

Step 9. We now return to equation (3.212a) and use here u ≡ 0 on Q(σ) from(3.239) to obtain


f(x)R(x, t) = utt(x, t)−∆u(x, t)− q(x)ut(x, t) ≡ 0, (x, t) ∈ Q(σ). (3.240)

We next invoke property (3.9) that [t0, t1]×Ω ⊂ Q(σ) ⊂ Q, and that T2∈ [t0, t1].

Thus, (3.240) in particular yields our desired goal stated in Orientation:

f(x)R

(x,T

2

)≡ 0, for all x ∈ Ω, (3.241)

Finally, we recall from (3.199) that∣∣R (x, T

2

)∣∣ ≥ r0 > 0, x ∈ Ω. Thus by (3.241)

f(x) ≡ 0 a.e. in Ω, (3.242)

as desired. Thus, with this step, Theorem 15 has been proved.

Stability of linear inverse problem for the u-problem (3.196):Proof of Theorem 16

Step 1. Let u(f) be the solution of problem (3.196), with dataq ∈ L∞(Ω), R,Rt, Rtt ∈ L∞(Q); Rt ∈ H2θ(0, T ;W θ,∞(Ω));

f ∈ Hθ0 (Ω),

∣∣∣∣R(x, T2)∣∣∣∣ ≥ r0 > 0, R

(· , T

2

)∈W 1,∞(Ω),

(3.243a)

(3.243b)

as implied by (3.202), 0 < θ ≤ 1, θ 6= 12

viewed as a function of the unknown term

f ∈ Hθ0 (Ω). Set

v = v(f) = ut(f). (3.244)

Then, differentiating in t problem (3.196), we see that v satisfies the following:vtt(x, t) = ∆v(x, t) + q(x)vt(x, t) + f(x)Rt(x, t) in Q;

v

(· , T

2

)= 0; vt

(· , T

2

)= f(x)R

(x,T

2

)in Ω;

v(x, t)|Σ = 0 in Σ,

(3.245a)

(3.245b)

(3.245c)

so that the I.C. vt|T2

= f(x)R(x, T

2

)∈ Hθ(Ω) ⊂ L2(Ω). Accordingly, by linearity,

we split v into two components:

v = ψ + z, (3.246)

where ψ satisfies the same problem as v, however, with homogeneous forcing termψtt(x, t) = ∆ψ(x, t) + q(x)ψt(x, t) in Q;

ψ

(· , T

2

)= 0; ψt

(· , T

2

)= f(x)R

(x,T

2

)in Ω;

ψ(x, t)|Σ = 0 in Σ,

(3.247a)

(3.247b)

(3.247c)

while z satisfies the same problem as v, however, with homogeneous I.C.:ztt(x, t) = ∆z(x, t) + q(x)zt(x, t) + f(x)Rt(x, t) in Q;

z( · , T2

) = 0; zt( · , T2 ) = 0 in Ω;

z(x, t)|Σ = 0 in Σ.

(3.248a)

(3.248b)

(3.248c)


Step 2. Theorem 19 Consider the ψ-system (3.247) with data q ∈ L∞(Ω), f ∈Hθ

0 (Ω), 0 ≤ θ ≤ 1, θ 6= 12, R(x, T

2) ∈ W 1,∞(Ω) satisfying, moreover, the positivity

condition (3.199). Then, the following inequality holds true, for 0 ≤ θ ≤ 1, θ 6= 12

‖f‖2Hθ0 (Ω) ≤ C2T,q,r0,θ

∫ T

0

∫Γ1

(∂(Dθ

tψ)

∂ν

)dΓ1dt = C2

T,q,r0,θ

∥∥∥∥∂ψ∂ν∥∥∥∥2

Hθ(0,T ;L2(Γ1))

(3.249)where the RHS of (3.249) is finite by Theorem 8, and where T > T0, see (3.49), asassumed. We note that in the case 1

2< θ ≤ 1, the required additional compatibility

conditions are satisfied since f ∈ Hθ0 (Ω), and so ψt( · , T2 )|Γ = 0 by (3.247b), as

needed. For 0 < θ < 12, Hθ

0 (Ω) = Hθ(Ω) ([LM72, p. 55]).

Proof. Step (i). We first prove inequality (3.249) for θ = 1: thus with f ∈ H10 (Ω).

Differentiate the ψ-system (3.247) in time to get

(ψt)tt(x, t) = ∆(ψt)(x, t) + q(x)(ψt)t(x, t) in Q;

(ψt)(· , T2 ) = f(x)R(x, T2

); in Ω

(ψt)t(· , T2 ) = q(x)f(x)R(x, T2

) in Ω;

ψ(x, t)|Σ = 0 in Σ,

(3.250a)

(3.250b)

(3.250c)

(3.250d)

where the I.C. satisfy the following regularity properties in the present case θ = 1:

(ψt)

(· , T

2

)= f(x)R

(x,T

2

)∈ H1

0 (Ω);

(ψt)t

(· , T

2

)= q(x)f(x)R

(x,T

2

)∈ L2(Ω), (3.251)

since R(·, T2

) ∈ W 1,∞(Ω) as in (3.243b). Then we can apply the COI contained inTheorem 8, inequality (3.34), to the ψt-system (3.250), as assumptions (3.32) aresatisfied. Accordingly, there is a constant CT > 0 such that a-fortiori via (3.251)∥∥∥∥f( · )R

(· , T

2

)∥∥∥∥2

H10 (Ω)

≤ C2T

∫ T

0

∫Γ1

(∂(ψt)

∂ν

)2

dΓ1 dt. (3.252)

We now recall that under the assumptions∣∣R (x, T

2

)∣∣ ≥ r0 > 0, x ∈ Ω andR(x, T

2) ∈W 1,∞(Ω), the LHS of inequality (3.221) holds true, whereby (3.252) then

yields

‖f‖2H10 (Ω) ≤ C

2T,r0

∫ T

0

∫Γ1

(∂(ψt)

∂ν

)2

dΓ1 dt, (3.253)

which is the desired inequality (3.249) with θ = 1.

Step (ii). We now take θ = 0. We can similarly apply the COI, Theorem 8,inequality (3.34), this time directly to the original ψ-system (3.247) and then useagain

∣∣R (x, T2

)∣∣ ≥ r0 > 0 to get

‖f‖2L2(Ω) ≤ C2T,q,r0

∫ T

0

∫Γ1

(∂(ψ)

∂ν

)2

dΓ1 dt, (3.254)


Step (iii). We now interpolate between (3.253) and (3.254) to get the desiredinequality (3.249): for 0 ≤ θ ≤ 1, θ 6= 1

2.

Step 3. Under Theorem 19, we then obtain from (3.249) by use of (3.246), thetriangle inequality and (3.244), still for 0 ≤ θ ≤ 1, θ 6= 1

2:

‖f‖Hθ0 (Ω) ≤ CT,q,r0,θ

∥∥∥∥∂ψ∂ν∥∥∥∥Hθ(0,T ;L2(Γ1))

≤ CT,q,r0,θ∥∥∥∥∂v∂ν − ∂z

∂ν

∥∥∥∥≤ CT,q,r0,θ

∥∥∥∥∂(ut)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

+ CT,q,r0,θ

∥∥∥∥∂z∂ν∥∥∥∥Hθ(0,T ;L2(Γ1))

(3.255)

Inequality (3.255) is the desired, sought-after RHS estimate (3.203) of Theo-rem 16, modulo (polluted by) the ∂z

∂ν-term. Such term will be next absorbed by

a compactness–uniqueness argument. To carry this through, we need the followingLemma 4 below.

Remark 18. The compactness-uniqueness argument, alluded to above, which is givennext critically requires that the map f → ∂z

∂ν|Γ1 be compact between suitable func-

tion spaces. In light of [LLT86, Theorem 2.4, p. 164], the map f ∈ Hθ(Ω) → ∂z∂ν∈

Hθ(Σ1), 0 ≤ θ ≤ 1, θ 6= 12, is only continuous, but not compact for the z-problem

(3.248) (for suitable Rt). This fact is a novel obstacle of the present stability anal-ysis for the Dirichlet B.C. case of problem (3.196) as opposed to the situation inSection 3.4. It is in order to overcome this obstruction that we shall work with thezt-problem (3.278) leading to the desired, sought-after compactness statement inCorollary 1 in Step 7 below, Eqns. (3.287), (3.288).

Remark 19. We remark that, since z|T2

= 0 by (3.248b), then an interpolation in-

equality and Poincare inequality (in the variable t) imply∥∥∥∥∂z∂ν∥∥∥∥Hθ(0,T ;L2(Γ1))

≤ Cθ∥∥∥∥∂zt∂ν

∥∥∥∥L2(Γ1×[0,T ])

. (3.256)

This inequality, once substituted into the RHS of (3.255) would allow us—through a compactness-uniqueness argument (Step 8 below)—to absorb the ∂zt

∂ν-

term so that (3.255) combined with (3.256) would yield the final desired RHS es-timate (3.203). This argument, however, requires higher regularity assumptions onthe data. The alternative procedure which follows uses interpolation and allows forrelaxed minimal regularity assumptions on the data, at a price of additional com-plications.

Step 4. Lemma 4 (a) Consider the z-system (3.248), with data

f(x)Rt(x, t) ∈ L2(0, T ;Hθ(Ω)); Dθt (f(x)Rt(x, t)) ∈ L2(0, T ;L2(Ω)), (3.257)

where 0 ≤ θ ≤ 1, θ 6= 12. Then, continuously

z, zt ∈ C([0, T ];H1+θ(Ω)×Hθ(Ω)), and∂z

∂ν∈ Hθ(Σ). (3.258)


(b) Checkable sufficient conditions for the regularity properties of (fRt) andDθt (fRt) in (3.257) to hold true are

f(x) ∈ Hθ0 (Ω); Rt ∈ L1(0, T ;W θ,∞(Ω)) ∩Hθ(0, T ;L∞(Ω)) (3.259)

so that, ultimately, as a sufficient condition on the data, it holds that

f(x) ∈ Hθ0 (Ω); Rt ∈ L2(0, T ;W θ,∞(Ω)) ∩Hθ(0, T ;L∞(Ω))⇒ ∂z

∂ν∈ Hθ(Σ)

(3.260)(of course, Rt ∈ Hθ(0, T ;W θ,∞(Ω)) implies Rt as on the LHS of (3.260)).

Proof. Part (a). Part (a) is precisely a special case of Theorem 16, where the requiredC.C. for 1

2< θ ≤ 1 are a-fortiori satisfied since the z-problem has homogeneous

Dirichlet B.C. as in (3.248c), as well as homogeneous I.C. as in (3.248b).Part (b). To show part (b), we shall rely on the theory of multipliers [MS85].

Remark 20. To gain preliminary insight we may start with the most direct butcrude (for the present analysis) case θ = 1 (while for future use of part (a) in thecompactness–uniqueness argument, θ > 0 needs only be arbitrarily small). Thus, inthe present Remark, let θ = 1.

First requirement (LHS of (3.257) for θ = 1). We take f ∈ H1(Ω) and seek apointwise multiplication operator γ(x) to yield f(x) ∈ H1(Ω) =⇒ γ(x)f(x) ∈ H1(Ω),

i.e., γ ∈M(H1(Ω)→ H1(Ω)) ([MS85]).

(3.261a)

(3.261b)

A checkable sufficient condition for (3.261) to hold true is

γ ∈W 1,∞(Ω), (3.262)

while a full characterization is given in [MS85, Theorem 1, m = l = 1, p=2, p. 243].Thus, with regard to our case on the LHS of (3.257) with θ = 1, where the role ofγ(x) is now played by Rt, we have by application of (3.262) f(x) ∈ H1(Ω); Rt(x, t) ∈ L2(0, T ;W 1,∞(Ω))

⇒ f(x)Rt(x, t) ∈ L2(0, T ;H1(Ω)).

(3.263a)

(3.263b)

Second requirement (RHS of (3.257) for θ = 1). Now we take f ∈ L2(Ω) andseek a pointwise multiplication operator γ(x) to yield f(x) ∈ L2(Ω) =⇒ γ(x)f(x) ∈ L2(Ω),

i.e., γ ∈M(L2(Ω)→ L2(Ω)).

(3.264a)

(3.264b)

A checkable sufficient condition for (3.264) to hold true is

γ ∈ L∞(Ω), (3.265)

while a full characterization is given in [MS85, Theorem 1, m = l = 0, p=2, p. 243].Thus, with regard to our case on the RHS of (3.257) with θ = 1, where the role ofγ is now played by Rtt, we have by application of (3.265)

3 Boundary control and boundary inverse theory 281 f(x) ∈ L2(Ω), Rtt(x, t) ∈ L2(0, T ;L∞(Ω)),

⇒ D1t (f(x)Rt(x, t)) = f(x)Rtt(x, t) ∈ L2(0, T ;L2(Ω)).

(3.266a)

(3.266b)

Case 0 < θ < 1. First requirement (LHS of (3.257) for 0 < θ < 1). We next considerthe case where θ is fractional: 0 < θ < 1. To this end, we shall invoke the fractionalderivative result in [MS85, Theorem 1, p. 123], as specialized to our case of interest

1 > m = l = θ > 0, l = [integer part of l] = 0; p = 2, (3.267)

whereby W θ2 (Ω) = Hθ(Ω) in our present Sobolev space notation. We accordingly

have [MS85, Theorem 1, p. 123] that

γ ∈M(W θ2 (Ω) = Hθ(Ω)→W θ

2 (Ω) = Hθ(Ω)), (3.268)

if and only if the following conditions hold true:(1)

γ ∈W θ2,loc, which is fulfilled if γ ∈ Hθ(Ω) = W θ

2 (Ω); (3.269)

(2)Dp=2,l=θ γ ∈M(W θ

2 (Ω)→ L2(Ω)); (3.270)

(3)γ ∈M(Wm−l=0

2 (Ω)→ Lp=2(Ω)) = M(L2(Ω)→ L2(Ω)), (3.271)

for which γ ∈ L∞(Ω) is a sufficient condition as in (3.264b), (3.265).It remains to clarify condition (2) in (3.270). To this end, we recall from [MS85,

p. 105, third line, top paragraph] that for l = 0 as in (3.267), then

(Dp=2,l=θ γ)(x) = |∇θγ(x)| = fractional (space) derivative of order θ. (3.272)

Thus, condition (2) is rewritten in the present case l = 0 as(2’) |∇θγ| ∈M(Hθ → L2); that is

g ∈ Hθ → |∇θγ|g ∈ L2,

(3.273a)

(3.273b)

a sufficient condition for which (absorbing the loss g ∈ Hθ ⊂ L2) is

|∇θγ| ∈ L∞(Ω); i.e., γ ∈W θ,∞(Ω). (3.274)

Thus, in conclusion, condition γ ∈ W θ,∞(Ω) in (3.274) is sufficient to guaran-tee all requirements (1)=(3.269), (2′)=(3.273), (3)=(3.271) cumulatively, hence thedesired result (3.268):

f ∈ Hθ(Ω), γ ∈W θ,∞(Ω) =⇒ γ(x)f(x) ∈ Hθ(Ω), 0 < θ < 1. (3.275)

The sufficient condition for γ in (3.275) for 0 < θ < 1 extends the sufficient condition(3.262) ⇒ (3.261) for θ = 1, as well as the sufficient condition (3.265) ⇒ (3.264) forθ = 0. Thus, with regard to our case on the LHS of (3.257), where the role of γ isnow played by Rt, we have by application of (3.274) f(x) ∈ Hθ(Ω), Rt(x, t) ∈ L2(0, T ;W θ,∞(Ω)), 0 < θ < 1

⇒ f(x)Rt(x, t) ∈ L2(0, T ;Hθ(Ω)).

(3.276a)

(3.276b)


an extension of (3.263a-b) (θ = 1) (and also of (3.267a-b) for θ = 0 for Rtt insteadof Rt).

Case 0 < θ < 1. Second requirement (RHS of (3.257)). We readily obtain f(x) ∈ L2(Ω), DθtRt ∈ L2(0, T ;L∞(Ω))

⇒ Dθt (f(x)Rt(x, t)) ∈ L2(0, T ;L2(Ω)).

(3.277a)

(3.277b)

by recalling the sufficient condition (3.265) ⇒ (3.264) in space regularity.Conclusion in the proof of part (b). We combine the sufficient conditions (3.276)

and (3.277). Since DθtRt ∈ L1(0, T ;L∞(Ω)) is a-fortiori guaranteed by Dθ

tRt ∈L2(0, T ;L∞(Ω)) or Rt ∈ Hθ(0, T ;L∞(Ω)), as assumed in (3.259), we then see thatcondition (3.259) guarantees both (3.276) and (3.277). Part (b) is proved.

Step 5. Returning to (3.248), we now differentiate the z-system in time and get(zt)tt(x, t) = ∆(zt)(x, t) + q(x)(zt)t(x, t) + f(x)Rtt(x, t) in Q;

(zt)

(· , T

2

)= 0; (zt)t

(· , T

2

)= f(x)Rt

(x,T

2

)in Ω;

(zt)(x, t)|Σ = 0 in Σ.

(3.278a)

(3.278b)

(3.278c)

Lemma 5. (a). Consider the zt-system (3.278), with dataq ∈ L∞(Ω); f(x)Rtt(x, t) ∈ L2(0, T ;Hθ(Ω));

Dθt (f(x)Rtt(x, t) ∈ L2(0, T ;L2(Ω)), f(x)Rt(x,

T2

) ∈ Hθ0 (Ω),

(3.279a)

(3.279b)

where 0 ≤ θ ≤ 1, θ 6= 12

. Then, continuously

∂(zt)

∂ν∈ Hθ(Σ). (3.280)

(b) Checkable sufficient conditions for the regularity properties of (fRtt) andDθt (fRtt) in (3.280) to hold true are for 0 ≤ θ ≤ 1, θ 6= 1

2

f(x) ∈ Hθ0 (Ω); Rtt ∈ L2(0, T ;W θ,∞(Ω)) ∩Hθ(0, T ;L∞(Ω)), (3.281)

so that, ultimately, as a sufficient condition on the data, still for 0 ≤ θ ≤ 1, θ 6= 12

,

f(x) ∈ Hθ0 (Ω); Rtt ∈ L2(0, T ;W θ,∞(Ω)) ∩Hθ(0, T ;L∞(Ω))⇒ ∂(zt)

∂ν∈ Hθ(Σ)

(3.282)(of course Rtt ∈ Hθ(0, T ;W θ,∞(Ω)) implies Rtt as on the LHS of (3.282)).

Proof. (a) Same proof as in Lemma 4(a): The forcing term f(x)Rt(x, t) in (3.248a)of the z-problem is now replaced by the forcing term f(x)Rtt(x, t) in (3.278a) of thezt-problem. Moreover, as the zt-problem has now non-zero I.C. zt|T

2as in (3.278b)—

unlike the I.C. of the z-problem which are homogeneous as in (3.248b)—we mustnow require f(x)Rt(x,

T2

) ∈ Hθ0 (Ω) as in (3.279b) and we now restrict to taking

f ∈ Hθ0 (Ω), so that for 1

2< θ ≤ 1 the compatibility conditions needed when

invoking Theorem 16 are satisfied, as then (zt)( · , T2 ) = 0, as required, by (3.278b).


(b) As in Lemma 4(b), the conditions in (3.281) imply those in (3.279), withthe following additional argument. We must justify that f(x)Rt(x,

T2

) ∈ Hθ0 (Ω).

But this is true by (3.275), since by (3.281), f ∈ Hθ0 (Ω), once we guarantee that

Rt(x,T2

) ∈ W θ,∞(Ω). But, in turn, this follows from (3.281) on Rtt which gives

Rt ∈ C([0, T ]; W θ,∞(Ω)).

Step 6. We now interpolate between (3.260) and (3.282) with the conditionf ∈ Hθ

0 (Ω) common to both. We obtain for q ∈ L∞(Ω), 0 ≤ β ≤ 1, 0 ≤ θ ≤ 1 andβ, θ 6= 1

2:

f(x) ∈ Hθ(Ω); Dβt Rt ∈ Hθ(0, T ;W θ,∞(Ω))

⇒ ∂(Dβt z)

∂ν∈ Hθ(Σ) continuously.

(3.283)

(3.284)

Step 7. Corollary 1 Consider the z-system (3.248), with data

q ∈ L∞(Ω), f ∈ Hθ0 (Ω), Rt ∈ Hθ+β(0, T ;W θ,∞(Ω)); (3.285)

0 ≤ θ ≤ 1, 0 ≤ β ≤ 1, θ, β 6= 12. Then for the operator K below, we have the

following properties:f ∈ Hθ

0 (Ω) ⇒ (Kf)(x, t) =∂z

∂ν∈ Hθ,θ+β(Σ)

≡ L2(0, T ;Hθ(Ω)) ∩Hθ+β(0, T ;L2(Γ )) continuous

⇒ ∂z

∂ν∈ Hθ−ε,θ+β−ε(Σ) compact.

(3.286)

(3.287)

for any 1 ≥ θ ≥ ε > 0. A-fortiori, selecting 0 < β = θ ≤ 1, then

q ∈ L∞(Ω), Rt ∈ H2θ(0, T ;W θ,∞(Ω))

f ∈ Hθ0 (Ω) ⇒ (Kf) =

∂z

∂ν∈ Hθ,2θ(Σ) continuous

⇒ (Kf) =∂z

∂ν∈ Hθ−ε,2θ−ε(Σ) compact,

(3.288)

(3.289)

(3.290)

for ε > 0. Take 0 < ε = θ ≤ 1, then, in particular, as desired

f ∈ Hθ0 (Ω)⇒ (Kf) =

∂z

∂ν∈ H0,θ(Σ) = Hθ(0, T ;L2(Γ1)) compact. (3.291)

Proof. This is a direct corollary of the implication from (3.283) to (3.284). First, theregularity properties in (3.285) are a restatement of those in (3.283): hence property(3.284) follows. Thus, with datum Rt as in (3.283)=(3.285), we obtain property(3.286), which is a restatement of (3.284). The other properties are obvious.

Step 8. Corollary 1 will allow us to absorb the terms∥∥∥∥Kf =∂z

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

, (3.292)

from the RHS of estimate (3.255), by a compactness–uniqueness argument, as usual.


Proposition 4. Consider the u-problem (3.196) with T > T0 in (3.49) under as-sumptions (3.202) (which imply (3.243)) for its data q( · ), f( · ) and R( · , · ), with Rsatisfying also (3.199), so that both estimate (3.255), as well as Corollary 1 hold true.Then, the term Kf = ∂z

∂ν|Σ1 measured in the Hθ(0, T ;L2(Γ1))-norm, 0 < θ ≤ 1,

θ 6= 12

, can be omitted from the RHS of inequality (3.255) (for a suitable constantCT,r0,... independent of the solution u), so that the desired RHS conclusion, equation(3.203), of Theorem 16 holds true:

‖f‖Hθ0 (Ω) ≤ CT,data

∥∥∥∥∂(ut)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

, 0 < θ ≤ 1, θ 6= 1

2. (3.293)

Proof. Step (i). Suppose, by contradiction, that inequality (3.293) is false. Then,there exists a sequence fn∞n=1, fn ∈ Hθ

0 (Ω), such that for 0 < θ ≤ 1, θ 6= 12

(i) ‖fn‖Hθ0 (Ω) ≡ 1, n = 1, 2, . . . ; (3.294a)

(ii) limn→∞

∥∥∥∥∂ut(fn)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

= 0. (3.294b)

where u(fn) solves problem (3.196) with f = fn:(u(fn))tt = ∆u(fn) + q(x)(u(fn))t + fn(x)R(x, t) in Q;

u(fn)

(· , T

2

)= 0; (u(fn))t

(· , T

2

)= 0 in Ω;

u(fn)(x, t)|Σ = 0 in Σ.

(3.295a)

(3.295b)

(3.295c)

In view of (3.294a), there exists a subsequence, still denoted by fn, such that:

fn converges weakly in Hθ0 (Ω) to some f0 ∈ Hθ

0 (Ω), 0 < θ ≤ 1, θ 6= 1

2. (3.296)

Moreover, since the operators K is compact as stated in Corollary 1, more specif-ically (3.291), it then follows by (3.296) that we have strong convergence:

limm,n→+∞

‖Kfn −Kfm‖Hθ(0,T ;L2(Γ1)) = 0, 0 < θ ≤ 1, θ 6= 1

2. (3.297)

Step (ii). On the other hand, since the map f → u(f) is linear, and recalling thedefinition of the operator K in (3.286), it follows from estimate (3.255) that

‖fn − fm‖Hθ0 (Ω) ≤ CT,r0,θ

∥∥∥∥∂ut(fn)

∂ν− ∂ut(fm)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

+ CT,r0,θ‖Kfn −Kfm‖Hθ((0,T ;L2(Γ1))

≤ CT,r0,θ

(∥∥∥∥∂ut(fn)

∂ν

∥∥∥∥+

∥∥∥∥∂ut(fm)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

)

+ CT,r0,θ‖Kfn −Kfm‖Hθ(0,T ;L2(Γ1)). (3.298)

for 0 < θ ≤ 1, θ 6= 12. It then follows from (3.294b) and (3.297) as applied to the

RHS of (3.298) that


limm,n→+∞

‖fn − fm‖Hθ0 (Ω) = 0. (3.299)

Therefore, fn is a Cauchy sequence in Hθ0 (Ω). By uniqueness of the limit,

recall (3.296), it then follows that the limit is f0 and

limn→∞

‖fn − f0‖Hθ0 (Ω) = 0, 0 < θ ≤ 1, θ 6= 1

2. (3.300)


‖f0‖Hθ0 (Ω) = 1, 0 < θ ≤ 1, θ 6= 1

2. (3.301)

Step (iii). We now apply to the the v-problem (3.245) with non-homogeneousinitial velocity (3.245b) the same trace regularity results Theorem 4 that we haveinvoked in Lemma 4(b) for the zt-problem (3.278) with non-homogeneous initialvelocity (3.278b); that is, as f ∈ Hθ

0 (Ω), Rt ∈ H2θ(0, T ;W θ,∞(Ω)) by assumption(3.243a), we have:

f(x)Rt(x, t) ∈ L2(0, T ;Hθ(Ω)), Dθt (f(x)Rt(x, t) ∈ L2(Q),

f(x)R

(x,T

2

)∈ Hθ

0 (Ω)⇒ ∂v

∂ν|Σ ∈ Hθ(Σ) continuously.

(3.302a)

(3.302b)

where vt(·, T2 ) = f(x)R(x, T2

) by (3.245b).Step (iv). We deduce from (3.302) with multiplier R(x, T

2) ∈ W 1,∞(Ω) as in

(3.275) that for 0 < θ ≤ 1, θ 6= 12

f(x) ∈ Hθ0 (Ω)→ ∂v(f)

∂ν|Σ =

∂ut(f)

∂ν|Σ ∈ Hθ(Σ) continuously,

i.e.

∥∥∥∥∂ut(f)

∂ν

∣∣∣∣Σ

∥∥∥∥Hθ(Σ)

≤ CR‖f‖Hθ0 (Ω)

(3.303a)

(3.303b)

As the map f → ut(f)|Σ is linear, it then follows in particular from (3.303b),since fn, f0 ∈ Hθ

0 (Ω) by (3.296):∥∥∥∥∂ut(fn)

∂ν|Σ1 −

∂ut(f0)

∂ν|Σ1

∥∥∥∥Hθ(Σ1)

≤ CR‖fn − f0‖Hθ0 (Ω). (3.304)

Recalling (3.299) on the RHS of (3.304), we conclude that

limn→∞

∥∥∥∥∂ut(fn)

∂ν|Σ1 −

∂ut(f0)

∂ν|Σ1

∥∥∥∥Hθ(Σ1)

= 0, (3.305)

This, combined with (3.294b), then yields

∂ut(f0)

∂ν

∣∣∣∣Σ1

≡ 0 in Hθ(Σ1), 0 < θ ≤ 1, θ 6= 1

2, (3.306a)

and hence∂u(f0)

∂ν

∣∣∣∣Σ1

= constant in t. (3.306b)

Step (v). For the u(fn)-problem (3.295) the interior regularity result Theorem 4yields for C = CT,R,θ > 0, standard trace theory, and (3.299):


‖u(fn), u(fn)t − u(f0), u(f0)t‖C([0,T ];H1+θ×Hθ) ≤ C‖fn − f0‖Hθ0 (Ω); (3.307a)

‖u(fn)|Γ − u(f0)|Γ ‖C([0,T ];H

12+θ

(Γ ))≤ C‖fn − f0‖Hθ0 (Ω) → 0. (3.307b)

Combining (3.307) with the homogeneous I.C. in (3.295b) and B.C. in (3.295c),we obtain

u(f0)

(· , T

2

)= 0; (u(f0))t

(· , T

2

)= 0 in Ω; u(f0)|Σ = 0, (3.308)

hence∂u(f0)

∂ν

(· , T

2

) ∣∣∣∣Γ

= 0 on Γ. (3.309)

Step (vi). Combining (3.306b) with (3.309) yields

∂u(f0)

∂ν(x, t)

∣∣∣∣Γ1

= 0. (3.310)

Step (vii). Ultimately, starting from (3.295), u(f0) satisfies weakly the followinglimit problem, via (3.307), (3.308), and (3.310),

utt(f0)−∆u(f0)− q(x)ut(f0) = f0(x)R(x, t) in Q;

u(f0)

(· , T

2

)= 0; ut(f0)

(· , T

2

)= 0 in Ω;

u(f0)|Σ = 0 and∂u(f0)

∂ν|Σ1 = 0 in Σ, Σ1,

(3.311a)

(3.311b)

(3.311c)

f0 ∈ Hθ0 (Ω); q ∈ L∞(Ω); R,Rt, Rtt ∈ L∞(Q); R

(x,T

2

)∈W 1,∞(Ω),

(3.312)0 < θ ≤ 1, θ 6= 1

2, by virtue of (3.306a) and assumption (3.202), moreover, assump-

tion (3.199) holds. Thus, the uniqueness Theorem 15 applies and yields the followingconclusion

f0(x) ≡ 0, a.e. x ∈ Ω. (3.313)

which contradicts (3.301). Thus, assumption (3.294) is false and inequality (3.293)holds true and Proposition 4, as well as the RHS inequality in (3.203) of Theorem16 are then established.

Step 9. The LHS inequality in (3.203) of Theorem 16 is a-fortiori contained in thefollowing proposition.

Proposition 5. Consider the v = ut-problem as in (3.245), with Rt ∈ L∞(Q) asin (3.243a), and f ∈ Hθ

0 (Ω), q ∈ L∞(Ω). Then the following inequality holds true:There exists C = CT,θ > 0 such that∥∥∥∥∂ut∂ν

∥∥∥∥Hθ(Σ)

≤ C‖f( · )‖Hθ(Ω), θ > 0. (3.314)

Proof. Apply the regularity result Theorem 4, Eqn. (3.20) to the v = ut-problem(3.245) for m = θ.


Uniqueness and stability of nonlinear inverse problem for thew-problem (3.193): Proof of Theorems 17 and 18

Step 1. Orientation. We return to the non-homogeneous w-problem (3.193). Letw(q), w(p) be solutions of problem (3.193) due to the damping coefficients q(·) andp(·), respectively. By the change of variable as in (3.195),

f(x) ≡ q(x)− p(x); u(x, t) = w(q)(x, t)−w(p)(x, t); R(x, t) = wt(p)(x, t), (3.315)

then the variable u satisfies problem (3.196), for which Theorems 15 and 16 pro-vide the corresponding uniqueness and stability results. We here seek to reduce the(nonlinear) uniqueness and stability results for the original w-problem (3.193) to the(linear) uniqueness and stability results for the u-problem (3.196), Theorems 15 and16. To this end, we need to verify for the term R(x, t) = wt(p)(x, t) in (3.315) the as-sumptions required in equations (3.198)+(3.199) of the uniqueness Theorem 15 andequations (3.202)+(3.199) of the stability Theorem 16. For this, since q, p ∈ L∞(Ω),by assumption in the uniqueness Theorem 17 and q, p ∈ Hθ(Ω) by assumption inthe stability Theorem 18, respectively, we then have f = q − p ∈ L∞(Ω) ⊂ L2(Ω)in the first case and, f = q− p ∈ Hθ(Ω)∩L∞(Ω) in the second case, as required in(3.198) and (3.202). Moreover, assumption (3.207) implies the positivity of R(·, T

2)

in (3.199), while ∇w1 ∈ L∞(Ω) in assumption (3.205) implies Rxi(x,T2

) ∈ L∞(Ω)in (3.198) and (3.202). Thus, in order to be able to invoke the uniqueness and stabil-ity results, Theorem 15 and 16, for the variable u = w(q)−w(p) in (3.315), solutionof problem (3.196), what is left is to verify the regularity properties (3.198) and(3.202) on R defined by (3.315), i.e., the following regularity properties:

wt(p), wtt(p), wttt(p) ∈ L∞(Q); wtt(p) ∈ H2θ(0, T ;W θ,∞(Ω)), 0 < θ <1

2, (3.316)

as a consequence of suitably smooth I.C. w0, w1 in (3.193b) [with q(·) replaced byp(·)] and, respectively, of suitably smooth Dirichlet boundary term g in (3.193c). ByTheorem 18, both conditions in (3.316) are needed for stability, while the first oneis needed for uniqueness via Theorem 17. This program will be accomplished below.

Step 2. Proposition 6 (a) With reference to the w-problem (3.193), with q ∈L∞(Ω), let (with m non-necessarily integer):

q ∈Wm,∞(Ω), w0, w1 ∈ Hm+1(Ω)×Hm(Ω) and g ∈ Hm+1(Σ), (3.317)

where all Compatibility Conditions (trace coincidence) which make sense are satis-fied. Then, the corresponding solution w(q) satisfies (a-fortiori) the following regu-larity properties: continuously,

wt(q), wtt(q), wttt(q) ∈ C([0, T ];Hm(Ω)×Hm−1(Ω)×Hm−2(Ω)). (3.318)

(b) In additional, let m > dimΩ2

+ 2. Then, continuously,

wt(q), wtt(q), wttt(q) ∈ L∞(Q). (3.319)

(c) Let now q ∈ Hθ(Ω) ∩ L∞(Ω), m be as in (b). Then

wtt(q) ∈ H2θ(0, T ;W θ,∞(Ω)), 0 < θ < min

1

2,m− 1− dimΩ

2

3

. (3.320)


Proof. (a) Step 1. We start with the following Φ-problem:Φtt(x, t) = ∆Φ(x, t) in Q;

Φ(· , T

2

)= Φ0(x) = w0(x); Φt

(· , T

2

)= Φ1(x) = w1(x) in Ω;

Φ(x, t)|Σ = g(x, t) in Σ.

(3.321a)

(3.321b)

(3.321c)

which corresponds to the w-problem (3.193) with q = 0. Its optimal regularityis given by [LLT86, Remark 2.10, p.167] (generalizing Theorem 1.5, p.164): underassumptions (3.317), along with all Compatibility Conditions (trace coincidence)which make sense (they invoke g, w0, w1), we obtain

Φ,Φt, Φtt, Φttt ∈ C([0, T ];Hm+1(Ω)×Hm(Ω)×Hm−1(Ω)×Hm−2(Ω)), (3.322)

along with ∂Φ∂ν|Σ ∈ Hm(Σ) (which is not needed in the present proof). Thus [LM72,

p. 45],

Φ,Φt, Φtt, Φttt ∈ L∞(Q), for m >dimΩ

2+ 2. (3.323)

Step 2. We sety = w − Φ, (3.324)

where then, by (3.193) and (3.321), y solves

ytt(x, t) = ∆y + q(x)yt + F in Q;

y

(· , T

2

)= y0(x) = 0; yt

(· , T

2

)= y1(x) = 0 in Ω;

y|Σ = 0 in Σ;

F = q(x)Φt in Q;

(3.325a)

(3.325b)

(3.325c)

(3.325d)[y(t)

yt(t)

]=

∫ t

0

eAq(t−s)

[0

q( · )Φt(s)

]ds; (3.326)

Aq =

[0 I

−AD q( · )

]; A0 =

[0 I

−AD 0,

](3.327)

ADh = −∆h, D(AD) = H2(Ω)×H10 (Ω). The following Lemma is readily shown:

Step 3. Lemma 6 Assume

q( · ) is a bounded operator: D(Am2D )→ D(A

m2D ); (3.328)

so that, then, with respect to (3.327), we have:

the operator Aq is a bounded perturbation of the operator A0 on the state

space D(A mq ) = D(A m

0 ) = D(Am+1

2D )×D(A

m2D ), hence with equal domains

on that space.(3.329)

Assume further that

q( · )Φt ∈ C([0, T ]; D(Am2D )), (3.330)


with Φt as in (3.322). Then y(t)

yt(t)

∈ C[0, T ]; D(A m

q ) = D(A m0 ) =

D(Am+1

2D )

D(A mD )

(3.331)

⊂ C

[0, T ];

Hm+1(Ω)

Hm(Ω)

. (3.332)

Remark 21. We shall collect at the end the assumptions on the coefficient q, that willensure that all the required assumptions—such as (3.328) and (3.330) and othersbelow—are satisfied.

Step 4. Lemma 7 Assume hypotheses (3.328) and (3.330) of Lemma 6, so that theregularity properties (3.332) hold true: y, yt ∈ C([0, T ];Hm+1(Ω)×Hm(Ω)). Letm, m+1 6= integer

2. Furthermore, with reference to Φ,Φt, Φtt ∈ C([0, T ];Hm+1(Ω)×

Hm(Ω)×Hm−1(Ω)) as in (3.322), assume [MS85]

q( · ) ∈M(Hm(Ω)→ Hm−1(Ω)); q( · ) ∈M(Hm−1(Ω)→ Hm−2(Ω)). (3.333)

Then, recalling (3.225a-d),

ytt = ∆y + q( · )yt + q( · )Φt ∈ C([0, T ];Hm−1(Ω)); (3.334)

yttt = ∆yt + q( · )ytt + q( · )Φtt ∈ C([0, T ];Hm−2(Ω)). (3.335)

Proof. The proof is immediate: For m, m + 1 6= positive integer2

, ∆y,∆yt ∈C([0, T ];Hm−1(Ω) × Hm−2(Ω)); moreover, qyt, qΦt ∈ C([0, T ];Hm−1(Ω)), qytt,qΦtt ∈ C([0, T ];Hm−2(Ω)).

Step 5. Corollary 2 Assume condition (3.317) for w0, w1, g, as well as the hy-potheses of Lemma 7. Then, with reference to (3.323), (3.324) and (3.335), we have

wttt = yttt + φttt ∈ L∞(Q), m >dimΩ

2+ 2. (3.336)

Step 6. Here we collect all requirements of Lemma 7 (or Corollary 2), which include(3.328), (3.330) and (3.333). Conditions (3.328), (3.330) of Lemma 6 include thefollowing regularity conditions

q ∈M(Hm(Ω)→ Hm(Ω)) [included in (3.328)], (3.337)

plus boundary compatibility conditions (B.C.C.) to be discussed below. Lemma 7does not impose additional restrictions. Thus, in terms of just regularity properties,we need to require:

For q : q ∈M(Hm(Ω)→ Hm(Ω)) as in (3.337) (3.338)

Conclusion #1: All the regularity properties can be fulfilled by assuming

q ∈Wm,∞(Ω) (3.339)

as in hypothesis (3.317).


Conclusion #2: In addition, boundary compatibility conditions (B.C.C.) re-lated to (3.328) and (3.330) need to be imposed. One can obtain the following results

Case dim Ω = 2. In this case the only boundary compatibility condition is

∇q tangential on Γ ; or∂q

∂ν

∣∣∣∣Γ

= 0. (3.340)

Case dim Ω = 3. In this case, we need, in addition to (3.340) also

∇∆q|Γ tangential to Γ ; or∂∆q

∂ν

∣∣∣∣Γ

= 0; [∂xi∇q]Γ = [∇qxi ]Γ = 0, i = 1, 2, 3.

(3.341)(b) Apply the usual embedding since m > dimΩ

2+ 2 to obtain part (b).

(c) Apply the intermediate derivative theorem [LM72, m = 1, l = θ, p. 15] onwtt and Dtwtt in (3.320), with C([0, T ]; ·) replaced by L2(0, T ; ·), we obtain

wtt(q) ∈ H2θ(0, T ;Hm−1−2θ(Ω)), 0 < θ <1

2, (3.342)

calling 2θ, 0 ≤ 2θ ≤ 1, the parameter of interpolation, to match the notationin (3.317). Comparing with wtt in (3.317), we see that we need to ascertain thatHm−1−2θ(Ω) ⊂ W θ,∞(Ω), or Hm−1−3θ(Ω) ⊂ L∞(Ω) which holds true providedm− 1− 3θ > dimΩ

2. Thus (3.320) is proved.

Completion of the Proof of Theorems 17 and 18. Having verified prop-erties (3.198) and (3.199), it follows via (3.317) that we have verified the properties(3.198) and (3.199) for the u-problem (3.196), with u defined by (3.315). Thus,Theorem 15 and 16 apply, and we then obtain uniqueness result q(x) = p(x) as inTheorem 17 and stability result of the conclusion of Theorem 18

c

∥∥∥∥∂wt(q)∂ν− ∂wt(p)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

≤ ‖q − p‖Hθ(Ω) ≤ C∥∥∥∥∂wt(q)∂ν

− ∂wt(p)

∂ν

∥∥∥∥Hθ(0,T ;L2(Γ1))

. (3.343)

3.5.4 Notes and literature

The present Section 3.5 is an improved version of [LT12] with the additional materialof Subsection 3.5.3 to obtain the final results on uniqueness and stability of theoriginal nonlinear inverse problem for system (3.193a-c) explicitly in terms of theproblem’s data.

On the uniqueness issue, the present Section 3.5, i.e. [LT12] refines prior liter-ature on various fronts; principally [BCI01] in the case of damping coefficient as inEqn. (3.193a); and [Yam99] in the case of the lower-order source or potential term.First, our Theorems 15 and 17 require weaker geometrical conditions on the sub-boundary Γ1 (where measurement takes place) than [BCI01], by relying on a moregeneral potential vector field h(x) = ∇d as in (3.2a) rather than the radial vectorfield h(x) = x− x0; while in [Yam99], Γ1 is the entire boundary Γ = ∂Ω. Next, ournonlinear uniqueness Theorem 17 is expressed in terms of (sharp) assumptions onthe non-homogeneous data (by critically invoking [LLT86], [LT81], [LT83]), while


both [BCI01, Theorem 2, p.29] in the damping coefficient case, as well as [Yam99,Theorem 2, p.68] in the source term case make assumptions (stronger than neces-sary) on the regularity of the respective solutions. However, it is on the stabilityissue that lie the main results of the present Section ([LT12]): Theorem 16 (linearinverse problem) and above all Theorem 18 (nonlinear inverse problem), which isexpressed–again–explicitly in terms of sharp assumptions on the data, not on theregularity of the solutions. Reference [BCI01] does not study the stability issue,while reference [Yam99] studies the stability issue by making (too strong) assump-tions on the regularity of the solutions. As in [Yam99], our stability estimate isoptimal (it is expressed by a double inequality, see (3.203) and (3.110)–this is done,again, by invoking the optimal regularity results of [LLT86], [LT81], [LT83] in onedirection). However, the Lipschitz estimate of our Theorem 16 and 18 are at theHθ-level, 0 < θ ≤ 1, θ 6= 1

2(For the stability of the nonlinear inverse problem, we

deliberately restrict to the range 0 < θ < 12

of greater interest. The case 12< θ ≤ 1

can be done likewise). A novelty–and difficulty of the present problem–is that forθ = 0, the required compactness regularity result fails, thus preventing the use ofa compactness-uniqueness argument to absorb lower-order terms (the ∂z

∂ν-term ap-

pearing in (3.255)). To overcome this obstacle and obtain the final Hθ(Ω)-Lipschitzstability estimate for all 0 < θ ≤ 1, θ 6= 1

2, heavy use is made of both optimal

regularity theory [LLT86] and multiplier theory [MS85]. The entire treatment of thepresent Section relies on the Carleman estimates for general second-order hyperbolicequations recalled in Section 1. As noted in Section 3.1, they have the double advan-tage of being expressed for H1,1(Q)-solutions on the entire cylinder Q = Ω × [0, T ],and moreover with explicit boundary traces as in (3.13). In contrast, in [BCI01] andin [Yam99], the Carleman estimates are expressed over the set Q(σ): this suffices foruniqueness with a lower order term as the source coefficient for one equation.

3.6 Inverse problems for a system of strongly coupledwave equations with Neumann boundary data: Globaluniqueness and Lipschitz stability

3.6.1 The coupled hyperbolic system with two unknown dampingcoefficients

Following [LT94], we consider the following coupled system of two second-orderhyperbolic equations in the unknowns w = w(x, t) and z = z(x, t) on Q = Ω× [0, T ]:

wtt = ∆w + q(x)zt; ztt = ∆z + p(x)wt in Q; (3.344a)

w

(· , T

2

)= w0(x), wt

(· , T

2

)= w1(x) in Ω; (3.344b)

z

(· , T

2

)= z0(x), zt

(· , T

2

)= z1(x) in Ω; (3.344c)

∂w

∂ν|Σ = µ1(x, t);

∂z

∂ν|Σ = µ2(x, t) in Σ. (3.344d)


Here q(x), p(x) are the time independent unknown damping coefficients. Instead,[w0, w1, z0, z1] are the given initial conditions (I.C.) and µ1, µ2 are the given Neu-mann boundary conditions (B.C.). We shall denote by w(q, p), z(q, p) the so-lution to problem (3.344) due to the damping coefficients q, p (and fixed dataw0, w1, z0, z1, µ1, µ2). A sharp interior and boundary regularity theory of the cor-responding coupled mixed problem (3.344a-d) may be given following the singleequation case from [LT83, LT89] (see also [LT90] and [LT96, Ch. 8, Sect. 8A, p. 755],and [Tat92], in part reported in Section 3.2.

We note that the map q, p → w, z is nonlinear and hence consider thefollowing nonlinear inverse problem.

I: Nonlinear inverse problem for system (3.344): Let w = w(q, p), z =z(q, p) be a solution to system (3.344). Under geometrical conditions on the unob-served part Γ0 of the boundary Γ , is it possible to retrieve q(x) and p(x), x ∈ Ω,from measurement of the Dirichlet traces of w(q, p) and z(q, p) on the observed partof the boundary Γ1 × [0, T ] over a sufficiently large time interval T? This problemcomprises two basic issues: uniqueness and stability. More precisely, we consider

I(1): Uniqueness in the nonlinear inverse problem for system w, zin (3.344). Let w = w(q, p), z = z(q, p) be the solution to system (3.344). Un-der geometrical conditions on Γ0, do the Dirichlet boundary traces w|Γ1×[0,T ] andz|Γ1×[0,T ] determine q(x) and p(x) uniquely? In other words,

does

w(q1, p1) = w(q2, p2)|Γ1×[0,T ]

z(q1, p1) = z(q2, p2)|Γ1×[0,T ]

imply

q1(x) = q2(x)

p1(x) = p2(x)a.e. in Ω? (3.345)

Assuming that the answer to the aforementioned uniqueness question (3.345) isin the affirmative, we then ask the following more demanding, quantitative estimate.

I(2): Stability in the nonlinear inverse problem for system w, z in(3.344). In the above setting, let w(q1, p1), z(q1, p1), w(q2, p2), z(q2, p2) be so-lutions to (3.344) due to corresponding damping coefficients q1, p1, and q2, p2and fixed common data w0, w1, z0, z1, µ0, µ1. Under geometric conditions on thecomplementary unobserved part of the boundary Γ0 = Γ \ Γ1, is it possible to esti-mate of the norms ‖q1 − q2‖L2(Ω), ‖p1 − p2‖L2(Ω) in terms of suitable norms of theDirichlet traces (w(q1, p1)− w(q2, p2))|Γ1×[0,T ] and (z(q1, p1)− z(q2, p2))|Γ1×[0,T ]?

II: The corresponding homogeneous linear inverse problem. As beforethe nonlinear inverse problem is converted into a linear inverse problem for anauxiliary, corresponding problem. Let

f(x) = q1(x)− q2(x), g(x) = p1(x)− p2(x); (3.346a)

R1(x, t) = zt(q2, p2)(x, t), R2(x, t) = wt(q2, p2)(x, t); (3.346b)

u(x, t) = w(q1, p1)(x, t)−w(q2, p2)(x, t), v(x, t) = z(q1, p1)(x, t)−z(q2, p2)(x, t).(3.346c)

Then u(x, t), v(x, t) satisfies the following homogeneous system:


utt(x, t)−∆u(x, t)− q(x)vt(x, t) = f(x)R1(x, t) in Q; (3.347a)

vtt(x, t)−∆v(x, t)− p(x)ut(x, t) = g(x)R2(x, t) in Q; (3.347b)

u( · , T

2) = 0, ut( · ,

T

2) = 0; v( · , T

2) = 0, vt( · ,

T

2) = 0 in Ω; (3.347c)

∂u

∂ν

∣∣∣∣Σ

= 0;∂v

∂ν

∣∣∣∣Σ

= 0 in Σ. (3.347d)

The above serves as a motivation. Henceforth, we shall consider the u, v-problem,with damping coefficients q, p ∈ L∞(Ω) as given, and terms R1(x, t), R2(x, t) fixedand suitable while the terms f(x), g(x) are unknown time-independent coefficients.The u, v-problem has three advantages over the original w, z-problem in (3.344):it has homogeneous I.C. and B.C. and, above all, the map f, g → u, v is linear.

II(1): Uniqueness in the linear inverse problem for system u, v in(3.347). Let u = u(f, g), v = v(f, g) be the solution to system (3.347). Undergeometrical conditions on Γ0, do the Dirichlet traces u|Γ1×[0,T ] and v|Γ1×[0,T ] deter-mine f(x) and g(x) uniquely? In other words, by linearity,

does

u(f, g)|Γ1×[0,T ] = 0

v(f, g)|Γ1×[0,T ] = 0imply =⇒

f(x) = 0

g(x) = 0a.e. in Ω? (3.348)

Assuming that the answer to the aforementioned uniqueness question (3.348) isin the affirmative, we then ask the following more demanding, quantitative estimate.

II(2): Stability in the linear inverse problem for system u, v in(3.347). In the above setting, let u(f, g), v(f, g) be solution to (3.347). Un-der geometric conditions on the complementary unobserved part of the boundaryΓ0 = Γ \ Γ1, is it possible to estimate of the norms ‖f‖L2(Ω), ‖g‖L2(Ω) in terms ofsuitable norms of the Dirichlet traces u(f, g)|Γ1×[0,T ] and v(f, g)|Γ1×[0,T ]?

3.6.2 Main results

Next we give answers to all above uniqueness and stability questions. We begin witha uniqueness result for the linear inverse problem (3.348) involving the u, v-system(3.347).

Theorem 20 (Uniqueness of linear inverse problem). Assume the preliminarygeometric assumptions (A.1): (3.2b), (3.3), (A.2) = (3.4). Let T > T0 be as in(3.49). With reference to the u, v-system (3.347), let the fixed data q, p, R1, R2and unknown terms f and g satisfy the following regularity properties

q, p ∈ L∞(Ω); Ri, Rit, Ritt ∈ L∞(Q), Ri

(· , T

2

)∈W 1,∞(Ω); f, g ∈ L2(Ω),

(3.349)i = 1, 2, j = 1, · · · , n, as well as the following positivity conditions∣∣∣∣R1

(x,T

2

) ∣∣∣∣ ≥ r1 > 0,

∣∣∣∣R2

(x,T

2

) ∣∣∣∣ ≥ r2 > 0, x ∈ Ω, (3.350)


for some positive constants r1, r2. If the solution u = u(f, g), v = v(f, g) to system(6.4) satisfies the additional homogeneous Dirichlet boundary trace condition

u(f, g)(x, t) = v(f, g)(x, t) = 0, x ∈ Γ1, t ∈ [0, T ], (3.351)

over the observed part Γ1 of the boundary Γ and over the time interval T as in(3.49), then, in fact

f(x) = g(x) ≡ 0, a.e. x ∈ Ω. (3.352)

Next, we provide the stability result for the linear inverse problem involving theu, v-system (3.347), and the determination of the terms f( · ), g( · ) in (3.347a-b).We shall seek f and g in L2(Ω).

Theorem 21 (Lipschitz stability of linear inverse problem). Assume the pre-liminary geometric assumptions (A.1) = (3.2b), (3.3), (A.2) = (3.4). Consider prob-lem (3.347) on [0, T ] with T > T0, as in (3.49) and data satisfying properties (3.349)where, moreover, R1, R2 satisfy the positivity condition (3.350) at the initial timet = T

2. Then there exists a constant C = C(Ω, T, Γ1, ϕ, q, p, R1, R2) > 0, i.e., de-

pending on the data of problem (3.347), but not on the unknown coefficients f andg, such that

‖f‖L2(Ω) + ‖g‖L2(Ω) ≤

C(‖ut(f)‖L2(Σ1)+‖utt(f)‖L2(Σ1)+‖vt(f)‖L2(Σ1)+‖vtt(f)‖L2(Σ1)

)(3.353)

for all f, g ∈ L2(Ω).

Next we give the corresponding uniqueness and stability results to the nonlinearinverse problem invoking the w, z-system (3.344).

Theorem 22 (Uniqueness of nonlinear inverse problem). Assume the pre-liminary geometric assumptions (A.1) = (3.2b), (3.3), (A.2) = (3.4). Let T be as in(3.49). Assume further the following a-priori regularity of the unknown coefficientsfor (3.344):

q1, q2, p1, p2 ∈Wm,∞(Ω). (3.354)

Assume further that the I.C. of (3.344) satisfy the following regularity and positivityconditions

w0, w1, z0, z1 ∈ H`+1(Ω)×H`(Ω), ` >dimΩ

2+ 2, w1xj , z1xj ∈ L

∞(Ω),

(3.355)|w1(x)| ≥ w1 > 0, |z1(x)| ≥ z1 > 0, x ∈ Ω, (3.356)

for some positive constants w1, z1, where AN is the operator in (3.454) below, andthe non-homogemeous B.C. satisfy

µi ∈ Hm(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12

+(m−1)(Γ )),

α =2


3


with Compatibility Relations (C.R.) trace coincidence that make sense.(3.357)

for


m >dimΩ

2+ 3− α. (3.358)

Finally, if the solutions w(q1, p1), z(q1, p1) and w(q2, p2), z(q2, p2) to system(6.1) have the same Dirichlet boundary traces on Σ1 = Γ1 × [0, T ]:

w(q1, p1)(x, t) = w(q2, p2)(x, t), z(q1, p1)(x, t) = z(q2, p2)(x, t), (3.359)

then, in fact,the respective damping coefficients coincide

q1(x) = q2(x), p1(x) = p2(x), a.e. x ∈ Ω. (3.360)

Finally, we state the stability result for the nonlinear inverse problem involvingthe w, z-problem (3.344) with damping coefficient q( · ) and p( · ).

Theorem 23 (Lipschitz stability of nonlinear inverse problem). Assume pre-liminary geometric assumptions (A.1): (3.2b), (3.3), (A.2) = (3.4). Consider prob-lem (3.344) on [0, T ], with T > T0 as in (3.49), one time with damping coefficientsq1, p1 ∈ Wm,∞(Ω), and one time with damping coefficients q2, p2 ∈ Wm,∞(Ω),subject to assumptions of Theorem 21, and let w(q1, p1), w(q2, p2) denote the corre-sponding solutions. Assume the regularity and positivity conditions (3.355), (3.356)on the initial data and regularity property (3.357) on the boundary data. Then, thefollowing stability result holds true for the w-problem (3.344): there exists a con-stant C = C(Ω, T, Γ1, ϕ,M,w0, w1, z0, z1, µ1, µ2) > 0, i.e., depending on the data ofproblem (3.344) and on the L∞(Ω)-norm of the damping coefficients such that

‖q1 − q2‖L2(Ω) + ‖p1 − p2‖L2(Ω)

≤ C(‖wt(q1, p1)− wt(q2, p2)‖L2(Σ1) + ‖wtt(q1, p1)− wtt(q2, p2)‖L2(Σ1)

+ ‖zt(q1, p1)− zt(q2, p2)‖L2(Σ1) + ‖ztt(q1, p1)− ztt(q2, p2)‖L2(Σ1)) (3.361)

for all coefficients q1, p1, q2, p2 ∈ q = q1, q2 ∈ L∞(Ω) × L∞(Ω)| ‖q‖L∞ ≤ M,for a fixed arbitrary constant M .

3.6.3 Proofs

Uniqueness of linear inverse problem for the u, v-system (3.347):Proof of Theorem 20

Step 1. Proposition 7 Assume (A.1): (3.2b), (3.3), (A.2) = (3.4) and T > T0,q, p ∈ L∞(Ω), Ri ∈ L∞(Q), i = 1, 2, f, g ∈ L2(Ω). Then, the following one-parameter family of energy estimates holds true for the u, v-system (3.347) satis-fying also the Dirichlet B.C. (3.351), for all τ > 0 sufficiently large:

C1,τ

∫Q

e2τϕ[|∇u|2 + u2t + |∇v|2 + v2

t ]dQ+ C2,τ

∫Q(σ)

e2τϕ[u2 + v2]dxdt

≤ Cp,q

∫Q

e2τϕ[u2t + v2

t ]dQ+ C1,T e2τσ

∫Q

[u2 + v2]dQ

+ 4

∫Q

e2τϕ[|fR1|2 + |gR2|2]dQ

+ cT τ3e−2τδ[Eu(0) + Eu(T ) + Ev(0) + Ev(T )]. (3.362a)


where C1,τ , C2,τ are the constants in (3.12), that is

C1,τ = τερ− 2CT , C2,τ = 2τ3β + O(τ2)− 2CT , (3.362b)

Proof. Under present assumptions on p, q, f, g, Ri, system (3.347), rewritten here asuttvtt

=

∆ 0

0 ∆

uv

+

0 q(x)

p(x) 0

utvt

+

f(x)R1(x, t)

g(x)R2(x, t),

(3.363)

with zero I.C. as in (3.347c) and homogeneous B.C. (3.347d) possesses a-fortiorithe regularity u, v ∈ H1(Q) × H1(Q). Moreover, also because of (3.351) andh · ν = 0 on Γ0 in (3.2b), we have that, in view of Theorem 2, we can applythe Carleman estimate (3.11) of Theorem 1 to the u-equation (3.347a) and thev-equation (3.347b) separately, where—to fit model (3.1a)—we have Fu(x, t) =q(x)vt(x, t) + f(x)R1(x, t) and F v(x, t) = p(x)ut(x, t) + g(x)R2(x, t) respectively.We then obtain

BT |Σ(u) + 2

∫Q

e2τϕ|qvt + fR1|2dQ+ C1,T e2τσ

∫Q

u2dQ

≥ C1,τ

∫Q

e2τϕ[u2t + |∇u|2]dQ

+ C2,τ

∫Q(σ)

e2τϕu2dxdt− cT τ3e−2τδ[Eu(0) + Eu(T )]; (3.364)

BT |Σ(v) + 2

∫Q

e2τϕ|put + gR2|2dQ+ C1,T e2τσ

∫Q

v2dQ

≥ C1,τ

∫Q

e2τϕ[v2t + |∇v|2]dQ

+ C2,τ

∫Q(σ)

e2τϕv2dxdt− cT τ3e−2τδ[Ev(0) + Ev(T )], (3.365)

respectively, with boundary terms defined by (3.13), which, in fact, now vanish by(3.347d) on Γ , (3.351) on Γ1, and h ·ν = 0 on Γ0. We obtain, recalling q, p ∈ L∞(Ω)

BT |Σ(u) ≡ 0;∫Q

e2τϕ|qvt + fR1|2dQ ≤ Cq∫Q

e2τϕ|vt|2dQ+ 2

∫Q

e2τϕ|fR1|2dQ; (3.366)

BT |Σ(v) ≡ 0;∫Q

e2τϕ|put + gR2|2dQ ≤ Cp∫Q

e2τϕ|ut|2dQ+ 2

∫Q

e2τϕ|gR2|2dQ. (3.367)

Adding (3.364) and (3.365), and taking into account (3.366), (3.367) yields(3.362a).

Step 2. We differentiate system (3.347) in t, supplemented by the over-determinedB.C. (3.351) and obtain, invoking also the I.C. (3.347c):


(ut)tt(x, t)−∆(ut)(x, t)− q(x)(vt)t(x, t) = f(x)R1t(x, t) in Q;

(vt)tt(x, t)−∆(vt)(x, t)− p(x)(ut)t(x, t) = g(x)R2t(x, t) in Q;

(ut)

(· , T

2

)= 0, (ut)t

(· , T

2

)= f(x)R1

(x,T

2

)∈ L2(Ω)in Ω;

(vt)

(· , T

2

)= 0, (vt)t

(· , T

2

)= g(x)R2

(x,T

2

)∈ L2(Ω)in Ω;

∂

∂ν(ut)(x, t) = 0,

∂

∂ν(vt)(x, t) = 0; ut = 0, vt = 0

(3.368a)

(3.368b)

(3.368c)

(3.368d)

(3.368e)

Proposition 8. Assume the hypotheses of Proposition 7 with, in addition, Rit ∈L∞(Q), Ri( · , T2 ) ∈W 1,∞(Ω), i = 1, 2. Then, the following one-parameter family ofenergy estimates holds true for the ut, vt-system (6.25), for all τ > 0 sufficientlylarge:

C1,τ

∫Q

e2τϕ[|∇ut|2 + u2tt + |∇vt|2 + v2

tt]dQ+ C2,τ

∫Q(σ)

e2τϕ[u2t + v2

t ]dxdt

≤ Cp,q

∫Q

e2τϕ[u2tt + v2

tt]dQ+ C1,T e2τσ

∫Q

[u2t + v2

t ]dQ

+ 4

∫Q

e2τϕ[|fR1t|2 + |gR2t|2]dQ

+ cT τ3e−2τδ[Eut(0) + Eut(T ) + Evt(0) + Evt(T )]. (3.369)

Proof. We now have that ut, vt ∈ H1(Q)×H1(Q), since f(x)R1(x, T2

), g(x)R2(x,T2

) ∈ L2(Ω) and f(x)R1t, g(x)R2t ∈ L2(Q), under present assumptions. Thus thesame proof of Proposition 7 applies, based on Theorem 2, as ut and vt both vanishon Γ1 × [0, T ] as in (6.25e).

Step 3. We differentiate system (3.368) in t one more time and obtain, invokingalso the I.C. (3.368c-d):

(utt)tt(x, t)−∆(utt)(x, t)− q(x)(vtt)t(x, t) = f(x)R1tt(x, t) (3.370a)

(vtt)tt(x, t)−∆(vtt)(x, t)− p(x)(utt)t(x, t) = g(x)R2tt(x, t) (3.370b)

(utt)( · ,T

2) = f(x)R1(x,

T

2) (3.370c)

(utt)t( · ,T

2) = f(x)R1t(x,

T

2) + q(x)g(x)R2(x,

T

2) ∈ L2(Ω) (3.370d)

(vtt)( · ,T

2) = g(x)R2(x,

T

2) (3.370e)

(vtt)t( · ,T

2) = g(x)R2t(x,

T

2) + p(x)f(x)R1(x,

T

2) ∈ L2(Ω) (3.370f)

∂

∂ν(utt)(x, t) = 0,

∂

∂ν(vtt)(x, t) = 0; utt = 0, vtt = 0 (3.370g)

We note that, under present assumptions f, g ∈ L2(Ω) and p, q, R1( · , T2

),R2( · , T

2) ∈ L∞(Ω) as in (3.349), we have:

(utt)t

(· , T

2

)∈ L2(Ω), (vtt)t

(· , T

2

)∈ L2(Ω), (3.371)


as desired; however, at least at a first glance, (utt)( · , T2 ) and (vtt)( · , T2 ) are onlyin L2(Ω), not in H1(Ω), as needed to invoke Carleman estimates. But as we haveshown in Lemma 1 for a single equation (Neumann case) and in Lemma 3 (Dirichletcase), the special over-determined structure of the utt, vtt-system (3.370) wouldboost the regularity of f and g to H1(Ω) (by invoking the COI (3.43)) of Theorem10, see also Remark 8. Hence, with the enhanced regularity f, g ∈ H1(Ω), see proofof (3.75) of Lemma 1, we have the following regularity for problem (3.370):

f, g ∈ H1(Ω), hence utt, vtt ∈ H1(Q)×H1(Q), (3.372)

so that we can apply the Carleman estimate Theorem 2 for problem (3.370). Weobtain

Proposition 9. Assume (A.1): (3.2b), (3.3), (A.2) = (3.4), q, p ∈ L∞(Ω), as wellas T > T0 and R1tt, R2tt ∈ L∞(Q), Ri( · , T2 ) ∈ L∞(Ω). Then, the following one-parameter family of energy estimates holds true for the utt, vtt-system (3.370), forall τ > 0 sufficiently large:

C1,τ

∫Q

e2τϕ[|∇utt|2 + u2ttt + |∇vtt|2 + v2

ttt]dQ+ C2,τ

∫Q(σ)

e2τϕ[u2tt + v2

tt]

≤ Cp,q

∫Q

e2τϕ[u2ttt + v2

ttt]dQ+ C1,T e2τσ

∫Q

[u2tt + v2

tt]dQ

+ 4

∫Q

e2τϕ[|fR1tt |2 + |gR2tt |

2]dQ

+ cT τ3e−2τδ[Eutt(0) + Eutt(T ) + Evtt(0) + Evtt(T )] (3.373)

Step 4. Under the assumptions of Propositions 7 through 9 cumulatively, that is,(3.349) and f, g ∈ H1(Ω), we sum up (3.362a), (3.369) and (3.373) to obtain

Proposition 10. Assume (A.1): (3.2b), (3.3), (A.2) = (3.4), T > T0, (3.349) andf, g ∈ L2(Ω) (which then are enhanced to H1(Ω) as in (3.372). Then the followingone-parameter family of energy estimates holds true for the u, v-system (3.347),for all τ > 0 sufficiently large:

C1,τ

∫Q

e2τϕ[|∇utt|2+|∇ut|2+|∇u|2+u2ttt+u

2tt+u

2t+|∇vtt|2+|∇vt|2+|∇v|2

+v2ttt+v

2tt+v

2t ]dQ+C2,τ

∫Q(σ)

e2τϕ[u2tt + u2

t + u2] + [v2tt + v2

t + v2]dx dt

≤ Cp,q

∫Q

e2τϕ[u2ttt + u2

tt + u2t ] + [v2

ttt + v2tt + v2

t ]dQ

+C1,T e2τσ

∫Q

[u2tt + u2

t + u2] + [v2tt + v2

t + v2]dQ

+4

∫Q

e2τϕ[|fR1tt|2+|fR1t|2+|fR1|2]+[|gR2tt|2+|gR2t|2+|gR2|2]dQ

+cT τ3e−2τδ[Eu,v]T0 ; (3.374)

[Eu,v]T0 = [Eutt(0) + Eutt(T )] + [Evtt(0) + Evtt(T )] + [Eut(0) + Eut(T )]

+ [Evt(0) + Evt(T )] + [Eu(0) + Eu(T )] + [Ev(0) + Ev(T )]. (3.375)


Step 5. As in Step 5 of Sections 3.4 and 3.5, in this step we derive the fol-lowing estimates on

∫Qe2τϕ|f |2dQ and

∫Qe2τϕ|g|2dQ, following an idea of [Isa06,

Thm. 8.2.2, p. 231]:

Proposition 11. With reference to the third integral term on the RHS of estimate(3.374), assume (3.349), as well as (3.350). Then we have:

(1) ∫Q

e2τϕ [|fR1|2 + |gR2|2 + |fR1t|2 + |gR2t|2 + |fR1tt|2 + |gR2tt|2]dQ

≤ CR

∫Q

e2τϕ[|f |2 + |g|2]dQ; (3.376)

(2)

∫Q

e2τϕ|f |2dQ ≤(

T

r21

)(2cTτ+1

)∫Ω

∫ T2

0

e2τϕ(x,s)|utt(x, s)|2dsdΩ

+T

r21

∫Ω

∫ T2

0

e2τϕ(x,s)|uttt(x, s)|2dsdΩ +T

r21

∫Ω

|utt(x, 0)|2dΩ; (3.377)

(3)

∫Q

e2τϕ|g|2dQ ≤(

T

r22

)(2cTτ+1

)∫Ω

∫ T2

0

e2τϕ(x,s)|vtt(x, s)|2dsdΩ

+T

r22

∫Ω

∫ T2

0

e2τϕ(x,s)|vttt(x, s)|2dsdΩ +T

r22

∫Ω

|vtt(x, 0)|2dΩ. (3.378)

Proof. (1) is obvious, recalling assumption (3.349) onRi, Rit, Ritt ∈ L∞(Q), i = 1, 2.For (2), we return to (3.347a-b), evaluate at the initial time T

2, use (3.347c) and

obtain

utt

(x,T

2

)= f(x)R1

(x,T

2

); vtt

(x,T

2

)= g(x)R2

(x,T

2

). (3.379)

Recalling assumption (3.350), we have

|f(x)| ≤ 1

r1

∣∣∣∣utt(x, T2) ∣∣∣∣; |g(x)| ≤ 1

r2

∣∣∣∣vtt(x, T2) ∣∣∣∣, x ∈ Ω. (3.380)

By virtue of (3.380), and recalling (3.84) from Section 3.4.1, Step 5.

∫Q

e2τϕ|f |2dQ ≤ T

r21

((2cTτ)

∫Ω

∫ T2

0

e2τϕ|utt|2dtdΩ

+

∫Ω

∫ T2

0

e2τϕ(|utt|2 + |uttt|)2dtdΩ +

∫Ω

|utt(x, 0)|2dΩ

). (3.381)

The proof of (3) is similar using (3.381) on g.


Step 6. By substituting estimates (3.376)–(3.378) on the RHS of (3.374), we obtain

Proposition 12. Assume the hypotheses (3.349), (3.350) of Theorem 20. Then, thefollowing one-parameter family of energy estimates holds true for the u, v-system(3.347), for all τ > 0 sufficiently large:

C1,τ

∫Q

e2τϕ

[|∇utt|2 + |∇ut|2 + |∇u|2

]+[|∇vtt|2 + |∇vt|2 + |∇v|2

]dQ

+

[C1,τ −

CRT

r20

− Cp,q] ∫

Q

e2τϕ[u2ttt + v2

ttt]dQ

+ [C1,τ − Cp,q]∫Q

e2τϕ[u2tt + u2

t + v2tt + v2

t ]dQ

+ C2,τ

∫Q(σ)

e2τϕ [u2tt + u2

t + u2] + [v2tt + v2

t + v2]dQ

≤ CRT (2cTτ + 1)

∫Q

e2τϕ[u2tt + v2

tt]dQ

+ C1,T e2τσku,v + autt,vtt + cT τ

3e−2τδ[Eu,v]T0 ; (3.382)

ku,v = ku,v;ut,vt;utt,vtt ≡∫Q

[(u2tt + u2

t + u2) + (v2tt + v2

t + v2)]dQ; (3.383)

CR = CR

(1

r21

+1

r22

); autt,vtt =

T

r21

∫Ω

|utt(x, 0)|2dΩ +T

r22

∫Ω

|vtt(x, 0)|2dΩ,

(3.384)constants depending on the solution u, v, r0 = minr1, r2.

Step 7. Recalling now e2τϕ < e2τσ on Q \ Q(σ) by (3.8), we obtain the followingestimate for the integral term on the RHS of inequality (3.382)∫

Q

e2τϕ[u2tt + v2

tt]dQ =

∫Q(σ)

e2τϕ[u2tt+v

2tt]dtdx+

∫Q\Q(σ)

e2τϕ[u2tt + v2

tt]dx dt

≤∫Q(σ)

e2τϕ[u2tt+v

2tt]dt dx+e2τσ

∫Q\Q(σ)

[u2tt + v2

tt]dx dt.

(3.385)

Substituting inequality (3.385) in the integral term on the RHS of estimate(3.382), we thus obtain the final sought-after estimate:

Proposition 13. Assume the hypotheses (3.349), (3.350) of Theorem 20. Then, thefollowing one-parameter family of energy estimates holds true for the u, v-system(3.347), for all τ > 0 sufficiently large:


C1,τ

∫Q

e2τϕ

[|∇utt|2 + |∇ut|2 + |∇u|2

]+[|∇vtt|2 + |∇vt|2 + |∇v|2

]dQ

+

[C1,τ −

CRT

r20

− Cp,q] ∫

Q

e2τϕ[u2ttt + v2

ttt]dQ

+ [C1,τ − Cp,q]∫Q

e2τϕ[u2tt + u2

t + v2tt + v2

t ]dQ

+[C2,τ − CRT (2cTτ + 1)

] ∫Q(σ)

e2τϕ [u2tt + u2

t + u2] + [v2tt + v2

t + v2]

≤ [C1,T + CRT (2cTτ + 1)]e2τσ

∫Q

[u2tt + u2

t + u2 + v2tt + v2

t + v2]dQ

+ autt,vtt + cT τ3e−2τδ[Eu,v]T0 . (3.386)

Step 8. The ‘final’ estimate (3.386) is more than we need to conclude the argument.First, as all coefficients of the integral terms on the LHS of estimate (3.386) arepositive for τ > 0 sufficiently large, recalling (3.362b), we can drop all these termssave the term

∫Q(σ)

and obtain:[C2,τ−CRT (2cTτ+1)−Cp,q

]∫Q(σ)

e2τϕ [u2tt+u

2t+u2]+[v2

tt+v2t +v2]

≤ (2cTτ + 1)e2τσku,v + cT τ

3e−2τδ[Eu,v]T0 (3.387)

ku,v = constant depending on solution u, v and data (3.388)

But on Q(σ), we have e2τϕ ≥ e2τσ by (3.8). Using this, in the LHS integralof (3.387) and dividing (3.387) across by (2cTτ + 1)e2τσ, we obtain for all τ > 0sufficiently large:

1

2cTτ + 1

[C2,τ − CRT (2cTτ + 1)− Cp,q

] ∫Q(σ)

[u2tt + u2

t + u2]

+[v2tt + v2

t + v2]dQ ≤ ku,v +

cT τ3e−2τδ

(2cTτ + 1)e2τσ[Eu,v]T0 ≤ Constu,v,data (3.389)

Letting τ → +∞ in (3.389), and recalling from (3.362b) that C2,τ grows as τ3,we obtain for (x, t) ∈ Q(σ)

u(x, t) = ut(x, t) ≡ utt(x, t) ≡ 0, v(x, t) = vt(x, t) = vtt(x, t) ≡ 0. (3.390)

Then (3.390) implies ∆u ≡ 0, ∆v ≡ 0 in Q(σ). Thus, returning to (3.347a-b),we then obtain

f(x)R1(x, t) ≡ 0, g(x)R2(x, t) ≡ 0 in Q(σ). (3.391)

Recalling now from (3.9) that [t0, t1]×Ω ⊂ Q(σ) and that t0 <T2< t1, we see

that (3.391) in particular implies:

f(x)R1

(x,T

2

)≡ 0; g(x)R2

(x,T

2

)≡ 0, x ∈ Ω. (3.392)

Thus, by use of assumption (3.350), (3.392) implies

f(x) ≡ 0, g(x) ≡ 0, a.e. x ∈ Ω. (3.393)

The proof of Theorem 20 is completed.


Stability of linear inverse problem for the u, v-system (3.347):Proof of Theorem 21

Step 1. Let u = u(f, g), v = v(f, g) be the solution of problem (3.347), with dataq, p ∈ L∞(Ω), f, g ∈ L2(Ω), Ri, Rit, Ritt ∈ L∞(Q), i = 1, 2;

|Ri(x,T

2

)| ≥ ri > 0, x ∈ Ω, Ri

(x,T

2

)∈W 1,∞(Ω),

(3.394a)

(3.394b)

from assumptions (3.349) and (3.350). Consider again the ut, vt-system (3.368),which we rewrite here for convenience:

(ut)tt(x, t)−∆(ut)(x, t)− q(x)(vt)t(x, t) = f(x)R1t(x, t)in Q (3.395a)

(vt)tt(x, t)−∆(vt)(x, t)− p(x)(ut)t(x, t) = g(x)R2t(x, t)in Q (3.395b)

(ut)( · ,T

2) = 0, (ut)t( · ,

T

2) = f(x)R1(x,

T

2) ∈ L2(Ω)in Ω (3.395c)

(vt)( · ,T

2) = 0, (vt)t( · ,

T

2) = g(x)R2(x,

T

2) ∈ L2(Ω)in Ω (3.395d)

∂

∂ν(ut)(x, t) = 0;

∂

∂ν(vt)(x, t) = 0in Σ (3.395e)

so that f(x)R1

(x, T

2

), g(x)R2

(x, T

2

)∈ L2(Ω). Accordingly, by linearity, we split

the problem ut, vt into two components:

ut = ut + ut; vt = vt + vt (3.396)

where ut, vt satisfies problem (3.395), however, with homogeneous forcing terms:

(ut)tt(x, t)−∆(ut)(x, t)− q(x)(vt)t(x, t) = 0 in Q; (3.397a)

(vt)tt(x, t)−∆(vt)(x, t)− p(x)(ut)t(x, t) = 0 in Q; (3.397b)

(ut)

(· , T

2

)= 0, (ut)t

(· , T

2

)= f(x)R1

(x,T

2

)in Ω; (3.397c)

(vt)

(· , T

2

)= 0, (vt)t

(· , T

2

)= g(x)R2

(x,T

2

)in Ω; (3.397d)

∂

∂ν(ut)(x, t) = 0;

∂

∂ν(vt)(x, t) = 0 in Σ, (3.397e)

while ut, vt satisfies the same problem (3.395), however, with homogeneous I.C.:

(ut)tt(x, t)−∆(ut)(x, t)− q(x)(vt)t(x, t) = f(x)R1t(x, t)in Q; (3.398a)

(vt)tt(x, t)−∆(vt)(x, t)− p(x)(ut)t(x, t) = g(x)R2t(x, t)in Q; (3.398b)

(ut)

(· , T

2

)= 0, (ut)t

(· , T

2

)= 0in Ω; (3.398c)

(vt)

(· , T

2

)= 0, (vt)t

(· , T

2

)= 0in Ω; (3.398d)

∂

∂ν(ut)(x, t) = 0;

∂

∂ν(vt)(x, t) = 0in Σ. (3.398e)


Step 2. Here we apply the continuous observability inequality, Theorem 10,Eqn. (3.43), to the ut, vt-problem (3.397), as assumptions (3.42) are satisfied. Ac-cordingly, there is a constant CT,q,p > 0 depending on T and on the L∞(Ω)-normof the datum q and p but not on f and g, such that∥∥∥∥f( · )R1

(· , T

2

)∥∥∥∥2

L2(Ω)

+

∥∥∥∥g( · )R2

(· , T

2

)∥∥∥∥2

L2(Ω)

≤ C2T,q,p

∫ T

0

∫Γ1

[u2t + u2

tt + v2t + v2

tt]dΓ1 dt, (3.399)

whenever the RHS is finite, where T > T0, as assumed. Since∣∣Ri (x, T2 )∣∣ ≥ ri > 0,

x ∈ Ω, i = 1, 2, by assumption (3.350), we then obtain from (3.399) by use of(3.396), the triangle inequality, with constant C = CT,q,p,r1,r2 :

‖f‖L2(Ω) + ‖g‖L2(Ω)

≤ C(‖ut‖L2(Γ1×[0,T ])+‖utt‖L2(Γ1×[0,T ])+‖vt‖L2(Γ1×[0,T ])+‖vtt‖L2(Γ1×[0,T ])

)≤ C

(‖ut − ut‖L2(Γ1×[0,T ])+‖utt − utt‖L2(Γ1×[0,T ])+‖vt − vt‖L2(Γ1×[0,T ])

+ ‖vtt − vtt‖L2(Γ1×[0,T ])

)≤ C

(‖ut‖L2(Γ1×[0,T ])+‖utt‖L2(Γ1×[0,T ])+‖vt‖L2(Γ1×[0,T ])+‖vtt‖L2(Γ1×[0,T ])

)+ C

(‖ut‖L2(Γ1×[0,T ])+‖utt‖L2(Γ1×[0,T ])+‖vt‖L2(Γ1×[0,T ])+‖vtt‖L2(Γ1×[0,T ])

)(3.400)

Inequality (3.400) is the desired, sought-after estimate (3.353) of Theorem 21,modulo (polluted by) the ut, utt- and vt, vtt-terms. Again such terms will be nextabsorbed by a compactness-uniqueness argument. To carry this through, we firstshow the following lemma.

Step 3. Lemma 8 Consider the ut, vt-system (3.398), with data

q, p ∈ L∞(Ω), f, g ∈ L2(Ω), Rit, Ritt ∈ L∞(Q), Ri

(· , T

2

)∈L∞(Ω), i = 1, 2;

(3.401)Define the following operators K, K1, L and L1:

(Kf, g)(x, t) = ut(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]);

(K1f, g)(x, t) = utt(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]);

(Lf, g)(x, t) = vt(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]);

(L1f, g)(x, t) = vtt(x, t)|Σ1: L2(Ω)→ L2(Γ1 × [0, T ]),

(3.402a)

(3.402b)

(3.402c)

(3.402d)

where ut, vt is the unique solution of problem (3.398). Then,

K, K1, L and L1 are compact operators. (3.403)

Proof. First, under present assumptions (3.401) with zero I.C. (3.398c-d) and ho-mogeneous B.C. (3.398e), system (3.398) with f(x)Rit(x, t) ∈ L2(Q), possesses a-fortiori the regularity


ut, vt ∈ H1(Q)×H1(Q). (3.404)

Moreover, differentiate the system (3.398) in time: we obtain the utt, vtt-system which contains the homogeneous B.C., forcing terms f(x)R1tt(x, t), g(x)R2tt(x, t) ∈L2(Q), and non-zero initial velocity f(x)R1t(x,

T2

) and g(x)R2t(x,T2

) ∈ L2(Ω) underthe present assumptions (3.401). Therefore a-fortiori we obtain also

utt, vtt ∈ H1(Q)×H1(Q). (3.405)

Preliminaries. We now invoke sharp Dirichlet trace theory results Theorem 5II,Eqn. (3.24) for the Neumann hyperbolic problem (3.398). More precisely, regardingthe ut, vt-problem (3.398), the following Dirichlet trace results hold true:

(a) Assumptions f(x), g(x) ∈ L2(Ω), Rit ∈ L∞(Q), i = 1, 2 as in (3.401) andproperties (3.404) imply

f(x)R1t(x, t)+q(x)vtt(x, t) ∈ L2(Q); g(x)R2t(x, t)+p(x)utt(x, t) ∈ L2(Q), (3.406)

and then [LT91], see below:

f(x)R1t(x, t)+q(x)vtt(x, t) ∈ L2(Q)⇒ut|Σ ∈ Hβ(Σ) continuously (3.407)

g(x)R2t(x, t)+p(x)utt(x, t) ∈ L2(Q)⇒vt|Σ ∈ Hβ(Σ) continuously (3.408)

(b) Assumption (3.401), as well as the regularity properties (3.405) imply

f(x)R1t(x, t) + q(x)vtt(x, t) ∈ H1(0, T ;L2(Ω)); (3.409a)

g(x)R2t(x, t) + p(x)utt(x, t) ∈ H1(0, T ;L2(Ω)), (3.409b)

and then

f(x)R1t(x, t)+q(x)vtt(x, t) ∈ H1(0, T ;L2(Ω))⇒ D1t ut = utt ∈ Hβ(Σ) (3.410)

g(x)R2t(x, t)+p(x)utt(x, t) ∈ H1(0, T ;L2(Ω))⇒ D1t vt = vtt ∈ Hβ(Σ) (3.411)

continuously with β the following constant (see 3.39):

β=3

5, for a generalΩ;β =

2

3, if Ω is a sphere;β=

3

4−ε, ifΩ is parallelepiped

(3.412)Then implications (3.410), (3.411) are immediate consequences of Theorem 5II,

Eqn. (3.24) (as implications (3.407) and (3.408)) for problem (3.398), as then oneapplies the regularity properties (3.407), (3.408) to utt, vtt, solution of the problemobtained from (3.398), after differentiating in time once with admissible data:

f(x)R1tt(x, t), g(x)R2tt(x, t) ∈ L2(Q); (3.413)

utt

(· , T

2

)= 0; (utt)t

(· , T

2

)= f(x)R1t

(x,T

2

)∈ L2(Ω); (3.414a)

vtt

(· , T

2

)= 0; (vtt)t

(· , T

2

)= g(x)R2t

(x,T

2

)∈ L2(Ω), (3.414b)

as needed in applying Theorem 5II, (3.24) to each equation. After these preliminar-ies, we can now draw the desired conclusions on the compactness of the operatorsK, K1, L, and L1 defined in (3.402).


Compactness of K, L According to (3.407), (3.408), (3.415), it suffices to haveRit ∈ L∞(Q), i = 1, 2, in order to have that the map

f, g ∈ L2(Ω)→ Kf, g = ut|Σ ∈ Hβ−ε(Σ),

Lf, g = vt|Σ ∈ Hβ−ε(Σ) are compact, (3.415)

∀ ε > 0 sufficiently small, for then f(x)R1t(x, t) + q(x)vtt(x, t), g(x)R2t(x, t) +p(x)utt(x, t) ∈ L2(Q) via (3.404), as required by (3.407) and (3.408).

Compactness of K1, L1. According to (3.410), (3.411), (3.405), it suffices to haveRit, Ritt ∈ L∞(Q), Ri( · , T2 ) ∈ L∞(Ω), in order to have that the map

f, g ∈ L2(Ω)→ K1f, g = utt ∈ Hβ−ε(Σ) compact (3.416)

f, g ∈ L2(Ω)→ L1f, g = vtt ∈ Hβ−ε(Σ) compact, (3.417)

for all ε > 0 sufficiently small, for then f(x)R1t(x, t) + q(x)vtt, g(x)Rtt(x, t) +p(x)utt(x, t) ∈ H1(0, T ;L2(Ω)), via (3.405), as required by (3.410), (3.411). Lemma8 is proved.

Remark 22 (A more refined analysis). By interpolation between (3.407) and (3.410),and between (3.408) and (3.411). one obtains, for 0 ≤ θ ≤ 1, still under the hypothe-ses (3.401) and hence regularity properties (3.404), (3.405):

fR1t + qvtt ∈ Hθ(0, T ;L2(Ω)) ⇒ Dθt ut

∣∣∣Σ∈ Hβ(Σ); (3.418)

gR2t + putt ∈ Hθ(0, T ;L2(Ω)) ⇒ Dθt vt

∣∣∣Σ∈ Hβ(Σ), (3.419)

continuously. In particular, for θ = 1− β

f(x)R1t(x, t) + q(x)vtt(x, t) ∈ H1−β(0, T ;L2(Ω))⇒utt|Σ ∈ L2(Σ); (3.420)

g(x)R2t(x, t) + p(x)utt(x, t) ∈ H1−β(0, T ;L2(Ω))⇒vtt|Σ ∈ L2(Σ), (3.421)

continuously.

Step 4. Lemma 8 will allow us to absorb the terms

‖Kf = ut‖L2(Σ1), ‖K1f = utt‖L2(Σ1), ‖Lf = vt‖L2(Σ1), ‖L1f = vtt‖L2(Σ1)

(3.422)on the RHS of estimate (3.400), by a compactness–uniqueness argument, as usual.

Proposition 14. Consider the u, v-problem (3.347) with T > T0 in (3.49) underassumption (3.349) = (3.401) for its data q( · ), p( · ), f( · ), g( · ) and Ri( · , · ), withRi satisfying also (3.350), so that both estimate (3.400), as well as Lemma 8 holdtrue. Then, the terms Kf = ut|Σ1 , K1f = utt|Σ1 , Lf = vt|Σ1 and L1f = vtt|Σ1

measured in the L2(Γ1 × [0, T ])-norm can be omitted from the RHS of inequality(3.400) (for a suitable constant CT,ri,... independent of the solution u, v), so thatthe desired conclusion (3.353) of Theorem 21 holds true:

‖f‖2L2(Ω) + ‖g‖2L2(Ω) ≤ CT,data

∫ T

0

∫Γ1

[u2t + u2

tt + v2t + v2

tt]dΓ1 dt, (3.423)

for all f, g ∈ L2(Ω), with CT,data independent of f and g.


Proof. Step (i). Suppose, by contradiction, that inequality (3.423) is false. Then,there exists sequences fn∞n=1, gn∞n=1, fn, gn ∈ L2(Ω), such that

(i) ‖fn‖L2(Ω) = ‖gn‖L2(Ω) ≡ 1, n = 1, 2, . . . ;

(ii) limn→∞

(‖ut(fn)‖L2(Σ1) + ‖utt(fn)‖L2(Σ1)

+ ‖vt(fn)‖L2(Σ1) + ‖vtt(fn)‖L2(Σ1)

)= 0,

(3.424a)

(3.424b)

where u(fn, gn), v(fn, gn) solves problem (3.347a-d) with f = fn, g = gn:

u(fn, gn)tt−∆u(fn, gn)−q(x)v(fn, gn)t = fn(x)R1(x, t); (3.425a)

v(fn, gn)tt−∆v(fn, gn)−p(x)u(fn, gn)t = gn(x)R2(x, t); (3.425b)

u(fn, gn)( · , T2

) = u(fn, gn)t( · ,T

2) = v(fn, gn)( · , T

2)

= vt(fn, gn)( · , T2

) = 0; (3.425c)

∂

∂νu(fn, gn)|Σ = 0;

∂

∂νv(fn, gn)|Σ = 0. (3.425d)

In view of (3.424a), there exists subsequences, still denoted by fn and gn, suchthat:

fn, gn converges weakly in L2(Ω) to some f0, g0 ∈ L2(Ω). (3.426)

Moreover, since the operators K, K1, L and L1 are all compact (Lemma 8), itthen follows by (3.426) that we have strong convergence

‖Kfn, gn −Kfm, gm‖L2(Σ1) = limm,n→+∞

‖K1fn, gn −K1fm, gm‖L2(Σ1)

= 0; (3.427a)

‖Lfn, gn − Lfm, gm‖L2(Σ1) = limm,n→+∞

‖L1fn, gn − L1fm, gm‖L2(Σ1)

= 0. (3.427b)

Step (ii). On the other hand, since the map f, g → u(f, g), v(f, g) is linear,and recalling the definition of the operators K, K1, L and L1 in (3.402), it followsfrom estimate (3.400) that

‖fn − fm‖L2(Ω)+‖gn − gm‖L2(Ω)

≤ C(‖ut(fn, gn)− ut(fm, gm)‖L2(Γ1)+‖utt(fn, gn)− utt(fm, gm)‖L2(Σ1)

+ ‖vt(fn, gn)− vt(fm, gm)‖L2(Σ1)+‖vtt(fn, gn)− vtt(fm, gm)‖L2(Σ1))

+C(‖Kfn, gn −Kfm, gm‖L2(Σ1)+‖K1fn, gn −K1fm, gm‖L2(Σ1)

+ ‖Lfn, gn − Lfm, gm‖L2(Σ1)+‖L1fn, gn − L1fm, gm‖L2(Σ1))

≤ C(‖ut(fn, gn)‖L2(Σ1)+‖utt(fn, gn)‖L2(Σ1)+‖ut(fm, gm)‖L2(Σ1)

+ ‖utt(fm, gm)‖L2(Σ1))+C(‖vt(fn, gn)‖L2(Σ1)+‖vtt(fn, gn)‖L2(Σ1)

+ ‖vt(fm, gm)‖L2(Σ1)+‖vtt(fm, gm)‖L2(Σ1))

+C(‖Kfn, gn −Kfm, gm‖L2(Σ1)+‖K1fn, gn −K1fm, gm‖L2(Σ1)

+ ‖Lfn, gn − Lfm, gm‖L2(Σ1)+‖L1fn, gn − L1fm, gm‖L2(Σ1)),

(3.428)


where again the constant C = CT,q,p,r1,r2 is independent of f and g. It then followsfrom (3.424b) and (3.427) as applied to the RHS of (3.428) that

limm,n→+∞

‖fn − fm‖L2(Ω) = 0, limm,n→+∞

‖gn − gm‖L2(Ω) = 0. (3.429)

Thus, fn, gn are Cauchy sequences in L2(Ω). By uniqueness of the limit,recall (3.426), it then follows that

limn→∞

‖fn − f0‖L2(Ω) = 0, limn→∞

‖gn − g0‖L2(Ω) = 0. (3.430)


‖f0‖L2(Ω) = ‖g0‖L2(Ω) = 1. (3.431)

Step (iii). We now apply to the the u, v-problem (3.347) the same trace the-orem Theorem 5 that we have invoked in (3.407) and (3.408) for the ut, vt-problem (3.398); that is, as f, g ∈ L2(Ω), Ri, Rit ∈ L∞(Q) by assumption andu, v, ut, vt ∈ H1(Q)×H1(Q) a-fortiori due to the L2 forcing terms and homo-geneous B.C. and I.C.:

fR1+qvt, gR2+put ∈ L2(Q)⇒ u, v ∈ Hβ(Σ)×Hβ(Σ); (3.432)

fR1+qvt, gR2+put ∈ H1(0, T ;L2(Ω))⇒ ut, vt ∈ Hβ(Σ)×Hβ(Σ), (3.433)

continuously, hence by interpolation (as in Eqns. (3.420), (3.421))

f(x)R1(x, t)+q(x)vt(x, t) ∈ H1−β(0, T ;L2(Ω))⇒ ut|Σ ∈ L2(Σ); (3.434)

g(x)R2(x, t)+p(x)ut(x, t) ∈ H1−β(0, T ;L2(Ω))⇒ vt|Σ ∈ L2(Σ) (3.435)

Here β is defined in (3.412).Step (iv). Thus, since Ri ∈ L∞(Q), i = 1, 2, we deduce from (3.432) that

f, g ∈ L2(Ω)→ u(f, g)|Σ ∈ Hβ(Σ), v(f, g)|Σ ∈ Hβ(Σ) continuously, (3.436)

i.e.,

‖u(f, g)|Σ‖Hβ(Σ), ‖v(f, g)|Σ‖Hβ(Σ) ≤ CR1,R2(‖f‖L2(Ω) + ‖g‖L2(Ω)), (3.437)

with CR1,R2 = max ‖R1‖L∞(Q), ‖R2‖L∞(Q).As the map f, g → u(f, g), v(f, g)|Σ is linear, it then follows in particular

from (3.437), since fn, gn, f0, g0 ∈ L2(Ω)

‖|u(fn, gn)−u(f0, g0)‖Hβ(Σ1) ≤ CR1,R2(‖fn−f0‖+ ‖gn−g0‖L2(Ω)) (3.438)

‖|v(fn, gn)−v(f0, g0)‖Hβ(Σ1) ≤ CR1,R2(‖fn−f0‖+ ‖gn−g0‖L2(Ω)) (3.439)

Recalling (3.430) on the RHS of (3.438) and (3.439), we conclude first that

limn→∞

‖u(fn, gn)− u(f0, g0)‖Hβ(Σ1) = limn→∞

‖v(fn, gn)− v(f0, g0)‖Hβ(Σ1) = 0,

(3.440)and next that

limn→∞

‖u(fn, gn)|Σ1 − u(f0, g0)|Σ1‖C([0,T ];L2(Γ1))

= limn→∞

‖v(fn, gn)|Σ1 − v(f0, g0)|Σ1‖C([0,T ];L2(Γ1)) = 0 (3.441)


since β > 12, so that Hβ(0, T ) embeds in C[0, T ].

Step (v). Similarly, from (3.433) and recalling (3.430), where in addition, Rit ∈L∞(Q), i = 1, 2, we deduce likewise in addition that

limn→∞

‖ut(fn, gn)|Σ1 − ut(f0, g0)|Σ1‖C([0,T ];L2(Γ1))

= limn→∞

‖vt(fn, gn)|Σ1 − vt(f0, g0)|Σ1‖C([0,T ];L2(Γ1)) = 0. (3.442)

Then, by virtue of (3.424b), combined with (3.442), we obtain in t ∈ [0, T ] that

ut(f0, g0)|Σ1 = vt(f0, g0)|Σ1 ≡ 0; or u(f0, g0)|Σ1

and v(f0, g0)|Σ1 functions of x ∈ Γ1, (3.443)

that is, constant in time.Step (vi). We return to problem (3.423): with fn, gn ∈ L2(Ω) and data q, p ∈

L∞(Ω), R1, R2 ∈ L∞(Q). We have the following regularity results, continuously:

u(fn, gn), ut(fn, gn), v(fn, gn), vt(fn, gn) ∈ C([0, T ]; (H1(Ω)× L2(Ω))2); (3.444)

u(fn, gn), v(fn, gn)|Σ ∈ Hβ(Σ)×Hβ(Σ). (3.445)

Again, the sharp trace regularity (3.445) is the same result noted in (3.407),(3.408), from Theorem 5. As a consequence of (3.430), we also have via (3.444),(3.445):

u(fn, gn), ut(fn, gn)→u(f0, g0), ut(f0, g0),

v(fn, gn), vt(fn, gn)→v(f0, g0), vt(f0, g0) in C([0, T ];H1(Ω)× L2(Ω));

(3.446)

u(fn, gn), v(fn, gn) → u(f0, g0), v(f0, g0) in Hβ(Σ)×Hβ(Σ), (3.447)

same as (3.441) and (3.440), respectively.On the other hand, recalling (3.425c), we have that u(fn, gn)(x, T

2) = v(fn, gn)(x,

T2

) ≡ 0, x ∈ Ω and hence

u(fn, gn)

(x,T

2

)= v(fn, gn)

(x,T

2

)≡ 0, x ∈ Γ1. (3.448)

in the sense of trace in H12 (Γ1). Then (3.448), combined with (3.440), (3.441) (or

(3.446), (3.447)) yields a-fortiori

u(f0, g0)

(x,T

2

)= v(f0, g0)

(x,T

2

)≡ 0, x ∈ Γ1, (3.449)

and next, by virtue of (3.443), the desired conclusion,

u(f0, g0)|Σ1 = v(f0, g0)|Σ1 ≡ 0. (3.450)

Here, u(f0, g0), v(f0, g0) satisfies weakly the limit problem, via (3.430), (3.446),(3.447) applied to (3.425):


utt(f0, g0)−∆u(f0, g0)− q(x)vt(f0, g0) = f0(x)R1(x, t); (3.451a)

vtt(f0, g0)−∆v(f0, g0)− p(x)ut(f0, g0) = g0(x)R2(x, t); (3.451b)

u(f0, g0)

(· , T

2

)= ut(f0, g0)

(· , T

2

)= 0; (3.451c)

v(f0, g0)

(· , T

2

)= vt(f0, g0)

(· , T

2

)= 0; (3.451d)

∂

∂νu(f0, g0)|Σ = 0;

∂

∂νv(f0, g0)|Σ = 0; (3.451e)

u(f0, g0)|Σ1 = v(f0, g0)|Σ1 = 0, (3.451f)

via also (3.450), where f0, g0 ∈ L2(Ω) and q, p,R1, R2 satisfy the assumptions(3.349), (3.350). By virtue of assumption (3.394) = (3.349)+(3.350). Thus, theuniqueness Theorem 20 applies and yields

f0(x) = g0(x) ≡ 0, a.e. x ∈ Ω. (3.452)

Then (3.452) contradicts (3.431). Thus, assumption (3.424) is false and inequal-ity (3.423) holds true. Proposition 14, as well as Theorem 21 are then established.

Uniqueness and stability of the nonlinear inverse problem for thew, z-system (3.344). Proof of Theorems 22 and 23

As usual, the proof of Theorem 22 (uniqueness of the nonlinear inverse problem forthe w, z-dynamics (3.344)) is reduced to Theorem 20 (uniqueness of the linearinverse problem for the u, v-dynamics (3.347)), and the proof of Theorem 23 (sta-bility of the nonlinear inverse problem for the w, z-dynamics (3.344)) is reduced toTheorem 21 (stability of the linear inverse problem for the u, v-dynamics (3.347))In fact, as in (3.346), set

f(x) = q1(x)− q2(x), g(x) = p1(x)− p2(x); (3.453a)

R1(x, t) = zt(q2, p2)(x, t), R2(x, t) = wt(q2, p2)(x, t); (3.453b)

u(x, t) = w(q1, p1)(x, t)−w(q2, p2)(x, t), v(x, t) = z(q1, p1)(x, t)− z(q2, p2)(x, t)(3.453c)

Then, as noted before, the variables u(x, t), v(x, t) solve problem (3.347). Byvirtue of assumption (3.354), we then have via (3.453) that f(x), g(x) ∈ L∞(Ω).

Step 1. We rewrite the coupled problem (3.344) as in (3.363):wttztt

=

−AN 0

0 −AN

wz

+

0 q( · )

p( · ) 0

wtzt

= −AN

wz

+Π

wtzt

, (3.454)

where AN = −∆ with Neumann B.C., non-negative self-adjoint on L2(Ω), Π is abounded perturbation of −AN which will not affect the regularity of the solutions

w(t) = C(t)w0 + S(t)w1; wt(t) = ANS(t)w0 + C(t)w1; (3.455)


wtt(t) = ANC(t)w0 +ANS(t)w1; wttt(t) = A32NA

12NS(t)w0 +ANC(t)w1, (3.456)

and similarly for z(t), zt(t), ztt(t), zttt(t). Here C(t) is the cosine operator onL2(Ω) × L2(Ω) generated by −AN + Π, D(AN ) = D(AN ) × D(AN ), and S(t)its corresponding sine operator [Fat85]. We have from (3.455), (3.456)

w0, w1, z0, z1 ∈ D(Ak+ 1

2N )×D(AkN ) ⊂ H2k+1(Ω)×H2k(Ω) (3.457)

implies⇒ wttt, zttt ∈ C([0, T ]; D(Ak−1N )) ⊂ C([0, T ];H2(k−1)(Ω)); (3.458)

wttt, zttt ∈ C([0, T ];C(Ω)), provided k >dimΩ

4+ 1. (3.459)

Step 2. Moreover, in terms of the boundary data, we have

Proposition 15. We return to the w, z-problem (3.344). (a) Under the followingassumptions on the data:

q( · ), p( · ) ∈ L∞(Ω); (3.460)

µi ∈ Hm(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12

+(m−1)(Γ )),

α =2


3


with Compatibility Relations (C.R.)

µi

(T

2

)= µi

(T

2

)= · · · = µ

(m−1)i

(T

2

)= 0, i = 1, 2.

(3.461)

(the regularity in (3.461) is a-fortiori implied by

µi ∈ Hm(2α−1),m(Σ) = L2(0, T ;Hm(2α−1)(Γ )) ∩Hm(0, T ;L2(Γ )), (3.462)

via Theorem 11. Then the solution w = w(q, p), z = z(q, p) satisfies the followingregularity property that continuously,

w,wt, wtt, wttt, z, zt, ztt, zttt

∈ C([0, T ];Hα+m(Ω)×Hα+(m−1)(Ω)×Hα+(m−2)(Ω)×Hα+(m−3)(Ω). (3.463)

(b) If, moreover,

m >dimΩ

2+ 3− α (3.464)

then a-fortiori, properties (3.461) are fulfilled.

wt, wtt, wttt, zt, ztt, zttt ∈ L∞(Q). (3.465)

Proof. (a) Again the result in (a) relies critically on sharp regularity results Theorem5. More precisely,

Case m = 1. Let

µi ∈ H1(0, T ;L2(Γ )) ∩ C([0, T ];Hα− 12 (Γ )), with C.R. µi

(T

2

)= 0. (3.466)

Then Theorem 6 implies that

w,wt, wtt, z, zt, ztt ∈ C([0, T ];Hα+1(Ω)×Hα(Ω)×Hα−1(Ω)), (3.467)


continuously. Eqn. (3.467) is result (a), Eqn. (3.463), for m = 1, except for wttt, zttt.Case m = 2. Let now

µi ∈ H2(0, T ;L2(Γ )) ∩ C([0, T ];Hα+ 12 (Γ )), with C.R. µi

(T

2

)= µi

(T

2

)= 0.

(3.468)Then Theorem 7 implies that

w,wt, wtt, z, zt, ztt ∈ C([0, T ];Hα+2(Ω)×Hα+1(Ω)×Hα(Ω)

), (3.469)

continuously, Eqn. (3.469) is result (a), Eqn. (3.463), for m = 2, except for wttt, zttt.General case m. As noted in [LT91], the general case is similar and yieldsµi as in (3.352)⇒

w,wt, wtt, z, zt, ztt ∈ C(

[0, T ];Hα+m ×Hα+(m−1) ×Hα+(m−2)(Ω)

),

(3.470)continuously, to which we add

wttt, zttt ∈ C([0, T ];Hα+(m−3)(Ω)), (3.471)

as the above theorems for the map µi → w,wt, wtt, z, zt, ztt (with zero I.C.) canbe applied now to the map µit → wt, wtt, wttt, zt, ztt, zttt (still with zero I.C.), asq( · ), p( · ) are time-independent. Thus (3.463) is proved.

(b) If α+ (m− 3) > dimΩ2

, then from [LM72, Corollary 9.1, p. 96] the followingembedding holds

Hα+(m−3)(Ω) → C(Ω) ⊂ L∞(Ω), (3.472)

which, along with properties (3.463), yields (3.465) under (3.464).

Step 3. Thus, under assumption (3.457), with k in (3.459) on the I.C. and assump-tion (3.461) with m in (3.464) on the B.C. we have that Ri(x, t), i = 1, 2 satisfyassumption (3.349); moreover, so do

R1

(x,T

2

)= zt(q2, p2)

(x,T

2

)= z1(x),

R2

(x,T

2

)= wt(q2, p2)

(x,T

2

)= w1(x). (3.473)

Thus, assumptions (3.354), (3.355), (3.357) of Theorem 22 implies assumption(3.349) of Theorem 20. Moreover, assumption (3.356) of Theorem 22 implies as-sumptions (3.350) of Theorem 20. In addition, the present assumption (3.359) that

w(q1, p1)(x, t)=w(q2, p2)(x, t), z(q1, p1)(x, t)=z(q2, p2)(x, t), x∈Γ1, t∈ [0, T ](3.474)

implies via (3.453c) that: u(f, g)(x, t) = 0, v(f, g)(x, t) = 0, x ∈ Γ1, t ∈ [0, T ].Therefore, Theorem 20 applies, and we conclude that f(x) = q1(x)− p1(x) = 0 andg(x) = q2(x) − p2(x) = 0; that is, that q1(x) = p1(x), q2(x) = p2(x) a.e. x ∈ Ω.Similarly, Theorem 21 also applies and we then obtain for f(x) = q1(x) − q2(x),g(x) = p1(x)− p2(x), we have the desired stability estimate (3.361).


3.6.4 Notes

This section is a streamlined improvement of [LT11d]. See also [LT11e]. Althoughthere is a wide literature dealing with the inverse problem of single hyperbolic equa-tion, there are only a few references (e.g., [BIY08], [IIY03], [IY05]) where coupledPDEs are considered. We note that our methodology can also readily treat evenstronger coupled systems with gradient terms even time and space dependent.

3.7 Recovery damping and source coefficients in oneshot by means of a single boundary measurement. TheDirichlet case.

3.7.1 Problem formulation

There is no space in these notes to consider the case of simultaneous recovery of twocoefficients—say, the damping and source coefficients—in one shot, by means of justone boundary measurement, for both the Neumann and the Dirichlet B.C. case, asa full treatment. The Neumann B.C. problem is studied in [LT11a]. Here we reportonly a condense treatment of the Dirichlet B.C. case, after [LT13].

I: The original hyperbolic problem subject to unknown damping co-efficient q1(x) and potential coefficient q0(x). On Ω we consider the followingsecond-order hyperbolic equation with zero initial position:

wtt(x, t) = ∆w(x, t) + q1(x)wt(x, t) + q0(x)w(x, t) in Q;

w(· , T

2

)= w0(x); wt

(· , T

2

)= w1(x) in Ω;

w(x, t)|Σ = µ(x, t) in Σ.

(3.475a)

(3.475b)

(3.475c)

Given data: The initial conditions w0, w1, as well as the Neumann boundaryterm µ are given.

Unknown terms: Instead, the damping and potential coefficients q1(x), q0(x) ∈L∞(Ω) are assumed to be unknown.

For dynamics (3.475a-c) we may define the uniqueness and stability problems,regarding the simultaneous recovery of both coefficients–q1(x) and q0(x). They arethe perfect analog to those of Section 3.4; thus, we need not state them.

II: The corresponding linear inverse problem. Next, we shall turn theabove inverse problems for the original w-system (3.475a-c) into a correspondinginverse problem for a related (auxiliary) problem. Define, in the above setting f(x) = q1(x)− p1(x), g(x) = q0(x)− p0(x)

u(x, t) = w(q1, q0)(x, t)− w(p1, p0)(x, t), R(x, t) = w(p1, p0)(x, t).(3.476)

Then, u = u(x, t) is readily seen to satisfy the following (homogeneous) mixedproblem

3 Boundary control and boundary inverse theory 313utt −∆u− q1(x)ut − q0(x)u = f(x)Rt(x, t) + g(x)R(x, t) in Q;

u(· , T

2

)= 0; ut

(· , T

2

)= 0 in Ω;

u(x, t)|Σ = 0 in Σ.

(3.477a)

(3.477b)

(3.477c)

Thus, in this setting (3.476), (3.477a-c), we haveGiven data: The coefficients q1, q0 ∈ L∞(Ω) and the term R( · , · ) are given,

subject to appropriate assumptions.Unknown term: The terms f( · ), g( · ) ∈ L∞(Ω) are assumed to be unknown.

3.7.2 Main results

We begin with a uniqueness result for the linear inverse problem involving the u-system (3.477a-c). It requires that the damping coefficient q1(x) be a-priori in afixed ball of L∞(Ω).

Theorem 24 (Uniqueness of linear inverse problem). Assume the preliminarygeometric assumptions (A.1): (3.2a), (3.3), and (A.2) = (3.4). Let T > T0, as in(3.49). With reference to the u-problem (3.477a-c), assume further the followingregularity properties on the fixed data q1( · ), q0( · ) and R( · , · ) and unknown termsf( · ), g( · ), where Q = Ω × [0, T ]:

q1, q0 ∈ L∞(Ω); ‖q1‖L∞(Ω) ≤M, for some fixed M > 0; f, g ∈ L2(Ω), (3.478a)

R,Rt, Rtt, Rttt, Rtttt ∈ L∞(Q), Rt

(x,T

2

)∈W 1,∞(Ω), (3.478b)

with M otherwise arbitrary, as well as the following property at the initial time T2

:

R

(x,T

2

)= 0,

∣∣∣∣Rt(x, T2)∣∣∣∣ ≥ r0 > 0, for some constant r0 and x ∈ Ω. (3.479)

If the solution to problem (3.477a-c) satisfies the additional homogeneous Neu-mann boundary trace condition

∂u(f, g)

∂ν(x, t) = 0, x ∈ Γ1, t ∈ [0, T ] (3.480)


f(x) = 0; g(x) = 0, a.e. x ∈ Ω. (3.481)

Theorem 25 (Lipschitz stability of linear inverse problem). Assume the pre-liminary geometric assumptions (A.1)=(3.2a), (3.3), (A.2) = (3.4). Consider prob-lem (3.477a-c) on [0, T ] with T > T0, as in (3.49), f ∈ H1

0 (θ), g ∈ Hθ(Ω) for somefixed but otherwise arbitrary θ, 0 < θ ≤ 1, θ 6= 1

2where of course Hθ

0 (Ω) = Hθ(Ω),0 < θ < 1

2[LM72, p. 55] and data

q1, q0 ∈ L∞(Ω), ‖q1‖L∞ ≤M ; R,Rt, Rtt, Rttt, Rtttt ∈ L∞(Q);

Rtt, Rttt ∈ Hθ(0, T ;W θ,∞(Ω)), Rt

(x,T

2

)∈W 1,∞(Ω), (3.482)


where, moreover, R(x, T2

) = 0 and Rt satisfies the positivity condition (3.479) at theinitial time t = T

2. Then there exist constants C = C(Ω, T, Γ1, ϕ, q, R, θ) > 0 and

likewise constant c > 0, i.e., depending on the data of problem (3.477a-c), but noton the unknown coefficient f , such that

c

∥∥∥∥∂utt(f, g)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

≤ ‖f‖H10 (Ω) + ‖g‖L2(Ω)

≤ C∥∥∥∥∂utt(f, g)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

, (3.483)

for all f ∈ H10 (Ω), g ∈ Hθ(Ω), 0 < θ ≤ 1, θ 6= 1

2fixed (More precisely, C, c depend

on the L∞(Ω)-norm of q1 and q0, more specifically the constant M).

We next give the corresponding uniqueness result to the nonlinear inverse prob-lem involving the determination of the damping and source coefficients q1( · ) andq0( · ) in the w-problem (3.475a-c).

Theorem 26 (Uniqueness of nonlinear inverse problem). Assume the prelim-inary geometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Let T > T0 asin (3.49). With reference to the w-problem (3.475a-c), assume the following a-prioriregularity of two damping/sources coefficients

q1, q0, p1, p0 ∈Wm,∞(Ω), (3.484)

plus boundary compatibility conditions depending on dim Ω. For dim Ω = 2, 3, theseare identified in the proof of Proposition 20 below.

Let w(q1, q0) and w(p1, p0) denote the corresponding solutions of problem (3.475a-c). Assume further the following regularity properties on the initial and boundarydata

w0, w1 ∈ Hm+1(Ω)×Hm(Ω), m >dimΩ

2+ 2, w1 ∈W 1,∞(Ω), (3.485)

µ ∈ Hm+1(Σ), (3.486)

along with all compatibility conditions (trace coincidence) which make sense. Letthe initial conditions w0, w1 in (3.475b) satisfy the following zero and positivityconditions:

w0(x) = 0, |w1(x)| ≥ v1 > 0, for some constant v1 > 0 and x ∈ Ω. (3.487)

Finally, if w(q1, q0) and w(p1, p0) have the same Neumann boundary traces on Σ1:

∂w(q1, q0)

∂ν(x, t) =

∂w(p1, p0)

∂ν(x, t), x ∈ Γ1, t ∈ [0, T ], (3.488)

over the observed part Γ1 of the boundary Γ and over the time interval T , as in(3.49), then, in fact, the two pairs of coefficients coincide

q1(x) ≡ p1(x), q0(x) ≡ p0(x), a.e. x ∈ Ω. (3.489)


Remark 23. Theorem 26 (refer to its proof in Section 3.7.4 below) requires additionalconditions on the coefficients q1, q0, which can be precise and explicit. They involvetwo aspects: (i) suitable regularity assumptions on q1, q0, which can be expressed interms of these coefficients as belonging to appropriate multipliers spaces [MS85], suchas q1 belongs to M(Hm(Ω)→ Hm(Ω)), and q0 belongs to M(Hm+1(Ω)→ Hm(Ω));as well as (ii) suitable boundary compatibility conditions, depending on dim Ω. Acheckable sufficient condition on the regularity requirements (i) is: q1, q0 belongto Wm,∞(Ω). A most direct condition on the boundary compatibility requirement(ii) for any dimension of Ω is that q1 and q0 have compact support on Ω. Morespecifically, for dim Ω = 2, the only C.C. is ∂q1

∂ν= 0 on Γ , and the same for q0.

For dim Ω = 3, these conditions have to be supplemented by ∂(∆q1)∂ν

= 0 on Γ ;∇(q1xi) = 0, and the same for q0. Details are provided in Section 3.7.5 below. Allthese conditions are needed to guarantee that the solution w, wt, wtt, wttt, wtttthave the same regularity as R, Rt, Rtt, Rttt, Rtttt as assumed in Theorem 24 and25.

Finally, we state the stability result for the nonlinear inverse problem involvingthe w-problem (3.475a-c) with damping and potential coefficients q1( · ), q0( · ).

Theorem 27 (Lipschitz stability of nonlinear inverse problem). Assumepreliminary geometric assumptions (A.1) = (3.2a), (3.3), (A.2) = (3.4). Considerproblem (3.475a-c) on [0, T ], with T > T0 as in (3.49), one time with coefficientsq1, q0, and one time with coefficients p1, p0 subject to the assumptions of Theo-rem 26, and let w(q1, q0), w(p1, p0) denote the corresponding solutions. Assumefurther the properties (3.485) and (3.486)) on the data and on m. Then, the fol-lowing stability result holds true for the w-problem (3.475a-c): There exists constantC = C(Ω, T, Γ1, ϕ, q1, q0, w0, w1, g) > 0 and likewise c > 0, depending on the data ofthe problem (3.475a-c) and on the L∞(Ω)-norm of the coefficient q1, p1 such that

c

∥∥∥∥∂wtt(q1, q0)

∂ν− ∂wtt(p1, p0)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

≤ ‖q1 − p1‖H10 (Ω) + ‖q0 − p0‖L2(Ω)

≤ C∥∥∥∥∂wtt(q1, q0)

∂ν− ∂wtt(p1, p0)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

, (3.490)

for all such coefficients p1 and p0, 0 < θ < min

12,m−1− dimΩ

23

. In particular, the

constant C in (3.490) may be thought of as dependent only on the radius (arbitrarilylarge) of an L∞(Ω)-ball, for all coefficients q1, q0 in such a ball, independent of thecoefficients p1, p0 ∈ L∞(Ω).

Remark 24. The sufficient condition R(x, T2

) ∈W 1,∞(Ω) in (3.478) of Theorem 24–as well as in (3.482) of Theorem 25–could be replaced by a weaker assumption:“R(x, T

2) is a multiplier H1(Ω) → H1(Ω),” or R(x, T

2) ∈ M(H1(Ω) → H1(Ω)),

of which [MS85, Theorem 1, m = l = 1, p=2, p. 243] provides a characterization.Similarly, in (3.485) of Theorem 26 as well as Theorem 27, the sufficient condition|∇w1| ∈ L∞(Ω) could be replaced by the weaker condition w1 ∈ M(H1(Ω) →H1(Ω)).


3.7.3 Proofs

Sketch of proof of uniqueness Theorem 24

Step 1. We follow the strategy of the proof of Theorem 15 in Section 3.5.3 (orTheorem 14 in Section 3.4.3), by applying the Carleman estimates (3.11)–(3.13)not only to the u-, (ut)-, (utt)-problems, but also to the (uttt)-problem. For thelatter two we invoke the COI (3.39) to boost the regularity of the respective I.C.(counterpart of Lemma 3 or of Lemma 1), so that in each case the solution utt, utttis in H1,1(Q).

Adding up all these four Carleman estimates yields

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + u2

tttt + |∇u|2 + |∇ut|2 + |∇utt|2 + |∇uttt|2]

+ C2,τ

∫Q(σ)

e2τϕ[u2 + u2t + u2

tt + u2ttt]dx dt

≤ 4

∫ T

0

∫Ω

e2τϕ

[|fRt|2 + |fRtt|2 + |fRttt|2 + |fRtttt|2 + |gR|2 + |gRt|2

+ |gRtt|2 + |gRttt|2]dQ+ [CT,u + CT,ut + CT,utt + CT,uttt ] e

2τσ

+ cT τ3e−2τδ

[Eu(0) + Eut(0) + Eutt(0) + Euttt(0)

+ Eu(T ) + Eut(T ) + Eutt(T ) + Euttt(T )

]. (3.491)

Next, we invoke one more time properties (3.478) for R,Rt, Rtt, Rttt, Rtttt ∈L∞(Q):

|f(x)Rt(x, t)|, |f(x)Rtt(x, t)|, |f(x)Rttt(x, t)|, |f(x)Rtttt(x, t)| ≤ CR|f(x)|, (3.492a)

|g(x)R(x, t)|, |g(x)Rt(x, t)|, |g(x)Rtt(x, t)|, |g(x)Rttt(x, t)| ≤ CR|g(x)|, (3.492b)

with CR = sup‖R(x, t)‖L∞(Q), ‖Rt(x, t)‖L∞(Q), ‖Rtt(x, t)‖L∞(Q),‖Rttt(x, t)‖L∞(Q), ‖Rtttt(x, t)‖L∞(Q).

Using (3.492) into the RHS of (3.491) yields finally

C1,τ

∫Q

e2τϕ [u2t + u2

tt + u2ttt + u2

tttt + |∇u|2 + |∇ut|2 + |∇utt|2 + |∇uttt|2]

+ C2,τ

∫Q(σ)

e2τϕ [u2 + u2t + u2

tt + u2ttt

]dx dt

≤CR,T,u∫

Q

e2τϕ (|f |2 + |g|2)dQ+ e2τσ + τ3e−2τδ[Eu(0) + Eut(0)

+ Eutt(0) + Euttt(0) + Eu(T ) + Eut(T ) + Eutt(T ) + Euttt(T )]

(3.493)

where CR,T,u is a positive constant depending on R, T , and u.


Step 2. We now evaluate utt( · , T2 ) and uttt( · , T2 ) at the initial time t = T2

from(3.477a), use the vanishing initial conditions (3.477b) as well as R

(x, T

2

)= 0 as

assumed in (3.479), to obtain:

utt

(x,T

2

)= f(x)Rt

(x,T

2

), x ∈ Ω, (3.494)

uttt

(x,T

2

)=q1(x)f(x)Rt

(x,T

2

)+f(x)Rtt

(x,T

2

)+g(x)Rt

(x,T

2

), x ∈ Ω.

(3.495)We combine (3.494), (3.495) in vectorial form in terms of f, g:utt

(x,T

2

)uttt

(x,T

2

) =

Rt

(x,T

2

)0

q1(x)Rt

(x,T

2

)+Rtt

(x,T

2

)Rt

(x,T

2

)f(x)

g(x)

= A(x)

f(x)

g(x)

. (3.496)

Since detA(x) = R2t (x,

T2

) ≥ r20, x ∈ Ω by (3.479), then the triangular matrix A(x)

is boundedly invertible, with inverse A−1(x)

f(x)

g(x)

= A−1(x)

utt(x,T

2

)uttt

(x,T

2

) ,

A−1(x) =1

detA(x)

Rt

(x,T

2

)0

−q1(x)Rt

(x,T

2

)−Rtt

(x,T

2

)Rt

(x,T

2

),

(3.497)

which is uniformly norm bounded in x ∈ Ω, by use also of hypotheses (3.475):Rt( · , T2 ), Rtt( · , T2 ) ∈ L∞(Ω) and, in particular, ‖q1‖L∞(Ω) ≤ M . Thus, (3.496),(3.497) yield the desired inversion on system (3.496)

|f(x)|2 + |g(x)|2 ≤ CR,M,r0,T

(∣∣∣∣utt(x, T2) ∣∣∣∣2 +

∣∣∣∣uttt(x, T2) ∣∣∣∣2

), (3.498)

which is uniformly in x ∈ Ω, where C = CR,M,r0,T denotes a constant depends onR, M , r0, T . This step generalizes the idea in [Isa06, Thm. 8.2.2, p. 231].

Remark 25. We remark that (3.498) is the step that reduces stability of f, g interms of invoking the COI for utt( · , T2 ), uttt( · , T2 ) (see Section 3.7.4 below).

Claim: Using (3.498) in the first integral term∫Qe2τϕ

(|f |2 + |g|2

)dQ on the

RHS of (3.227) yields the following estimate:

318 Shitao Liu and Roberto Triggiani∫Q

e2τϕ (|f |2 + |g|2)dQ =

∫ T

0

∫Ω

e2τϕ (|f |2 + |g|2)dΩ dt

≤ CR,M,r0,T

∫ T

0

∫Ω

e2τϕ

(|utt(x,

T

2)|2 + |uttt(x,

T

2)|2)dΩ dt

≤ CR,M,r0,T

(2Tcτ + 2)

∫Ω

∫ T/2

0

e2τϕ(x,s) (|utt(x, s)|2 + |uttt(x, s)|2)ds dΩ

+

∫Ω

∫ T/2

0

e2τϕ(x,s)|utttt(x, s)|2ds dΩ +

∫Ω

(|utt(x, 0)|2 + |uttt(x, 0)|2

).

(3.499)

The proof of this claim is given in (3.84–3.87). The rest of the uniqueness proofleading to f = g ≡ 0 a.e. in Theorem 24 can then be obtained by proceeding alongthe proof of Steps 6–9 in Section 3.4.3.

Stability of linear inverse problem for the u-problem (3.477):Proof of Theorem 25

(a) Preparatory material: The inversionutt(· , T

2

), uttt

(· , T

2

)⇒ f( · ),

g( · ) of (3.496) at various topological levels. Preliminary stability esti-mates polluted by `.o.t.

Step 1. We rewrite system (3.497) explicitly

f(x) =1

Rt(x, T

2

)utt(x, T2

)(3.500a)

g(x) =

(− q1(x)

Rt(x, T

2

) − Rtt(x, T

2

)R2t

(x, T

2

) )utt(x, T2

)+uttt

(x, T

2

)Rt(x, T

2

) (3.500b)

g(x) = −q1(x)f(x)−Rtt

(x, T

2

)Rt(x, T

2

) f(x) +1

Rt(x, T

2

)uttt (x, T2

). (3.500c)

In this section, we shall consider, under different sets of hypotheses, the inversionof the map utt( · , T2 ), uttt( · , T2 ) → f(x), g(x) of (3.496) at various topologicallevels.

Proposition 16. (a) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω as in (3.479). Then

‖f‖2L2(Ω) ≤1

r20

∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

. (3.501)

(b) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, Rt(x,T2

) ∈ W 1,∞(Ω), as in (3.478),(3.479). Then

‖f‖2H1(Ω) ≤ CT,r0,R∥∥∥∥utt( · , T2

)∥∥∥∥2

H1(Ω)

. (3.502)


(c) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, ‖q1‖L∞(Ω) ≤ M , Rtt(x,T2

) ∈ L∞(Ω),as in (3.478), (3.479). Then

‖g‖2L2(Ω) ≤(M

r0

)2 ∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

+

(CR,Tr20

)2 ∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

+1

r20

∥∥∥∥uttt( · , T2)∥∥∥∥2

L2(Ω)

. (3.503)

(Notice that the upper bound on ‖g‖L2(Ω) in (3.503) implies the upper bound on‖f‖L2(Ω) in (3.501)).

(d) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, ‖q1‖L∞(Ω) ≤ M , Rtt(x,T2

) ∈ L∞(Ω)as in case (c). Then

‖f‖2L2(Ω) + ‖g‖2L2(Ω) ≤(M

r0

)2 ∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

+

(CR,Tr20

)2 ∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

+1

r20

∥∥∥∥uttt ( · , T2)∥∥∥∥2

L2(Ω)

. (3.504)

(e) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, ‖q1‖L∞(Ω) ≤M , Rt( · , T2 ) ∈W 1,∞(Ω),Rtt(x,

T2

) ∈ L∞(Ω) as in cases (b) and (c). Then

‖f‖2H1(Ω) + ‖g‖2L2(Ω) ≤ CT,r0,R∥∥∥∥utt( · , T2

)∥∥∥∥2

H1(Ω)

+

(M

r0

)2 ∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

+1

r20

∥∥∥∥uttt ( · , T2)∥∥∥∥2

L2(Ω)

. (3.505)

(f) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, ‖q1‖W1,∞(Ω) ≤ M1, Rt( · , T2 ) ∈W 1,∞(Ω), Rtt(x,

T2

) ∈W 1,∞(Ω). Then

‖g‖2H1(Ω) ≤ CT,r0,R,M1

∥∥∥∥utt( · , T2)∥∥∥∥2

H1(Ω)

+CT,r0,R

∥∥∥∥uttt ( · , T2)∥∥∥∥2

H1(Ω)

. (3.506)

(g) Assume |Rt(x, T2 )| ≥ r0 > 0, x ∈ Ω, ‖q1‖W1,∞(Ω) ≤ M1, Rt( · , T2 ) ∈W 1,∞(Ω), Rtt(x,

T2

) ∈ W 1,∞(Ω), thus combining the assumptions of cases (b) and(f). Then

‖f‖2H1(Ω) + ‖g‖2H1(Ω) ≤ CT,r0,R,M1

∥∥∥∥utt( · , T2)∥∥∥∥2

H1(Ω)

+ CT,r0,R

∥∥∥∥uttt ( · , T2)∥∥∥∥2

H1(Ω)

. (3.507)

Proof. (a) Eqn. (3.500a) yields |f(x)| ≤ 1r0|utt(x, T2 )|, by the hypothesis. Squaring

this and integrating over Ω yields (3.501).(b) Differentiating f(x) given by (3.500a) in xi, and using the hypotheses yields

the inequality |∂xif(x)|2 ≤ CT,r0,R[|utt(x, T2 )|2 + |∂xiutt(x, T2 )|2

]. Integrating over


Ω this inequality, as well as the one on |f(x)|2 obtained in the proof of (a), yields(3.502), as desired.

(c) We return to (3.500b), use here the hypotheses, integrate over Ω and obtain(3.503).

(d) Case (d) combines cases (a) and (c) (with M > 1).(e) Case (e) combines cases (b) and (c).(f) Differentiating g(x) given by (3.500b) in xi and using the hypotheses—in

particular q1(x), ∂xiq1(x) ∈ L∞(Ω)—yields

|∂xig(x)|2 ≤ CT,r0,R,M1

[∣∣∣∣utt(x, T2)∣∣∣∣2 +

∣∣∣∣∂xiutt(x, T2)∣∣∣∣2

+

∣∣∣∣uttt(x, T2)∣∣∣∣2 +

∣∣∣∣∂xiuttt(x, T2)∣∣∣∣2].

Integrating this inequality, as well as the inequality on |g(x)|2 obtained in the proofof (c), yields (3.506), as desired.

(g) Case (g) combines cases (b) and (f).

Step 2. The RHS of Eqns. (3.501)–(3.507) point out that we need to return to theutt-problem or even the corresponding uttt-problem obtained from differentiatingin t the original u-problem (3.477) (these will be repeated below). In fact, theirI.C. are precisely the terms on the RHS of Eqns. (3.501)–(3.507) at the appropriatetopological level. To this end, we return to the u-problem (3.477), differentiate in tonce or twice or three times, set for convenience

v = ut(f, g) = ψ + z; utt = vt = ψt + zt, uttt = ψtt + ztt. (3.508)

By linearity, we decompose each problem in a subproblem with homogeneous RHSand a subproblem with homogeneous I.C. Thus we find differentiating in t the u-problem (3.477)

(ut)tt−∆(ut)−q1(x)(ut)t−q0(x)(ut) = f(x)Rtt+g(x)Rt in Q;

(ut)

(· , T

2

)= 0; (ut)t

(· , T

2

)= f(x)Rt

(x,T

2

)in Ω;

(ut)|Σ = 0 in Σ,

(3.509a)

(3.509b)

(3.509c)

which is decomposed by (3.508) (left) as the ψ-problemψtt −∆ψ − q1(x)ψt − q0(x)ψ = 0 in Q;

ψ

(· , T

2

)= ut

(· , T

2

)= 0; ψt

(· , T

2

)= utt

(· , T

2

)in Ω;

ψ|Σ = 0 in Σ,

(3.510a)

(3.510b)

(3.510c)

and the z-problem

3 Boundary control and boundary inverse theory 321ztt −∆z − q1(x)zt − q0(x)z = f(x)Rtt + g(x)Rt in Q;

z

(· , T

2

)= 0; zt

(· , T

2

)= 0 in Ω;

z|Σ = 0 in Σ.

(3.511a)

(3.511b)

(3.511c)

Similarly, the utt-problem obtained from (3.477)(utt)tt −∆(utt)− q1(x)(utt)t − q0(x)(utt) = f(x)Rttt + g(x)Rtt;

(utt)

(· , T

2

)= f(x)Rt

(x,T

2

); (utt)t

(· , T

2

);

(utt)|Σ = 0,

(3.512a)

(3.512b)

(3.512c)

is decomposed as utt = ψt + zt, where(ψt)tt −∆(ψt)− q1(x)(ψt)t − q0(x)(ψt) = 0 in Q;

(ψt)

(· , T

2

)= utt

(· , T

2

); (ψt)t

(· , T

2

)= uttt

(· , T

2

)in Ω;

(ψt)|Σ = 0 in Σ;

(3.513a)

(3.513b)

(3.513c)(zt)tt−∆(zt)−q1(x)(zt)t−q0(x)(zt) = f(x)Rttt+g(x)Rtt in Q;

(zt)

(· , T

2

)= 0; (zt)t

(· , T

2

)= 0 in Ω;

(zt)|Σ = 0 in Σ.

(3.514a)

(3.514b)

(3.514c)

Similarly, the uttt-problem obtained from (3.477)(uttt)tt −∆(uttt)− q1(x)(uttt)t − q0(x)(uttt) = fRtttt + gRttt;

(uttt)

(· , T

2

); (uttt)t

(· , T

2

);

(uttt)|Σ = 0,

(3.515a)

(3.515b)

(3.515c)

is decomposed as uttt = ψtt + ztt where(ψtt)tt −∆(ψtt)− q1(x)(ψtt)t − q0(x)(ψtt) = 0 in Q;

(ψtt)

(· , T

2

)= uttt( · , T2 ); (ψtt)t

(· , T

2

)in Ω;

(ψtt)|Σ = 0 in Σ;

(3.516a)

(3.516b)

(3.516c)(ztt)tt −∆(ztt)− q1(x)(ztt)t − q0(x)(ztt) = f(x)Rtttt + g(x)Rttt;

(ztt)

(· , T

2

)= 0; (ztt)t

(· , T

2

)= 0;

(ztt)|Σ = 0.

(3.517a)

(3.517b)

(3.517c)

Step 3. In this step we systematically invoke the COI (3.39) of Theorem 9 to anappropriate ψ-, ψt-, or ψtt-problem to majorize the RHSs of (3.501)–(3.507) bycorresponding Neumann boundary trace terms.


Proposition 17. We have:(i) With reference to the ψ-problem (3.510) and the companion z-problem (3.511)

and ut-problem (3.509) via (3.508)∥∥∥∥utt( · , T2)∥∥∥∥2

L2(Ω)

≤ CT∫ T

0

∫Γ1

(∂ψ

∂ν

)2

dΣ1

≤ CT

[∫ T

0

∫Γ1

(∂ut∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂z

∂ν

)2

dΣ1

]. (3.518)

(ii) With reference to the ψt-problem (3.513) and the companion zt-problem (3.514)and utt-problem (3.512) , via (3.508)∥∥∥∥utt( · , T2

)∥∥∥∥2

H1(Ω)

+

∥∥∥∥uttt( · , T2)∥∥∥∥2

L2(Ω)

≤ CT∫ T

0

∫Γ1

(∂ψt∂ν

)2

dΣ1

≤ CT

[∫ T

0

∫Γ1

(∂utt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

]. (3.519)

(iii) With reference to the ψtt-problem (3.516) and the companion ztt-problem(3.517) and uttt-problem (3.515), via (3.508)∥∥∥∥uttt( · , T2

)∥∥∥∥2

H1(Ω)

≤ CT∫ T

0

∫Γ1

(∂ψtt∂ν

)2

dΣ1

≤ CT

[∫ T

0

∫Γ1

(∂uttt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂ztt∂ν

)2

dΣ1

]. (3.520)

Proof. We apply the COI (3.39) of Theorem 9 to the ψ-problem (3.510) to get(i)=(3.518); to the ψt-problem (3.513) to get (ii)=(3.519); and to the ψtt-problem(3.516) to get (iii)=(3.520), via (3.508).

Step 4. In this step we combine Propositions 16 and 17 to get the sought-after tracebounds for the pair f(x), g(x), obtained by inversion of (3.496) in terms of thepair utt( · , T2 ), uttt( · , T2 ).

Theorem 28. (a) Assume the hypotheses of (a) in Proposition 16. Then, via(3.518),

‖f‖2L2(Ω) ≤CTr20

[∫ T

0

∫Γ1

(∂ut∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂z

∂ν

)2

dΣ1

]. (3.521)

(b) Assume the hypotheses of (b) in Proposition 16. Then, via (3.519),

‖f‖2H10 (Ω) ≤ CR,r0,T

[∫ T

0

∫Γ1

(∂utt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

]. (3.522)

(c) Assume the hypotheses of (c) in Proposition 16. Then a-fortiori, via (3.519),


‖g‖2L2(Ω) ≤ CR,r0,T,M

[∫ T

0

∫Γ1

(∂utt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

]. (3.523)

(d) Assume the hypotheses of (d) in Proposition 16. Then a-fortiori, via (3.519),

‖f‖2L2(Ω) + ‖g‖2L2(Ω) ≤ CR,r0,T,M

[∫ T

0

∫Γ1

(∂utt∂ν

)2

+

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

].

(3.524)(e) Assume the hypotheses of (e) in Proposition 16. Then by (3.522), (3.523)

‖f‖2H10 (Ω) + ‖g‖2L2(Ω) ≤ CR,r0,T,M

[∫ T

0

∫Γ1

(∂utt∂ν

)2

+

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

].

(3.525)(f) Assume the hypotheses of (f) in Proposition 16. Then, via (3.506), (3.519),

and (3.520),

‖g‖2H1(Ω) ≤ CR,r0,T,M1

[∫ T

0

∫Γ1

(∂utt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂zt∂ν

)2

dΣ1

]

+CT,R,r0

[∫ T

0

∫Γ1

(∂uttt∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂ztt∂ν

)2

dΣ1

]. (3.526)

(g) Assume the hypotheses of (g) in Proposition 16. Then, by (3.525) and(3.526),

‖f‖2H10 (Ω) + ‖g‖2H1(Ω) ≤ CR,r0,T

[∫ T

0

∫Γ1

(∂ztt∂ν

)2

+

∫ T

0

∫Γ1

(∂uttt∂ν

)2]

+CR,r0,T,M,M1

[∫ T

0

∫Γ1

(∂zt∂ν

)2

+

∫ T

0

∫Γ1

(∂utt∂ν

)2]

. (3.527)

Proof. (a) We use (3.518) on the RHS of (3.501).(b) We use (3.519) (only for utt( · , T2 ) in H1(Ω)) on the RHS of (3.502).(c) We use a-fortiori (3.519) on the RHS of (3.503).(d) We use (3.519) a-fortiori on (3.504).(e) We combine (3.522) with (3.523) or use (3.519) on (3.505).(f) We use (3.519) (for utt( · , T2 ) in H1(Ω)) and (3.520) in the RHS of (3.506).(g) We combine (3.522) with (3.526) or use (3.519) and (3.520) on the RHS of

(3.505) and (3.506).

Corollary 3. Let 0 ≤ θ ≤ 1, θ 6= 12

.(1) Assume assumptions in (a) and (b) of Theorem 28. Then:

‖f‖2Hθ0 (Ω) ≤ CR,r0,T,θ

[∫ T

0

∫Γ1

(∂Dθ

t (ut)

∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂Dθ

t z

∂ν

)2

dΣ1

]

= CR,r0,T,θ

[∥∥∥∥∂ut∂ν

∥∥∥∥2

Hθ(0,T ;L2(Γ1))

+

∥∥∥∥∂z∂ν∥∥∥∥2

Hθ(0,T ;L2(Γ1))

]. (3.528a)


(2) Assume assumptions in (c) and (f) of Theorem 28. Then:

‖g‖2Hθ(Ω) ≤ CR,r0,T,θ,M,M1

[∫ T

0

∫Γ1

(∂Dθ

t (ut)

∂ν

)2

+

∫ T

0

∫Γ1

(∂Dθ

t (utt)

∂ν

)2]

+

[∫ T

0

∫Γ1

(∂Dθ

t z

∂ν

)2

dΣ1 +

∫ T

0

∫Γ1

(∂Dθ

t (zt)

∂ν

)2

dΣ1

]. (3.528b)

Proof. (1) We interpolate between inequalities (3.521) and (3.522) for f .(2) We interpolate between inequalities (3.523) and (3.526) for g. For 0<θ< 1

2,

the assumptions of (b) for (3.528a) and the assumptions of (f) for (3.528b) can bedispensed with in the interpolation process.

Remark 26. Theorem 28 and Corollary 3 provide the desired stability estimates atdifferent topological levels, modulo (polluted by) lower-order terms. These are theNeumann traces for the z, zt, ztt variables. In the next section, we shall absorb suchtraces by a compactness–uniqueness argument, as usual.

(b): Absorbing the lower order terms

In this section, we provide the compactness–uniqueness argument of absorbingthe lower order terms in the stability estimates (3.521)–(3.527). For simplicity wewill only show the case e) above where the stability is at the H1

0 × L2 level forf ∈ H1

0 (Ω) and g ∈ Hθ(Ω). The extra θ-derivative in g is due to a compactnesslemma that we need (Lemma 9). The cases where f and g are at other topologicallevels can be studied similarly.

Step 1. In this step we state a lemma which will allow us to absorb the secondterm on the RHS of (3.524) or (3.525). This lemma is the perfect counterpart ofLemma 3 to which we shall refer for a detailed proof.

Lemma 9. (a). Consider the z-system (3.511) with data q0, q1 ∈ L∞(Ω); f(x)Rtt(x, t) + g(x)Rt(x, t) ∈ L2(0, T ;Hθ(Ω));

Dθt (f(x)Rtt(x, t) + g(x)Rt(x, t)) ∈ L2(0, T ;L2(Ω)),

(3.529a)

(3.529b)

0 ≤ θ ≤ 1, θ 6= 12

. Then, continuously on the data, we have

z, zt ∈ C([0, T ];H1+θ(Ω)×Hθ(Ω)), and∂z

∂ν∈ Hθ(Σ). (3.530)

(b) Consider now the zt-system (3.514) with data q0, q1 ∈ L∞(Ω); f(x)Rttt(x, t)+g(x)Rtt(x, t) ∈ L2(0, T ;Hθ(Ω));

Dθt (f(x)Rttt(x, t) + g(x)Rtt(x, t)) ∈ L2(0, T ;L2(Ω)),

(3.531a)

(3.531b)

0 ≤ θ ≤ 1, θ 6= 12

. Then, continuously on the data, we have

zt, ztt ∈ C([0, T ];H1+θ(Ω)×Hθ(Ω)), and∂(zt)

∂ν∈ Hθ(Σ). (3.532)

(c) Checkable sufficient conditions for the regularity properties in (3.529), (3.531)to hold true are for 0 ≤ θ ≤ 1, θ 6= 1

2,


f(x), g(x) ∈ Hθ(Ω); Rtt, Rttt ∈ L2(0, T ;W θ,∞(Ω)) ∩Hθ(0, T ;L∞(Ω)), (3.533)

so that, ultimately, as a sufficient condition on the data, still for 0 ≤ θ ≤ 1, θ 6= 12

,

Regularity properties (3.533)⇒ ∂z

∂ν,∂(zt)

∂ν∈ Hθ(Σ) (3.534)

(of course, Rtt, Rttt ∈ Hθ(0, T ;W θ,∞(Ω)) implies Rtt, Rttt as on the LHS of(3.534)).

(d) If we define the operators K,K1 : H10 (Ω)×Hθ(Ω)⇒ L2(Γ1 × [0, T ]) by

Kf, g =∂zt∂ν|L2(Γ1×[0,T ]), K1f, g =

∂z

∂ν|L2(Γ1×[0,T ]). (3.535)

Then K and K1 are compact operators for θ > 0 in assumption (3.533), hence inconclusion (3.534).

Proof. The conclusions (a) and (b) are the standard regularity results Theorem 2,as applied to the z- and zt-problems (3.511), (3.514). A detailed proof of part (c)was given in Lemma 3. For (d), by the regularity results of (a) and (b) and thesufficient conditions in (c), we have for θ > 0, by (3.534):

f ∈ H10 (Ω), g ∈ Hθ(Ω) ⇒ (Kf, g) =

∂zt∂ν∈ Hθ(Σ) continuous; (3.536)

⇒ (Kf, g) =∂zt∂ν∈ Hθ−ε(Σ) compact, (3.537)

for any ε > 0 small, since now θ > 0 by assumption in (3.533). The same argumentalso applies to K1.

Step 2. Lemma 9 will allow us to absorb the term∥∥∥∥Kf, g =∂zt∂ν

∥∥∥∥L2(Γ1×[0,T ])

. (3.538)

on the RHS of estimate (3.525), by the compactness–uniqueness argument. Unique-ness rests on Theorem 24.

Proposition 18. Consider the u-problem (3.477) with T > T0 in (3.49) under as-sumption of Theorem 25: f ∈ H1

0 (Ω), g ∈ Hθ(Ω), 0 < θ ≤ 1, θ 6= 12

, q1, q0, Rsatisfying (3.478), (3.479), supplemented by the additional assumptions (3.533) for0 < θ ≤ 1, θ 6= 1

2implied by (3.482). Then the term Kf, g = ∂zt

∂ν|Σ1 measured in

the L2(Γ1 × [0, T ])-norm, can be omitted from the RHS of inequality (3.525) (for asuitable new constant CR,M,r0,T independent of the solution u), so that the desiredconclusion on the RHS of Theorem 25 holds true:

‖f‖2H10 (Ω) + ‖g‖2L2(Ω) ≤ CT,data

∫ T

0

∫Γ1

(∂utt∂ν

)2

dΓ1dt. (3.539)

Proof. Step (i). Suppose, by contradiction, that inequality (3.539) is false. Then,there exist two sequences fn∞n=1 and gn∞n=1, fn ∈ H1

0 (Ω), gn ∈ Hθ(Ω), suchthat


(i) ‖fn‖2H10 (Ω) + ‖gn‖2L2(Ω) = ‖fn, gn‖2H1

0 (Ω)×L2(Ω) ≡ 1; (3.540a)

(ii) limn→∞

∥∥∥∥∂utt(fn, gn)

∂ν

∥∥∥∥L2(Γ1×[0,T ])

= 0. (3.540b)

where u(fn, gn) solves problem (3.477a-c) with f = fn, g = gn:

(u(fn, gn))tt = ∆u(fn, gn) + q1(x)(u(fn, gn))t + q0(x)(u(fn, gn))

+fn(x)Rt(x, t) + gn(x)R(x, t) in Q; (3.541a)

u(fn, gn)

(· , T

2

)= 0; (u(fn, gn))t

(· , T

2

)= 0 in Ω; (3.541b)

u(fn, gn)(x, t)|Σ = 0 in Σ. (3.541c)

In view of (3.540a), there exists a subsequence of fn, gn ∈ H10 (Ω) × L2(Ω),

still denoted by fn, gn, such that: fn converges weakly in H10 (Ω) to some f0 ∈ H1

0 (Ω);

gn converges weakly in L2(Ω) to some g0 ∈ L2(Ω).

(3.542a)

(3.542b)

Moreover, since the operator K is compact under assumption (3.533) presentlyin force (Lemma 9), it then follows by (3.542) that we have strong convergence forKfn, gn:

limm,n→+∞

‖Kfn, gn −Kfm, gm‖L2(Γ1×[0,T ]) = 0; (3.543)

Step (ii). On the other hand, since the map f, g → u(f, g) is linear, andrecalling the definition of the operator K in (3.535), it follows from estimate (3.525)that

‖fn − fm‖H10 (Ω) + ‖gn − gm‖L2(Ω)

≤ CR,M,r0,T

(∥∥∥∥∂utt(fn, gn)

∂ν− ∂utt(fm, gm)

∂ν

∥∥∥∥L2(Γ1×[0,T ])

)+CR,M,r0,T

(‖Kfn, gn −Kfm, gm‖L2(Γ1×[0,T ])

). (3.544)

It then follows from (3.540b) and (3.543) as applied to (3.544) that

limm,n→∞

‖fn − fm‖H10 (Ω) = 0; lim

m,n→∞‖gn − gm‖L2(Ω) = 0. (3.545)

Thus, fn, gn is a Cauchy sequence in H10 (Ω)×L2(Ω). By uniqueness of the limit,

recall (3.542a-b), it then follows that

limn→∞

‖fn − f0‖H10 (Ω) = 0; lim

n→∞‖gn − g0‖L2(Ω) = 0. (3.546)


‖f0‖2H10 (Ω) + ‖g0‖2L2(Ω) = 1. (3.547)


Step (iii). We now apply to the the utt-problems (3.512) the trace regularity ofTheorem 5

f(x)Rttt + g(x)Rtt ∈ L2(0, T ;L2(Ω)),

utt

(x,T

2

)∈ H1

0 (Ω), (utt)t

(x,T

2

)∈ L2(Ω)

⇒ ∂utt∂ν

∣∣∣∣Σ

∈ L2(Σ) (3.548)

where in fact, by (3.509b) we have utt( · , T2 ) = f( · )Rt( · , T2 ) ∈ H10 (Ω), since

f( · ) ∈ H10 (Ω) and Rt( · , T2 ) ∈ W 1,∞(Ω); while by (3.509a) (utt)t( · , T2 ) =

q1( · )f( · )Rt( · , T2 ) +g( · )Rt( · , T2 ) + f( · )Rtt( · , T2 ) ∈ L2(Ω) by virtue of (3.478).Step (iv). We deduce from (3.548) and the considerations below (3.548), via

(3.478), thatf(x) ∈ H1

0 (Ω), g(x) ∈ L2(Ω)→ ∂utt(f, g)

∂ν|Σ ∈ L2(Σ)

i.e.

∥∥∥∥∂utt(f, g)

∂ν|Σ∥∥∥∥L2(Σ)

≤ CR(‖f‖H1

0 (Ω) + ‖g‖L2(Ω)

).

(3.549a)

(3.549b)

As the map f, g → utt(f, g)|Σ is linear, it then follows in particular from(3.549b), since fn, gn, f0, g0 ∈ H1

0 (Ω)× L2(Ω) by (3.542):∥∥∥∥∂utt(fn, gn)

∂ν|Σ1 −

∂utt(f0, g0)

∂ν|Σ1

∥∥∥∥L2(Σ1)

≤ CR(‖fn − f0‖H1

0 (Ω) + ‖gn − g0‖L2(Ω)

). (3.550)

Recalling (3.545) on the RHS of (3.550), we conclude that

limn→∞

∥∥∥∥∂utt(fn, gn)

∂ν|Σ1 −

∂utt(f0, g0)

∂ν|Σ1

∥∥∥∥L2(Σ1)

= 0, (3.551)

This, combined with (3.540b), then yields

∂utt(f0, g0)

∂ν

∣∣∣∣Σ1

≡ 0 in L2(Σ1), (3.552a)

and hence∂ut(f0, g0)

∂ν

∣∣∣∣Σ1

= constant in t. (3.552b)

Step (v). For the u(fn, gn)-problem (3.541) the standard interior regularity resultTheorem 5 yields for C = CT,R > 0, standard trace theory, and (3.545), via (3.478):

‖u(fn, gn), u(fn, gn)t − u(f0, g0), u(f0, g0)t‖C([0,T ];H1×L2)

≤ C(‖fn − f0‖H1

0 (Ω) + ‖gn − g0‖L2(Ω)

)→ 0; (3.553a)

‖u(fn, gn)|Γ − u(f0, g0)|Γ ‖C([0,T ];H

12 (Γ ))

≤ C(‖fn − f0‖H1

0 (Ω) + ‖gn − g0‖L2(Ω)

)→ 0. (3.553b)


Combining (3.553) with the homogeneous I.C. and zero Dirichlet B.C. of theu(fn, gn)-problem (3.550), we obtain

u(f0, g0)

(· , T

2

)≡ 0; (u(f0, g0))t

(· , T

2

)≡ 0 in Ω; u(f0, g0)|Σ = 0, (3.554)

hence∂u(f0, g0)

∂ν

(· , T

2

) ∣∣∣∣Γ

= 0 on Γ. (3.555)

Similarly, for the ut(fn, gn)-problem (3.509) corresponding to (3.541), we have from(3.554)

∂ut(f0, g0)

∂ν

(· , T

2

) ∣∣∣∣Γ

= 0 on Γ. (3.556)

Step (vi). Combining (3.552b) with first (3.556) and next (3.555) yields

∂u(f0, g0)

∂ν(x, t)

∣∣∣∣Γ1

= 0. (3.557)

Step (vii). Ultimately, u(f0, g0) satisfies weakly the following limit problem, via(3.553), (3.554), and (3.557),

utt(f0, g0)−∆u(f0, g0)− q1(x)ut(f0, g0)− q0(x)u(f0, g0)

= f0(x)Rt + g0(x)R in Q;

u(f0, g0)

(· , T

2

)= 0; ut(f0, g0)

(· , T

2

)= 0 in Ω;

u(f0, g0)|Σ = 0 and∂u(f0, g0)

∂ν|Σ1 = 0;

(3.558a)

(3.558b)

(3.558c)

f0 ∈ H10 (Ω), g0 ∈ Hθ(Ω) ⊂ L2(Ω); 0 < θ ≤ 1, θ 6= 1

2, (3.559)

with data satisfying (3.478), (3.479). Thus, the uniqueness Theorem 24 applies andyields the following conclusion

f0(x) = g0(x) ≡ 0, a.e. x ∈ Ω. (3.560)

which contradicts (3.547). Thus, assumption (3.540) is false and inequality (3.539)holds true and Proposition 18, as well as the RHS inequality (3.483) of Theorem 25are then established.

Step 3. The LHS inequality in (3.483) of Theorem 25 is a-fortiori contained in thefollowing proposition.

Proposition 19. Consider the utt-problem as in (3.512), with Rt, Rtt, Rttt ∈ L∞(Q)as in (3.478), and f ∈ H1

0 (Ω), g ∈ L2(Ω), q1, q0 ∈ L∞(Ω). Then the following in-equality holds true: There exists C = CT > 0 such that∥∥∥∥∂utt∂ν

∥∥∥∥L2(Σ1)

≤ C(‖f‖H1

0 (Ω) + ‖g‖L2(Ω)

). (3.561)


Proof. This is a direct application of the regularity result in Theorem 4, Eqn. (3.20)to the utt-problem (3.512). In fact, as to the RHS of (3.512a) via (3.478)

‖fRttt + gRtt‖2L1(0,T ;L2(Ω)) ≤ CT[‖f‖2H1

0 (Ω) + ‖g‖2L2(Ω)

], (3.562)

while regarding the I.C., we have (see below (3.548) with R( · , T2

) ∈W 1,∞(Ω))∥∥∥∥utt( · , T2)

= f( · )Rt(· , T

2

)∥∥∥∥2

H10 (Ω)

≤ CT ‖f‖2H10 (Ω); (3.563)

∥∥∥∥uttt( · , T2)∥∥∥∥2

L2(Ω)

≤ CT[‖f‖2H1

0 (Ω) + ‖g‖2L2(Ω)

], (3.564)

by invoking (3.478).

Uniqueness and stability of nonlinear inverse problem for thew-problem (3.475): Proof of Theorems 26 and 27

Step 1. Orientation. We return to the non-homogeneous w-problem (3.475).Let w(q1, q0), w(p1, p0) be solutions of problem (3.475) due to the coefficientsq1(·), q0( · ) and p1(·), p0( · ), respectively. By the change of variable as in (3.476),

f(x) = q1(x)− p1(x), g(x) = q0(x)− p0(x); (3.565a)

u(x, t) = w(q1, q0)(x, t)− w(p1, p0)(x, t); R(x, t) = w(p1, p0)(x, t), (3.565b)

then the variable u satisfies problem (3.477), for which Theorems 24 and 25 pro-vide the corresponding uniqueness and stability results. We here seek to reduce the(nonlinear) uniqueness and stability results for the original w-problem (3.475) to the(linear) uniqueness and stability results for the u-problem (3.477), Theorems 24 and25. To this end, we need to verify for the term R(x, t) = w(p1, p0)(x, t) in (3.565b)the assumptions required in Eqns. (3.478), (3.479) of the uniqueness Theorem 24,and in addition, Eqn. (3.482) for the stability Theorem 25 which implies (3.533). Forthis, since q1, q0, p1, p0 ∈ L∞(Ω), by assumption in the uniqueness Theorem 26 andq1, p1 ∈ H1

0 (Ω), q0, p0 ∈ Hθ(Ω), 0 < θ ≤ 1, θ 6= 12, by assumption in the stability

Theorem 27, respectively, we then have f = q1− p1, g = q0− p0 ∈ L∞(Ω) ⊂ L2(Ω),in the first case and, f = q1−p1 ∈ H1

0 (Ω)∩L∞(Ω), g = q0−p0 ∈ Hθ(Ω)∩L∞(Ω) inthe second case, as required in (3.478) and (3.482). Moreover, assumptions (3.489):w0 = 0, |w1(x)| ≥ v1 > 0 are the counterpart of the assumptions (3.479) for R(x, T

2)

and Rt(x,T2

) via (3.565b), while w1 ∈W 1,∞(Ω) in assumption (3.487) is the coun-terpart of assumption R(x, T

2) ∈W 1,∞(Ω) in (3.478) and (3.482). Thus, in order to

be able to invoke the uniqueness and stability results, Theorem 24 and 25, for thevariable u = w(q1, q0)−w(p1, p0) in (3.565), solution of problem (3.477), what is leftis to verify the regularity properties (3.478) and (3.482) on R defined by (3.565b),i.e., the following properties:

w(p1, p0), wt(p1, p0), wtt(p1, p0), wttt(p1, p0), wtttt(p1, p0) ∈ L∞(Q);

wttt(p1, p0) ∈ Hθ(0, T ;W θ,∞(Ω)), (3.566)


as a consequence of suitably smooth I.C. w0, w1 in (7.1b) [with q1(·), q0( · )replaced by p1(·), p0( · )] and, respectively, of suitably smooth Dirichlet boundaryterm µ in (3.475c).

The L∞-regularity conditions in (3.566) are needed for uniqueness via the corre-sponding regularity conditions in (3.477b) of Theorem 24. The additional regularityHθ(0, T ;W θ,∞(Ω)) in (3.566) is needed for stability via the corresponding condi-tion (3.482) of Theorem 25. In addition, the reduction procedure from u to w willfurther require that the coefficients q1( · ), q0( · ) to be recovered be actually a-prioriin a smoother class than L∞(Ω), and moreover satisfy suitable boundary compat-ibility conditions (B.C.C.), depending on dim Ω. This is illustrated in Proposition20 below.

Proposition 20. (a) With reference to the w-problem (3.475), with q1, q0 ∈ L∞(Ω)originally, let, in fact (with m non-necessarily integer):

q1, q0 ∈Wm,∞(Ω), w0, w1 ∈ Hm+1(Ω)×Hm(Ω) and µ ∈ Hm+1(Σ), (3.567)

where all compatibility conditions (trace coincidence) which make sense are satis-fied. These are identified in the subsequent proof for dim Ω = 3 and dim Ω = 2.Then, the corresponding solution w(q) satisfies (a-fortiori) the following propertiescontinuously:

wt(q1, q0), wtt(q1, q0), wttt(q1, q0), wtttt(q1, q0) ∈

C([0, T ];Hm(Ω)×Hm−1(Ω)×Hm−2(Ω)×Hm−3(Ω)). (3.568)

(b) In addition, let m > dimΩ2

+ 3. Then, continuously [LM72, p. 45],

wt(q1, q0), wtt(q1, q0), wttt(q1, q0), wtttt(q1, q0) ∈ L∞(Q). (3.569)

(c) Moreover,

wttt(q1, q0) ∈ Hθ(0, T ;W θ,∞(Ω)), 0 < θ < min

1

2,m− 2− dimΩ

2

2

. (3.570)

Proof. (a) Step 1. We start with the following Φ-problem:Φtt(x, t) = ∆Φ(x, t) in Q;

Φ

(· , T

2

)= Φ0(x) = w0(x);Φt

(· , T

2

)= Φ1(x) = w1(x) inΩ;

Φ(x, t)|Σ = µ(x, t) in Σ.

(3.571a)

(3.571b)

(3.571c)

which corresponds to the w-problem (3.475) with q1 = q0 = 0. Its optimal regularityis given by Theorem 6: under assumptions (3.567), along with all CompatibilityConditions (trace coincidence) which make sense (they invoke µ, w0, w1), we obtain

Φ,Φt, Φtt, Φttt, Φtttt ∈

C([0, T ];Hm+1(Ω)×Hm(Ω)×Hm−1(Ω)×Hm−2(Ω)×Hm−3(Ω)), (3.572)

along with ∂Φ∂ν|Σ ∈ Hm(Σ) (which is not needed in the present proof). Thus [LM72,

p. 45],


Φ,Φt, Φtt, Φttt, Φtttt ∈ L∞(Q), for m >dimΩ

2+ 3. (3.573)

Step 2. We sety = w − Φ, (3.574)

where then, by (3.475) and (3.571), y solves

ytt(x, t) = ∆y + q1(x)yt + q0(x)y + F in Q;

y

(· , T

2

)= y0(x) = 0; yt

(· , T

2

)= y1(x) = 0 in Ω;

y|Σ = 0 in Σ;

F = q1(x)Φt + q0(x)Φ in Q;

(3.575a)

(3.575b)

(3.575c)

(3.575d)[y(t)

yt(t)

]=

∫ t

0

eAq(t−s)

[0

q1( · )Φt(s) + q0( · )Φ(s)

]ds; (3.576)

Aq =

[0 I

−AD + q0( · ) q1( · )

]; A0 =

[0 I

−AD 0,

](3.577)

ADh = −∆h, D(AD) = H2(Ω)×H10 (Ω). The following Lemma is readily shown:

Step 3. Lemma 10 Assume

q1( · ) is a bounded operator: D(Am2D )→ D(A

m2D ); (3.578a)

q0( · ) is a bounded operator: D(Am+1

2D )→ D(A

m2D ). (3.578b)

so that, then, with respect to (3.577), we have:

the operator Aq is a bounded perturbation of the operator A0 on the state

space D(A mq ) = D(A m

0 ) = D(Am+1

2D )×D(A

m2D ), hence with equal domains

on that space.(3.579)

Assume further that

q1( · )Φt + q0( · )Φ ∈ C([0, T ]; D(Am2D )), (3.580)

with Φ,Φt as in (3.572). Then y(t)

yt(t)

∈ C[0, T ]; D(A m

q ) = D(A m0 ) =

D(Am+1

2D )

D(A mD )

(3.581)

⊂ C

[0, T ];

Hm+1(Ω)

Hm(Ω)

. (3.582)

Remark 27. We shall collect at the end the assumptions on the coefficients q =q1, q0, that will ensure that all the required assumptions—such as (3.578) and(3.580) and others below—are satisfied.


Step 4. Lemma 11 Assume hypotheses (3.578) and (3.580) of Lemma 10, so thatthe regularity properties (3.582) hold true: y, yt ∈ C([0, T ];Hm+1(Ω)×Hm(Ω)). Let m, m+1 6= integer

2. Furthermore, with reference to Φ,Φt, Φtt ∈

C([0, T ];Hm+1(Ω)×Hm(Ω)×Hm−1(Ω)) as in (3.572), assume [MS85]

q1( · )∈M(Hm(Ω)→ Hm−1(Ω)); q1( · )∈M(Hm−1(Ω)→ Hm−2(Ω)); (3.583)

q0( · )∈M(Hm+1(Ω)→ Hm−1(Ω)); q0( · )∈M(Hm(Ω)→ Hm−2(Ω)). (3.584)

Then, recalling (3.575a-d),

ytt = ∆y + q1( · )yt + q0( · )y + q1( · )Φt + q0( · )Φ ∈ C([0, T ];Hm−1(Ω)); (3.585)

yttt = ∆yt+ q1( · )ytt+ q0( · )yt+ q1( · )Φtt+ q0( · )Φt ∈ C([0, T ];Hm−2(Ω)). (3.586)

Proof. The proof is immediate: For m, m + 1 6= positive integer2

, ∆y,∆yt ∈C([0, T ];Hm−1(Ω) × Hm−2(Ω)); moreover, q1yt, q1Φt ∈ C([0, T ];Hm−1(Ω)), q0y,q0Φ ∈ C([0, T ];Hm(Ω)); q1ytt, q1Φtt ∈ C([0, T ];Hm−2(Ω)), q0yt, q0Φt ∈ C([0, T ];Hm−2(Ω)).

Step 5. Lemma 12 Assume the hypotheses of Lemma 11, so that ytt, yttt ∈C([0, T ];Hm−1(Ω)×Hm−2(Ω)), as dictated by (3.585), (3.586). Let m−1 6= integer

2.

Furthermore, with reference to Φtt, Φttt ∈ C([0, T ];Hm−1(Ω)×Hm−2(Ω)), assume

q1( · ) ∈M(Hm−2(Ω)→ Hm−3(Ω)); q0( · ) ∈M(Hm−1(Ω)→ Hm−3(Ω)). (3.587)

Then

ytttt = ∆ytt + q1( · )yttt + q0( · )ytt + q1( · )Φttt + q0( · )Φtt ∈ C([0, T ];Hm−3(Ω)),(3.588)

hence

ytttt ∈ L∞(Q) for m >dimΩ

2+ 3. (3.589)

Proof. The proof of (3.588) is immediate, from which (3.589) follows by a standardresult [LM72, p. 45].

Step 6. Corollary 4 Assume condition (3.567) for w0, w1, µ, as well as the hy-potheses of Lemma 12. Then, with reference to (3.573), (3.574), and (3.589), wehave

wtttt = ytttt + φtttt ∈ L∞(Q), m >dimΩ

2+ 3. (3.590)

Step 7. Here we collect all requirements of Lemma 12 (or Corollary 4), which include(3.578), (3.580), (3.583), (3.584) and (3.587). Conditions (3.578), (3.580) of Lemma9 include the following regularity conditions

q1 ∈M(Hm(Ω)→ Hm(Ω)) [included in (3.578a)]; (3.591)

q0 ∈M(Hm+1(Ω)→ Hm(Ω)) [included in (3.578b)], (3.592)

plus boundary compatibility conditions (B.C.C.) to be analyzed below. Lemmas 11and 12, moreover, require the following regularity properties:


q1 ∈ M(Hm(Ω)→ Hm−1(Ω)) (3.593)

∈ M(Hm−1(Ω)→ Hm−2(Ω)) (3.594)

∈ M(Hm−2(Ω)→ Hm−3(Ω)); (3.595)

q0 ∈ M(Hm+1(Ω)→ Hm−1(Ω)) (3.596)

∈ M(Hm(Ω)→ Hm−2(Ω)) (3.597)

∈ M(Hm−1(Ω)→ Hm−3(Ω)). (3.598)

Thus, in terms of just regularity properties, we need to require:

For q1 : q1 ∈ M(Hm(Ω)→ Hm(Ω)) as in (3.591) (which thenimplies (3.593)), (3.594), (3.595);

(3.599)

For q0 : q0 ∈ M(Hm+1(Ω) → Hm(Ω)) as in (3.592) (whichthen implies (3.596)), (3.597), (3.598).

(3.600)

Conclusion #1: All the regularity properties can be fulfilled by assuming

q1 ∈Wm,∞(Ω), q0 ∈Wm,∞(Ω) (3.601)

as in hypothesis (3.567).Conclusion #2: In addition, boundary compatibility conditions (B.C.C.) re-

lated to (3.578) and (3.580) need to be imposed.Case dim Ω = 3. For example, consider the B.C.C. for m = 5, as to cover

readily the case dim Ω = 3: 5 = m > 32

+ 3. We must analyze the B.C.C. for q1:

bounded operator D(A52D )→ D(A

52D ), already knowing the regularity (3.601)

q1 ∈W 5,∞(Ω). (3.602)

Thus, let f ∈ D(A52D ), which means

(1a)f |Γ = 0 : to have f ∈ D(AD); (3.603a)

(1b)∆f |Γ =0 : to have ADf=−∆f ∈ D(AD) or f ∈ D(A 2

D); (3.603b)

(1c)∆2f |Γ = 0 : to have A 2Df = ∆2f ∈ D(A

12D ) = H1

0 (Ω). (3.603c)

Then require q1f ∈ D(A52D ) which means

(1’a) q1f |Γ = 0; (3.604a)

(1’b)∆(q1f)|Γ = 0; (3.604b)

(1’c)∆2(q1f)|Γ = 0. (3.604c)

We now compare (1a), (1b), and (1c) with (1’a), (1’b), and (1’c). We first seethat (1a) implies automatically (1’a). Next, (1’b) implies via (1a), (1b):

∆(q1f)|Γ = [(∆

q1)f + q1

(∆f) + 2∇q1 · ∇f ]Γ = 0⇐⇒ ∂q1∂ν

∣∣∣∣Γ

= 0, (3.605)

since f |Γ = 0 in (1a) makes ∇f |Γ orthogonal to Γ .


*@@I

Γ∇q1

∇f

f ≡ 0 on Γ

*@@I

Γ∇∆q1

∇∆f

∆f ≡ 0 on Γ

Thus, (3.605) is the only condition required to satisfy (1’a) and (1’b)(this is alsothe only B.C.C. needed for the situation of dim Ω = 2 when taking m = 4 + ε sincehere we have m > 2

2+ 3 = 4, see case dim Ω = 2 below). Next, to satisfy (1’c), we

compute

∆2(q1f) = ∆(∆(q1f)) = ∆[(∆q1)f + q1∆f + 2∇q1 · ∇f ]. (3.606)

Thus, by virtue of (1a), (1b), and (1c), we obtain

[∆2(q1f)]|Γ = [(∆2

q1)f + (∆q1)

∆f + 2∇∆q1 · ∇f +∆

q1∆f

+ q1∆2f + 2∇q1 · ∇∆f + 2∆(∇q1 · ∇f)]|Γ . (3.607)

We recall again that (1a) and (1b) imply that ∇f |Γ and ∇∆f |Γ are orthogonalto Γ . Thus, in order to satisfy condition (1’c) = (3.604c): [∆2(q1f)]Γ = 0, we imposeat first that

[∇∆q1 · ∇f ]Γ = 0 and [∇q1 · ∇∆f ]Γ = 0; i.e., (3.608a)

that ∇∆q1]Γ and [∇q1]Γ be tangential to Γ ; or (3.608b)

∂∆q1∂ν

∣∣∣∣Γ

= 0 and∂q1∂ν

∣∣∣∣Γ

= 0, (3.608c)

so that by (3.607) and (3.608a) we obtain and impose

1

2[∆2(q1f)]|Γ = |∆(∇q1 · ∇f)]|Γ =

[∆

(3∑i

q1xi · ∇fxi

)]Γ

=

[∇(∆q1) · ∇f +∇

q1 · ∇∆f + 2

3∑i

∇q1xi · ∇fxi

]Γ

= 0 on Γ, (3.609)

as the tangential vectors [∇q1 Γ , respectively [∇∆q1]Γ are the orthogonal to thenormal [∇∆f ]Γ , respectively [∇f ]Γ , on Γ ; or

1

4[∆2(q1f)]Γ = 2

∂x1∇q1· · ·∂x3∇q1

· ∂x1∇f· · ·∂x3∇f

∣∣∣∣∣∣Γ

= 0, (3.610)

which requires[∂xi∇q1]Γ = [∇q1xi ]Γ = 0, i = 1, 2, 3. (3.611)

Case dim Ω = 2. In this case we take m > 22

+ 3, or m = 4 + ε′. Then, we needto require

q1 bounded D(A2+εD )→ itself, (3.612)


so that we still need (1’a) and (1’b), but not (1’c), since we now require A2D(q1f) =

∆2(q1f) ∈ D(AεD) = H2ε(Ω), which is a requirement of extra regularity without,however, an additional boundary condition. Thus, now f satisfies only (1a) and (1b),and then (q1f) must satisfy only (1’a), (1’b).

In conclusion, for dim Ω = 2, the only boundary compatibility condition is

∇q1 tangential on Γ ; or∂q1∂ν

∣∣∣∣Γ

= 0. (3.613)

(b) The usual embedding applies on (3.568) to yield part (b) since m > dimΩ2

+3.(c) Applying the intermediate derivative theorem [LM72, m = 1, l = θ, p. 15] on

wttt and Dtwttt in (3.568), with C([0, T ]; · ) replaced by L2(0, T ; · ), we obtain

wttt(q1, q0) ∈ Hθ(0, T ;Hm−2−2θ(Ω)), 0 < θ <1

2. (3.614)

Comparing with wttt in (3.566), we see that we need to ascertainHm−2−2θ(Ω) ⊂ W θ,∞(Ω), or Hm−2−3θ(Ω) ⊂ L∞(Ω), which holds true providedm− 2− 3θ > dimΩ

2. Thus (3.570) is proved.

Completion of the Proof of Theorems 26 and 27 Having verified in Propo-sition 20 properties (3.560) and (3.570)—that is, (3.566)—it follows that we haveverified the properties (3.478), (3.479), and (3.492) for the u-problem (3.477), withu defined by (3.565a). Thus, Theorems 24 and 25 apply, and we then obtain unique-ness result q(x) = p(x) as in Theorem 26 and stability result of the conclusion ofTheorem 27:

c

∥∥∥∥∂wtt(q1, q0)

∂ν− ∂wtt(p1, p0)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

≤ ‖q1 − p1‖H10 (Ω) + ‖q0 − p0‖L2(Ω)

≤ C∥∥∥∥∂wtt(q1, q0)

∂ν− ∂wtt(p1, p0)

∂ν

∥∥∥∥L2(0,T ;L2(Γ1))

.

3.7.4 Notes

The present section is after [LT13]. We are not aware of works which recover simul-taneously two coefficients (damping and source) in one shot via a single boundarymeasurement.

Appendix: Admissible geometrical configurations in theNeumann B.C. case

Here we present some examples in connection to the main geometrical assumptions(A.1), (A.2). We refer to [LTZ00] for more details.

Ex. #1 (Any dimension ≥ 2): Γ0 is flat.


• • •1

measurement on Γ1x Ω

x0 6Γ0

Γ1

Let x0 ∈ hyperplane containing Γ0, then.

d(x) = ‖x− x0‖2; h(x) = ∇d(x) = 2(x− x0).

Ex. #2 (A ball of any dimension ≥ 2): d(x) in [LTZ00, Theorem. A.4.1, p. 301].

Measurement on Γ1 >12

circumference (as in the Dirichlet case), same as for con-trollability.

Ex. #3 (Generalizing Ex #2: a domain Ω of any dimension ≥ 2 with unob-served portion Γ0 convex, subtended by a common point x0): d(x) in [LTZ00, The-orem. A.4.1, p. 301].

HHHH

HHH

•

•

∇`

@@IΓ0 convex

Γ1

x0

Γ0 = `(x) = level set

(x− x0) · ∇` ≤ 0 on Γ0

Ex. #4 (A domain Ω of any dimension ≥ 2 with unobserved portion Γ0 concave,subtended by a common point x0): d(x) in [LTZ00, Theorem. A.4.1, p. 301].


HHHHHHH

•

•

Γ1

*Γ0 concave

x0

Ex. #5 (dim = 2): Γ0 neither convex or concave. Γ0 is described by graph

y =

f1(x), x0 ≤ x ≤ x1, y ≥ 0;

f2(x), x2 ≤ x ≤ x1, y < 0,

f1, f2 logarithmic concave on x0 < x < x1, e.g., sinx+ 1, −π2< x < π

2; cosx+

1, 0 < x < π

Function d(x) is given in [LTZ00, Eqn. (A.2.7), p. 289].

Acknowledgments

This first author was partially supported by the Academy of Finland Project 141075.The second author was partially supported by the National Science Foundationunder Grant DMS-0104305 and by the Air Force Office of Scientific Research underGrant FA 9550-09-1-0459. The second named author wishes to thank V. Isakov,Wichita State University, for bringing [Isa06, Theorem 8.2.2, p. 231] to his attention,and for useful insight and conversations on inverse hyperbolic problems. In fact, ourproofs of linear uniqueness also takes advantage of a convenient tactical route “post-Carleman estimates” provided by [Isa06, Theorem 8.2.2, p. 231]. The authors alsowish to thank the referee for suggestions of expository nature.

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[LTY03] Lasiecka, I., Triggiani, R., Yao, P. F.: Carleman estimates for a plateequation on a Riemann manifold with energy level terms. In: Begehr,H., Gilbert, R., Wang, M. W. (eds) Analysis and Applications. Kluwer,199–236 (2003)


[LTZ00] Lasiecka, I., Triggiani, R., Zhang, X.: Nonconservative wave equationswith unobserved Neumann B. C.: Global uniqueness and observability inone shot. Contemp. Math., 268, 227–325 (2000)

[LTZ04a] Lasiecka, I., Triggiani, R., Zhang, X.: Global uniqueness, observabilityand stabilization of nonconservative Schrodinger equations via pointwiseCarleman estimates. Part I: H1(Ω)-estimates. J. Inverse Ill-Posed Probl.,12, 43–123 (2004)

[LTZ04b] Lasiecka, I., Triggiani, R., Zhang, X.: Global uniqueness, observabilityand stabilization of nonconservative Schrodinger equations via pointwiseCarleman estimates. Part II: L2(Ω)-estimates. J. Inverse Ill-Posed Probl.,12, 182–231 (2004)

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[Liu11] Liu, S.: Inverse problem for a structural acoustic interaction. NonlinearAnal., 74, 2647–2662 (2011)

[LT11a] Liu, S., Triggiani, R.: Global uniqueness and stability in determiningthe damping and potential coefficients of an inverse hyperbolic problem.Nonlinear Anal. Real World Appl., 12, 1562–1590 (2011)

[LT11b] Liu, S., Triggiani, R.: Global uniqueness and stability in determiningthe damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B. C. through an additional Dirichlet boundarytrace. SIAM J. Math. Anal., 43, 1631–1666 (2011)

[LT11c] Liu, S., Triggiani, R.: Global uniqueness in determining electric poten-tials for a system of strongly coupled Schrodinger equations with mag-netic potential terms. J. Inverse Ill-Posed Probl., 19, 223–254 (2011)

[LT11d] Liu, S., Triggiani, R.: Recovering the damping coefficients for a systemof coupled wave equations with Neumann BC: uniqueness and stability.Chin. Ann. Math. Ser B, 32, 669–698 (2011)

[LT11e] Liu, S., Triggiani, R.: Determining damping and potential coefficients ofan inverse problem for a system of two coupled hyperbolic equations. PartI: Global uniqueness. In: Dynamical Systems and Differential Equations,DCDS Supplement 2011. Proceedings of the 8th AIMS International Con-ference (Dresden, Germany), 1001–1014

[LT12] Liu, S., Triggiani, R.: Global uniqueness and stability in determiningthe damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B. C. through an additional localized Neumannboundary trace. Applicable Analysis, Special Issue on Direct and InverseProblems, 91(8), 1551–1581 (2012)

[LT13] Liu, S., Triggiani, R.: Recovering damping and potential coefficients foran inverse non-homogeneous second-order hyperbolic problem via a lo-calized Neumann boundary trace. Discrete Contin. Dyn. Syst. Ser. A,33(11/12), 5217–5252 (2013)


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aimsciences.org

This is the first of two volumes containing the lecture notes of some ofthe courses given during the intensive trimester HCDTE, Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: analysis and control, held at SISSA, Trieste (Italy) from May 16th to July 22nd, 2011.

The lectures covered a number of different topics within the fields of hy-perbolic equations, fluid dynamic, dispersive and transport equations, measure theory and control and they were primarily intended for PhD students and young researchers at the beginning of their career.

I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponentsJ.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problemsS. Liu and R. Triggiani, Boundary control and boundary inverse theory for non-homogeneous second order hyperbolic equations: a common Carleman estimates approach

Giovanni AlbertiFabio AnconaStefano BianchiniGianluca CrippaCamillo De LellisAndrea MarsonCorrado Mascia (Eds.)


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