Florea-Some Methods of Infinite Dimensional Analysisi in Hydrodynamics_An Introduction

8/6/2019 Florea-Some Methods of Infinite Dimensional Analysisi in Hydrodynamics_An Introduction

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Lecture Notes in Mathematics 1942

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Sergio Albeverio

Franco FlandoliYakov G. Sinai

SPDE in Hydrodynamic:

Recent Progress andProspects

Lectures given at theC.I.M.E. Summer Schoolheld in Cetraro, ItalyAugust 29September 3, 2005

Editors:

Giuseppe Da PratoMichael Rckner

123


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Sergio AlbeverioInstitut fr Angewandte MathematikUniversitt BonnWegelerstr. 653115 [email protected]

Franco Flandoli

Dipartimento di Metamatica ApplicataU.Dini

Universit di PisaVia Buonarroti 156127 [email protected]

Yakov G. Sinai

Department of MathematicsPrinceton UniversityFine Hall, Washington RoadPrinceton N.J. [email protected]

Giuseppe Da PratoScuola Normale SuperiorePiazza dei Cavalieri 756126 [email protected]

Michael Rckner

Fakultat fr MathematikUniversitat BielefeldPostfach 10013133501 [email protected]

ISBN: 978-3-540-78492-0 e-ISBN: 978-3-540-78493-7DOI: 10.1007/978-3-540-78493-7

Lecture Notes in Mathematics ISSN print edition: 0075-8434

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Library of Congress Control Number: 2008921567

Mathematics Subject Classification (2000): 76D03, 35Q05, 76F02, 60H15

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Preface

It is a commonly accepted fact in the mathematical scientific community thatthe rigorous understanding of turbulence and related questions in hydrody-namics is one of the most important problems in mathematics and one ofthe challenging tasks for the future development of the theory of partial dif-ferential equations in particular, but also of analysis in general. One of thecentral open problems, namely the well posedness of the 3D-NavierStokessystem has been selected as one of the millennium problems and has resistedall attempts to solve it up to the present day.

Over more than half a century a lot of deep mathematics was developedto tackle these problems. One of the approaches was to use stochastic analysisbased on modifying the equations (as e.g. Euler, NavierStokes and Burgers)

adding a noise term. The idea here was to use the smoothing effect of thenoise on the one hand, but also to discover new phenomena of stochasticnature on the other hand. In addition, this was also motivated by physicalconsiderations, aiming at including perturbative effects, which cannot be mod-elled deterministically, due to too many degrees of freedom being involved, oraiming at taking into account different time scales of the components of theunderlying dynamics. Today we look back on 30 years of mathematical workimplementing probabilistic ideas into the area. During the last few years ac-tivities have become even more intense and several new groups in the worldworking on probability have turned their attention to these classes of highlyinteresting SPDEs of fundamental importance in Physics.

In a sentence, one of the purposes of the course was to understand the link

between the Euler and NavierStokes equations or their stochastic versionsand the phenomenological laws of turbulence. The idea can be better under-stood by analogy with Feynmans description of statistical mechanics: in thatfield on the one hand one has the Hamilton (or Schr odinger) equations for thedynamics of molecules and, on the other hand, the macroscopic laws of ther-modynamics. In between there is the concept of Gibbs measures, so the theorylooks like the ascent of a mountain, from Hamilton equations to Gibbs mea-sures (here ergodicity is a central topic), and a descent from Gibbs measures


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VI Preface

to thermodynamics. Translating this viewpoint in the realm of turbulent fluiddynamics, on the one side we have the NavierStokes equations as dynamicalequations (that we commonly accept as essentially correct). On the other side,we know a number of phenomenological laws, like the Kolmogorovs scaling(which he proposed in 1941, therefore called K41 scaling) or the multifractalscalings, which fit experimental and numerical data to some extent, but missa rigorous foundation and presumably require some correction. If we aim atan analogy with statistical mechanics, the missing point is a concept of Gibbsmeasure linking these two extreme parts of the theory.

Three courses of eight hours each were delivered to develop these ideas,both for the deterministic and the stochastic case.

Sergio Albeverio presented an approach to (deterministic) Euler and sto-

chastic NavierStokes equations in two dimensions based on invariant mea-sures and renormalization methods. His last lecture was devoted to asymptoticmethods for functional integrals.

Franco Flandoli started from some basic results on NavierStokesequations in three dimensions, discussing topics as existence of martingalesolutions, construction of a transition semigroup, ergodicity and continuousdependence on initial conditions. One of the main results was a preliminarystep to prove well posedness of the stochastic 3D-NavierStokes equations byshowing the existence of a Markov selection. He also has presented a reviewof the Kolmogorov K41 scaling law and some rigorous results on it for thestochastic NavierStokes equations.

Finally, Yakov Sinai described some rigorous mathematical results ford-dimensional (determinisitic) NavierStokes systems. In this direction he ex-

plained the power series and diagrams method for the Fourier transform ofNavierStokes equations and the Foias-Temam Theorem. He also presentedsome recent results on the one-dimensional Burgers equation with randomforcing, that is, in the stochastic case.

Afternoon sessions were devoted to research seminars delivered by theparticipants.

We thank the lecturers and all participants for their contributions to thesuccess of this Summer School.

Finally, we thank the CIME Scientific Committee for giving us the op-portunity to organize this meeting and the CIME staff for their efficient andcontinuous help.

Bielefeld and Pisa 2008 Giuseppe Da PratoMichael Rockner


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Contents

Some Methods of Infinite Dimensional Analysis

in Hydrodynamics: An Introduction

Sergio Albeverio and Benedetta Ferrario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Euler Equation, its Invariants and Associated

Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 The Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 The Euler Equation in Terms of Vorticity . . . . . . . . . . . . . . . . . . . . 32.3 The Conserved Quantities for the Euler Equation . . . . . . . . . . . . . 42.4 Heuristic Invariant Measures Associated

with the Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The Euler Equation in Terms of the Fourier Components

of the Stream Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 The Necessity of Looking for Singular Solutions. Divergence

of the Energy with Respect to the Enstrophy Measure . . . . . . . . . 92.7 Relation of the Enstrophy Measure

with the Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 The Infinitesimal Invariance of. Relation with the Hopf

Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 The Question of Uniqueness of Generators of the Euler Flow . . . . 122.10 An Euler Flow in a Sobolev Space of Negative Index . . . . . . . . . . . 142.11 Some Remarks on the Vortex Model and its Relations

with the Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Stochastic NavierStokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 The NavierStokes Equation with Space-Time White Noise . . . . . 173.2 The Gaussian Invariant Measure Given by the Enstrophy(and the Viscosity Parameter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Pathwise Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Some Additional Remarks and Complements . . . . . . . . . . . . . . . . . 333.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


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VIII Contents

An Introduction to 3D Stochastic Fluid DynamicsFranco Flandoli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 Abstract Framework and General Preliminaries . . . . . . . . . . . . . . . . . . . 523 Finite Dimensional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1 Introduction and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 A Priori Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Comparison of Two Solutions and Pathwise Estimates . . . . . . . . . 713.4 Existence and Uniqueness, Markov Property . . . . . . . . . . . . . . . . . . 733.5 Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.6 Galerkin Stationary Measures for the 3D Equation . . . . . . . . . . . . 79

4 Stochastic NavierStokes Equations in 3D . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Concepts of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Existence of Solutions to the Martingale Problem . . . . . . . . . . . . . 864.3 Technical Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.4 An Abstract Markov Selection Result . . . . . . . . . . . . . . . . . . . . . . . . 954.5 Markov Selection for the 3D Stochastic NSEs. . . . . . . . . . . . . . . . . 1034.6 Continuity in u0 of Markov Solutions . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Some Topics on Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.1 Introduction and a Few Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.2 K41 Scaling Law: Heuristics and Unclear Issues . . . . . . . . . . . . . . . 1265.3 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.4 Brownian Eddies and Random Vortex Filaments . . . . . . . . . . . . . . 1335.5 Necessary Conditions for K41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.6 A Condition Equivalent to K41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Mathematical Results Related to the NavierStokes System

Yakov G. Sinai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1511 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512 Power Series and Diagrams for the NavierStokes-System . . . . . . . . . . . 1543 Foias-Temam Theorem for 2DNavierStokes System with Periodic

Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584 Burgers System and 1D Inviscid Burgers Equation

with Random Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165


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Some Methods of Infinite Dimensional

Analysis in Hydrodynamics: An Introduction

Sergio Albeverio1 and Benedetta Ferrario2

1 Institute of Applied Mathematics, University of Bonn, Wegelerstr. 6, 53115Bonn, Germany [email protected]

2

Dipartimento di Matematica F. Casorati, Universita di Pavia, Via Ferrata 1,27100 Pavia, Italy [email protected]

1 Introduction

Mathematical modeling and numerical simulation for the study of fluids aretopics of great interest, both for our understanding of the phenomena relatedto fluids and for applications. In fact, we are still far from a complete under-standing of fluid phenomena. There is nowadays an increasing interplay of ap-proaches based on deterministic modeling and on stochastic modeling. AlreadyLeonardo da Vinci was fascinated, observed carefully and made drawings ofthe vortex formation in turbulent fluids. L. Euler formulated the equation ofmotion for the ideal case of inviscid fluids, H. Navier in 1822 and C.H. Stokes

in 1845 introduced the most studied of fluid models, namely the one describedby the NavierStokes equations. These equations constitute a challengingprototype for non linear parabolic differential equations. At the same timethey are the starting point for the building of discrete models used in numer-ical simulation of fluids. A deep mathematical analysis of the NavierStokesequations was initiated by J. Leray (1933). He, N. Kolmogorov and othersalso introduced and developed concepts used in what can be called a the-ory of turbulence. N. Wiener, according to his autobiographical account,developed a theory of Brownian motion as a first step for constructing aninfinite dimensional analysis, capable eventually to handle problems of tur-bulence. Developments in the study of fluids, in particular in their turbulentbehaviour, are connected with areas like non linear functional analysis, the

theory of dynamical systems, ergodic theory, the study of invariant measuresand stochastic analysis. We shall not discuss here the derivation of NavierStokes equations, but just mention some recent work on it. There are indeedattempts of deriving the NavierStokes equations from microdynamics, butthe theory is far from complete. The most ambitious starts from quantumdynamics, derives in a certain limit a suitable Hamiltonian reversible dynam-ics for particles, then from these by certain limit operations the (irreversible)Boltzmann equation and the NavierStokes equations; see, e.g., [ABGS00],


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2 S. Albeverio and B. Ferrario

[NY03], [LM01]. The program of understanding turbulence phenomena start-ing from the NavierStokes equations is also still widely open, see, e.g. [Fri95].

In these lectures we shall concentrate on certain mathematical resultsconcerning the case of deterministic Euler and stochastic NavierStokes equa-tions for incompressible fluids. For general references we refer to [Fri95],[Tem83], [CF88], [VF88], [VKF79], [Lio96], [MP94], [NPe01], [Che96b],[Che98], [Che04], [CK04a], [Con95], [Con94], [Con01a], [Con01b], [FMRT01],[LR02], [MB02] [Tem84] and [Bir60] and for a discussion of challengingopen problems see, e.g. [Fef06], [Can00], [Can04], [CF03], [Cho94], [Con95],[Con01a], [FMRT01], [Gal01], [ES00a], [Hey90], [Ros06] and [FMB03]. Weshall concentrate particularly on the study of invariant measures associatedwith the above equations for fluids. On the one hand, this follows an anal-

ogy with the statistical mechanical approach to classical particle systemsand ergodic theory, see, e.g. [Min00], [Rue69]. On the other hand, it followsKolmogorovs suggestion, see e.g. [ER85], of adding small stochastic pertur-bations (noise) in classical dynamical systems, so to construct invariantmeasures and then study what happens when removing the noise.

The content of our lecture is as follows: in Section 2 we shall study thedeterministic Euler equation and construct certain natural invariant measuresfor it. We also relate this analysis with the study of a certain Hamiltoniansystem describing vortices (vortex models). In Section 3 we shall study thestochastic NavierStokes equation with Gaussian space-time white noise andits invariant measure. We also provide brief comments and bibliographicalreferences concerning recent work in directions which are complementary tothose described here.

2 The Euler Equation, its Invariants and Associated

Invariant Measures

2.1 The Euler Equation

An Euler fluid is the particular case for vanishing viscosity of a fluid describedby the NavierStokes equation (for an incompressible, i.e. divergenceless andhomogeneous, i.e. constant density fluid). It is also called a perfect fluid.

Its equations of motion express the conservation of mass ( +div(u) = 0)which reduces for = constant in space and time to div(u) = 0, and Newtons

law. Here one takes into account that one has a continuum of fluid particlesmoving with coordinates x(t) in ddimensional space Rd associated with the(smooth) velocity field u: x(t) = u(t, x(t)) Rd (t being the time and x(t)taking values in a subset ofRd which contains the fluid). The correspondingacceleration is given by:


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Some Methods of Infinite Dimensional Analysis in Hydrodynamics 3

x =t

u(t, x(t)) +di=1

uxi

(t, x(t))ui(t, x(t))

=

tu(t, x(t)) + (u(t, x(t)) )u(t, x(t))

D

Dtu(t, x(t)).

(1)

DDt is the material derivative. The total force acting on the particles of thefluid can be decomposed in a stress force p (p being the pressure and the gradient in Rd) and an external force f. The Newton equation for fluidsis then

(2)D

Dtu = p + f,

where u = (u1, . . . , ud) = u(t, x), x Rd

, t 0. This together with thediv u = 0 condition constitutes Eulers equation of motion for a fluid. One hasto specify the boundary conditions, usually taken to be u n = 0 (u n beingthe component of u normal to ), which is natural when p is interpreted asthe pressure in the fluid, or u periodic in the space variables, if is identifiedwith a torus. Moreover one has to specify initial conditions at t = 0.

Remarks 2.1.

(i). Eulers equation is an equation for an ideal fluid, in particular it con-tains no viscosity term (which would be present if the stress tensor wouldbe more realistic. . . ). The condition div u = 0 leads by Liouvilles theo-rem (cf. e.g. [AK98]) to volume (and thus also mass) conservation.

(ii). The form (2) of a Newtons equation leads naturally to a geometrical pic-

ture of Eulers equation as a Hamiltonian system. This has been exploitedby P. and T. Ehrenfest and V. Arnold, see, e.g. [AK98], [MEF72], inparticular through work by Ebin and Marsden to prove existence results for smooth solutions. The geometrical picture is further exploited e.g.[Ebi84] in [AK98] [Gli03] [Rap02a, Rap03, Rap05].

(iii). Results on existence respectively uniqueness of solutions of Eulers equa-tion in various spaces and various degrees of smoothness of initial condi-tions and of the force f have been established. The results are particularlystrong for d = 2, see e.g. [Ebi84], [Kat67].

2.2 The Euler Equation in Terms of Vorticity

Let us first proceed heuristically assuming that there exist solutions of theEuler equation in the class of vector fields one is interested in. Set rot u ( is called vorticity function) and from now on assume f is a gradient field(which is natural due to the Newtons equation point of view), i.e., there existsa scalar function : R+ R so that f = . Then Eulers equation canbe written as

(3)D

Dt = ( )u


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with = rot u, and u n = 0 on (resp. u periodic if is a torus) and initialconditions. (3) is called vorticity equation. In fact, to see that (2) implies(3) it suffices to realize that 1

2(u u) = u rot u + (u )u, hence

(u )u =1

2(u u) u rot u

Moreover

rot(u ) = ( )u ( u) (u ) + u( )

= ( )u (u ),

where we used that u = 0 by the incompressibility condition and = 0.Taking then rot of (2) we get, observing that rot 1

2

(u u) = 0:

t = rot(u ) = ( )u (u ),

hence DDt = ( )u.For d = 2 and for a simply connected domain we can set, using div u = 0,

u = ,

with a scalar function, called stream function, = (2, 1). Moreover,for d = 2 the vorticity vector has only one non vanishing component. We write for this scalar quantity (for d = 2): = u. Since u = = ,we get = , so ( )u = ( ) = 0. In this case, (3) becomes

(4) DDt

= 0

This expresses the conservation of vorticity for d = 2. As an equation for, (4) reads:

(5)

t =

().

(where we used = , (u ) = (( ))).

Remark 2.1. For a corresponding treatment in the case of non simply con-nected domains see [AHK89].

2.3 The Conserved Quantities for the Euler Equation

Proposition 2.1. Let be either Rd or a bounded open subset of Rd withsmooth boundary (in which case one requires that the boundary conditionon the Euler velocity u, u n = 0, is satisfied) or the d-dimensional torus Td.Letu be a classical solution of the Euler equation (2) (see, e.g. [Kat67, Ebi84,BA94]). Then the following functionals of u are time independent (i.e. areconserved):


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(i). the energy

E(u) =1

2

u2dx

(for all d 1)(ii). for d = 2: the enstrophy

S(u) =1

2

(rot u)2dx

(iii). for d = 2: the gfunctionals of the vorticity

Sg(u) = g(rot u) dx,

for anyg C(R), such that the integral exists.

Proof. (a):

tE(u) =

uu

tdx =

u((u.)u p)dx,

where we used (2); integrating by parts and using the boundary conditionsand div u = 0, we obtain tE(u) = 0.(b), (c):

tSg(u) = g

(rot u)

trot u dx

=

g(rot u)(u )rot u dx =

(u )g(rot u) dx,

where we used Leibniz rule g(h) = g(h)h together with (4); integrat-ing by parts and using the boundary conditions and div u = 0, we obtaintSg(u) = 0.

Remarks 2.2.

(i). The integral Sg for g() = is called circulation.(ii). The above are essentially all known conserved quantities, called also in-

variants, see e.g. [Ser84a, Ser84b, Cip99].(iii). Sg can be expressed in terms of the vorticity respectively stream func-

tion as follows:

Sg(u) =

g()dx =

g()dx.

E can be expressed by the stream function as

E(u) =1

2

()2dx.


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2.4 Heuristic Invariant Measures Associatedwith the Euler Equation

The first observation is that the heuristic flat measure du on the space ofsolutions at time t of the Euler equation (2) does not depend on t, becauseof the incompressibility conditions div u = 0. This is a heuristic expression ofthe rigorous fact that the Euler flow t u(t, x) preserves volumes (see, e.g.[AK98]).

Let I(u) be any of the conserved quantities of Proposition 2.1. HeuristicallyI(du) = Z

1 exp(I(u))du, with Z

eI(u)du and being a spaceof solutions of (2), is an invariant (i.e. time independent) probability measureassociated with (2).1

Remark 2.2. From its heuristic expression in the case I(u) = E(u) we seethat E(du(0, )) can be realized rigorously as the Gaussian white noise mea-sure (i.e. the cylinder measure) associated withL2(,Rd) (with mean zero andunit covariance) (see, e.g. [HKPS93]). Whether this measure is indeed invari-ant in some sense under the Euler flow t : u(0, ) u(t, ) is a subtle openquestion, due to the bad support properties of E, t being understood as amap in a space of generalized functions (in the support of E). Henceforth weshall concentrate on the case d = 2, and for I(u) of the form b) in Proposition2.1 (which are less singular).

Let us consider for simplicity the case = T2 (a 2dimensional torus, identi-fiable with [0, 2] [0, 2]), with space periodic boundary conditions for thesolution u of (2) (the general case of a bounded is discussed in [AHK89];

a corresponding explicit discussion for = R2 seems to be lacking). Intro-duce the complex Hilbert space L2(). For L2() we have the Fourierexpansion

(6) (x) =kZ2

keikx

2,

with (k) 2(Z2) in the sense that k C for all k Z2 and 22(Z2) kZ2 |k|

2 < .

Remarks 2.3.

(i). Correspondingly we have u =

kukek, with ek(x)

eikx

2k

|k| , k Z2,

k = 0, k = (k2, k1). Then uk = i|k|k. The reality of u is equivalentwith uk = uk, k Z2, since k = k (with meaning complexconjugation).

1 This heuristics appears originally in work by L. Onsager and T.D. Lee and ex-panded in [Gal76], which provided the inspiration for the first rigorous work onthis line in [ARdFHK79], [BF80], [BF81], to which we refer for further references,see also [Gla81], [Gla77], [KM80], [RS91] for further physical discussions alongthese lines.


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(ii). The modification of by an additive constant does not change the relationu = . By this we see that we can assume, without losing generality,that

(x)dx = 0. This corresponds to taking 0 = 0.

Hence the above expansion (6) for can be written

(7) (x) =kZ20

keikx

2,

with k = k, Z20 Z2\{0}. Let us express the invariants E, Sof Proposition

2.1 in terms of the variables (k)kZ20 (denoting them again by the symbolsE, S):

(8) E() =12

kZ20

|k|2|k|2 ,

(9) S() =1

2

kZ20

|k|4|k|2.

Let us moreover remark that at least for the simple case where I = E orI = S a rigorous meaning can be immediately given to the measure I asGaussian product of measures, for any > 0. In fact in these cases

I() = kZ2+

|k|(I)|k|2

with (I) = 2 for I = E and (I) = 4 for I = S. We have set Z2+ =k Z20 : k1 > 0 or {k1 = 0, k2 > 0}

, because it is enough to consider half

of the indices k. Then

I(d) =kZ2+

k(dk),

where k is the Gaussian measure on C= R R given by

k(dk) = Z1k e

|k|(I)|k|2dk,

Zk =

C

e|k|(I)|k|2dk

(|k|2 = x2k + y2k, dk = dxk dyk for k = xk + iyk; xk, yk R). I

can be realized as a cylinder probability measure on CZ2+ . It is identifiable

with the standard centered Gaussian measure N(0, | |2H(I)), with H(I)

the complex Hilbert space H(I)

= (k)kZ2+ : I() <

, with scalar


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product (, )I kZ2+ |k|(I)kk. It is well known that to I one

can give a meaning as a additive probability measure on a larger spaceH(I) H(I), the embedding being an HilbertSchmidt operator (see, e.g.[Kuo75], [DPZ92]). The nontrivial problem is then to show that these rigorousmeasures are indeed invariant under the Euler flow in some rigorous sense,see Sections 2.82.10. In the next section we shall discuss the Euler equationin terms of the Fourier variables , in order to later on discuss the invarianceof above I under the Euler flow.

2.5 The Euler Equation in Terms of the Fourier Componentsof the Stream Function

Proposition 2.2. Let (t) be a classical solution of the Euler equation (5)on the 2-torus T2, with initial condition (0) = 0. Let (t) = (k(t))kZ20 bethe Fourier components of (t):

(t)(x) =kZ20

k(t)eikx

2, x T2,

with 0(x) =kZ20 k(0)

eikx

2.

Then

(10)d

dtk(t) = Bk((t)), k Z

20

with

Bk() 1

2

hZ20h=k

ch,khkh

ch,k (h k)(h k)

|k|2+

h k2

for any h = 0, h = k.

Viceversa, if the sequence (k) satisfies (10), then satisfies (5).

Proof. This is easily seen by computation. For details see, e.g. [ARdFHK79].

Proposition 2.3. Let Bk be as in Proposition 2.2. Then

(i). k Bk() = 0

(ii). Bk() = Bk(), for all k Z20

Proof. These properties are immediate consequences of the definition of Bk.

Remark 2.3. These equations hold for = (t), for all t 0.

Proposition 2.4. Let (t) be as in Proposition 2.2. Then for all t 0:


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(i).kZ20

|k|2Bk((t))k(t) = 0, if E((0)) <

(ii).kZ20

|k|4Bk((t))k(t) = 0, if S((0)) < .

Proof.

(i). Follows by computation from ddtE((t)) = 0, bearing in mind (8).

(ii). Follows by computation from ddtS((t)) = 0, bearing in mind (9).

Remarks 2.4.

(i). Independently of the time independence of E() andS(), properties (i),(ii) can also be seen to hold by exploiting the particular form of Bk.

(ii). A corresponding result holds for the Galerkin approximation to the Eulerequation, obtained by taking the equation system (4) only for k IN (INis the subset ofZ20 so that |k| N and IN = IN), i.e.

d

dtk(t) = B

N

k ((t)) for k Z20, |k| N

with BNk () =1

2

h

0 0), andthus Bk() has a meaning for almost all in the support of . Before doingso, let us however point out that the support of is bad in the sense thatthe energy E() is infinite for all in the support of . In fact we have thefollowing

Proposition 2.5. Let EN() 1

2

kZ200


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Proof. E (EN()) = 12

kZ2

0

0 (see[ARdFHK79, AHK89, CC95]). It follows then that

, (d) e:E:()(d)

e:E:()(d)

is a probability measure associated with both the energy and enstrophy. ,is heuristically invariant under the Euler flow (0) (t) (being constructedfrom invariant functions and the heuristic invariant flat measure). In the next

section we shall discuss more closely the invariance of = 0, (and ,).

2.7 Relation of the Enstrophy Measure with the Euler

Equation

Proposition 2.6. For any k Z20, Bk is the L2()limit for N of its

Galerkin approximations BNk ().

Proof. Set for simplicity E E.For = (k)kZ2

0 supp , the k are independent and

kdistributed

random variables, i.e. Gaussian centered with covariance E(kk) =

2|k|4

kk .

We have

E

|BNk |2

=1

(2)2

Nh,h

ch,kch,kE (hkhhkh)

where the sum is over the indices h, h such that 0 < |h| N, 0 < |k h| N, 0 < |h| N, 0 < |k h| N. But it is well known that all the evenmoments of a Gaussian process can be computed in terms of its covariance.


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Hence we get

E

|BNk |2

= 8(2)2

h h=k|h|Nc2h,k

1|h|4|kh|4

8(2)2

hZ20h=k

c2h,k1

|h|4|kh|4 <

(see [ARdFHK79] for details). Hence BNk L2(). Similarly one shows that

(BNk )NN is a Cauchy sequence in L2(), which proves the proposition.

Remark 2.6. One can also show that Bk Lp(), for any 1 p < . In

fact Cipriano [Cip99] has shown thatsupkE |Bk|

p1/(2p)

cp, < . From

this one sees thatk

|k|2bE |Bk|2p < , for all b < 1.

2.8 The Infinitesimal Invariance of. Relationwith the Hopf Approach

Hopf in [Hop52] introduced a general (heuristic) approach to the study of theequations of hydrodynamics, lifting the evolution equation from individualsolutions to statistical solutions. A rigorous implementation of this approachcan be obtained as follows (see [ARdFHK79, AHK89]).

Let F Cb be the space of finitely based functions (cylinder functions)of = (k) (smooth and bounded with all derivatives, on the base). ThusF F Cb iffF C

b (C

n), for some n N, so that F() = F(k1 , . . . , kn),

for some ki Z2

+, i = 1, . . . , n. The following proposition can easily be proved.Proposition 2.7. If (t) satisfies the Euler equation ddtk(t) = Bk((t)),k Z20, and F F C

b , then

d

dtF((t)) = BF((t)),

with B k Bk

k

(defined on F Cb ).

Remarks 2.5.

(i). One calls B the Liouville operator in L2() (associated with the Eulerequation). This describes the dynamics when looked through its action on

cylindric smooth functions F. Proposition 2.6 assures that B is well de-fined on the setF Cb . One calls the equation in Proposition 2.7 Liouvilleequation (associated with the Euler equation).

(ii). For F F Cb , BF =k Bk

Fk

and the sum is finite.

Definition 2.1 (Infinitesimal invariance). A probability measure on thespace of sequences = (k)kZ2+ is called infinitesimal invariant with respectto B (or with respect to the Euler flow) if

BFd = 0 for all F F Cb .


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Equivalent to this is B1 = 0, where 1 is the function identically one in L2()and is the adjoint in L2(), as seen from

BFd = 1, BFL2() = B1, FL2().

Proposition 2.8.

(i). B B, i.e. (B , F C b ) is skew-symmetric in L2().

(ii). is infinitesimal invariant with respect to B.

Proof.

(i). We have B = k Bkk

on F Cb . But Bkk

k

Bk on

F Cb withk

k + |k|4k. We know from Proposition 2.3,ii) that Bk = Bk; then B

k

k

Bk+k |k|

4kBk. Bearingin mind Proposition 2.4, ii) and Proposition 2.3, i), we prove (i).

(ii). From the definition, we have infinitesimal invariance iff B1 = 0. By (i)B1 = B1 = 0 (the latter step is an easy consequence of the specific formof B).

Remarks 2.6.

(i). N

0


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was: how many different selfadjoint extensions of L do exist? Let us firstremark that any such extension L, L being selfadjoint, by Stones theorem

generates a strongly continuous unitary group (eitL)tR in L2(). One callsF UtF = e

itLF, F L2() a generalized Euler flow (associated withthe Euler equation in the L2()-sense: this is a form of the HopfKoopmanvon Neumann approach to the study of evolution equations and ergodic ques-tions. See, e.g, [GGM80]).

Remark 2.7. If L is essentially selfadjoint on F Cb in L2() (i.e. its clo-

sure L is already selfadjoint), then there is only one selfadjoint extensionof L, namely L itself. One can show, see [AF04b, AF03] that the generated

Euler flowF eitLF comes from ameasurable point flow t, in the sensethat there exists a measurable flow t : (0) t((0)) (t), which is

preserving and so that (eitLF)() = F(1t ()), for a.e. .

The problem of essential selfadjointness ofL seems to be still open. Partialresults have however been obtained:

(a). In [AF02b] it is shown that L is dominated by a related selfadjoint oper-ator H of the Schrodinger type, in the sense that there exist constantsb > 1, Cb > 0 so that

2|F,LFL2()| Cb F,HFL2(),

with H kZ2+ |k|

2b( k ) k

+ V(), V kZ2+ |k|

2b|Bk|2.(b). Essential selfadjointness ofL is a subtle question, certain finite dimen-

sional analogues ofL are indeed not essential selfadjoint on correspond-

ing domains. E.g. i(x2 ddx + ddxx2) on C0 (R) is symmetric, real (withrespect to Jf(x) = f(x)), but not essential selfadjoint in L2(R, dx),having defect indices (0, 1) although it is dominated by the selfadjoint

Schrodinger operator d2

dx2 + x4, see [RS75, AF02b]. Similar results hold

for the corresponding operators in L2(R, e12x2

2

dx). Related problems

of essential selfadjointness arise for certain operators of quantum fieldtheory, see [AFY04].

(c). The finite-dimensional operators LN = iBN (i.e. the Galerkin approxi-mations of Section 2.5) are essentially self-adjoint in L2(N ), when de-

fined on Cb , and have N (d) =

0


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2.10 An Euler Flow in a Sobolev Space of Negative Index

Let us consider for any p R the Sobolev space

Hp2

= (k)kZ2+ :

kZ2+

|k|2p|k|2 <

;

for p = 2 this is a Hilbert space with scalar product ( , )H22 = S() (Sbeing the enstrophy introduced in Sections 2.3, 2.4). For any > 0 the triple(H22, H

12 , ) constitutes an abstract Wiener space (in L. Gross sense); in

fact the embedding H22 H12 is HilbertSchmidt, for the proof of this see

[AC90], [HKPS93]. One has (H12) = 0 (another expression of the fact that

the energy E is a.s. infinite) and (H22) = 0, but (H

1

2 ) = 1. One cantake for support of the enstrophy measure the space

>0 H

12 .

By the way, the support of the energy measure E , defined in Section 2.4,is>0 H

2 (for d = 2).

The following theorem was established in [AC90]:

Theorem 2.1. There exists a probability space (, A, P) and a pointwiseflow (s, ) (k(s, ))kZ2+, s R, so that (, ) C(R; H

12 )

for any > 32 and k(t, ) = k(0, ) +t0

Bk ((s, )) ds for Pa.e. ,k Z2+.Moreover is invariant under , in the sense that

F((t, ))P(d) =

E F t, F F Cb

Proof. It uses essentially the estimate Bk L2(), for details see [AC90].

Remarks 2.7.

(i). One shows that : E : L2() is an invariant function under .(ii). According to the Remark 2.6, the Hb2-norm of (Bk)kZ2+ is inL

2() for

any b < 1.(iii). Theorem 2.1 gives the existence of solutions of the Euler equation

almost everywhere in a weak probabilistic sense (implying, in particular,a change of probabilistic space).

2.11 Some Remarks on the Vortex Model and its Relationswith the Euler Equation

Consider the vorticity = rot u concentrated in a finite number n of distinctpoints:

(t, x) =

nj=1

jxj(t)(x)


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j is the intensity of the j-th vortex, x T2 the 2D-torus. For an ideal fluid,the time evolution of these point vortices in the vortex model is given by

(11) jd

dtxj(t) =

xj

n1

h, lh=l

hl G(xh(t) xl(t)), j = 1, . . . , n

G is the Greens function of on the torus.

Remark 2.8. Equation (11) has the structure of an Hamiltonian system, withHamiltonian function

H(x) =1

2 n

1j, lj=l

jlG(xj xl).

It has been studied by itself, see, e.g. [DP82], [Caf89]. Particular attentionhas been given to the case where the j take only 2 values, say , > 0.In this case there is a close relation (due to the special logarithmic singularityof G) with the Coulomb gas in 2 dimensions, which has been studied in con-nection with statistical mechanics [AHK73], [FS76], [CLMP92] (and plasmaphysics [AHKM85]) . Existence of Gibbs states Z1eH(x)dx1 . . . d xn withany number of vortices (resp. particles) for 0 < < 42 has been shown, seee.g. [FR83], [Lio98].

We are particularly interested here in the relation of the vortex model with the

Euler equation. This has been studied by Marchioro and Pulvirenti [MP94]. Asfar as we are concerned with invariant measures, this comes about through theconsideration of invariance of the Lebesgue measure for the Euler equation, seethe first Remark 2.1, as well as for (11), because of its Hamiltonian structure.Let us construct a concrete invariant measure considering any number ofvortices. For this we define the compound Poisson measure on the space of configurations ofT2. Let for any n N:

(n)

=

nl=1

lxl : l R0, xl T2, xl = xk for l = k

(where R0 R\{0}). (n) is looked upon as a space ofnpoint configurations

in T2. Let (n) = (i, xi) (R0 T2)n : i = 1, , n , xi = xk for i = k.There is a bijection J(n) of (n) mod S(n) into (n), where S(n) is the per-mutation group over {1, , n}. Let be a finite positive measure on B(R0)such that

R0

(1 2)(d) < . Set =R0

(d). Consider the measure

n ( )n on B((n) mod S(n)), being the Lebesgue measure onR2. Let n be the image ofn on (n). Set (0) = {}, 0 = {}. The spaceof configurations is by definition = n=0

(n), defined as disjoint union oftopological spaces with the corresponding Borel -algebra (see, e.g. [AKR98b],


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[AKR98a]). The compound Poisson measure on Borelians of is definedby = e(2)

2 n=0

nn! . In [AF03] it is shown that is invariant with

respect to a unique vortex flow, well defined for a.e. initial data. This isbased on results by Durr and Pulvirenti [DP82]. In fact in each component(n) of there is a npreserving flow, because the Lebesgue measure is in-variant and the vortex intensities are constant during the motion. Hence thereis a unique strongly continuous positivity preserving unitary group on L2().Under the assumption

R0

2(d) < one has Bk Lp(), 1 p < ,Bkk

= 0 a.e. , Bk() = Bk() a.e. , k Z20. Thus the Liouvilleoperator L = i

k Bk

k

in L2() is Markov unique in the sense that thereexists only one selfadjoint extension which generates a positivity preservingstrongly continuous unitary group Ut in L

2(), see [GGM80], [AF03]. is

invariant unter Ut.

Remarks 2.8.

(i). and are singular, see [AF03], and () = 0.(ii). See also [AF02a], [AF02b] for other measures of the Poisson type,

which are heuristically invariant for the 2D Euler flow, also discussedin [BF80], [BF81], [AHKM85], [CDG85].

(iii). Stochastic perturbations of the vortex model are mentioned in [AF02a].It would be interesting to study them in more details. Let us mentionin passing that stochastic perturbations of 2dimensional Euler equa-tions have been studied in particular in [BF99], [Bes99], [MV00], [BP01],[Kim02], and in [CC99] (using nonstandard analysis, see [AHKFL86]).

Stochastic models for the study of formation of vortices have been devel-oped [FG04], [FG05]. For related work see also, e.g., [BLS05].

3 Stochastic NavierStokes Equation

The study of the deterministic NavierStokes equation is well known to presentchallenging problems, especially in the case of 3 dimensions. E.g. the globalexistence and uniqueness of classical solutions of this equations with smoothinitial data is a famous open problem, see, e.g. [Sin05b, Sin05a], [Fef06],[Con01a], [CF03], [Soh01], [Zgl03]. In 2 space dimensions the situation is betterunderstood, see, e.g. [BA94], [Can04], [Tem83], [Tem84], [KT01]. However theproblem of the existence of invariant measures (different from those concen-trated on stationary solutions [VF88]) is widely open. In 2 space dimensionsthere are results on the construction of invariant measures for NavierStokesequations stochastically perturbed by Gaussian white noise, which we are go-ing to describe in detail. We are interested in the stochastic NavierStokesequation with a space-time white noise. We consider, as in Section 2, thespatial domain to be the torus T2 = [0, 2]2 (hence periodic boundary con-ditions are assumed); we point out that for other finite spatial domains, the


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boundary conditions for the Euler equation and for the NavierStokes equa-tion are different (see e.g. [Bir60]). In Section 3.1 we present the stochasticNavierStokes equation. In Section 3.2 we introduce an infinitesimal invari-ant measure associated to this stochastic equation, as done in [AC90]; thisis a Gaussian measure , with covariance given in terms of the enstrophy(and of the viscosity parameter ). Existence of a solution of this stochasticNavierStokes equation has been proved in two different ways: [AC90] con-siders a weak solution and [DPD02] a strong solution (weak and strong areto be understood in the probabilistic sense). We present the second approachin Section 3.3. Uniqueness of these strong solutions is given in Section 3.4,following [AF04b]. In the Appendix a technical lemma is presented.

3.1 The NavierStokes Equation with Space-Time White Noise

We consider an homogeneous incompressible viscous flow in T2 with periodicboundary conditions. Displaying the external force on the right hand side ofthe equation, we have

(1)

t u u + [u ]u + p = f

u = 0

u|t=0 = u0

The unknowns are u = u(t, x), p = p(t, x). The definition domains of thevariables are t 0, x T2. > 0 is the viscosity coefficient.

The mathematical setting is as in Section 2. We expand in Fourier seriesany periodic divergence-free vector u (see Section 2.4):

(2) u(x) =kZ20

ukek(x), uk C, uk = uk

Note that {ek}kZ20 is a complete orthonormal system of the eigenfunctions(with corresponding eigenvalues |k|2) of the operator in [Ldiv2 (T

2)]2 ={u [L2(T2)]2 : u = 0,

T2

u(x) dx = 0, with the normal component of ubeing periodic on T2}. With respect to the Fourier components, the energyis given by E = 12

kZ20 |uk|

2 and the enstrophy by S = 12kZ20 |k|

2|uk|2.Each ek is a periodic divergence-free C-vector function. The convergence

of the series (2) depends on the regularity of the vector function u, and can

be used to define Sobolev spaces as in the following definition.Let U be the space of zero mean value periodic divergence-free vectordistributions. Any element u U is uniquely defined by the sequence of thecoefficients {uk}kZ2+ ; indeed, by duality, uk = u, ek. In the following weoften identify the space of vectors u and the space of sequences {uk}, foru =

k ukek.

Following [BL76], we define the periodic divergence-free vector Sobolevspaces (s R, 1 p )


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Hsp = { u =kZ20

ukek U :k

uk|k|sek() Lp(T2) }

and the periodic divergence-free Besov spaces as real interpolation spaces

Bspq = (Hs0p , H

s1p ),q, s R, 1 p, q

s = (1 )s0 + s1, 0 < < 1

In particular, Bs2 2 = Hs2, the Hilbert spaces already defined in Section 2.10.

(Notice, however, that we dealt there with the Euler equation in the unknownstream function , whereas here we deal with the stochastic NavierStokesequation in the unknown velocity u.) Moreover, U = sR,1pHsp with theinductive topology.

Remark 3.1. For u =kZ20 ukek, if we define

ju =

2j1


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on B are given in Besov spaces (see, e.g., [Che96b, Che98]). The (optimal)regularity of B is the key point to solve the NavierStokes equation, both inthe deterministic and in the stochastic case.

We shall very often write shortly B(u) for the quadratic term B(u, u).The stochastic NavierStokes equation we are interested in, has the fol-

lowing abstract Ito form

(4)

d u(t) + [Au(t) + B(u(t))] dt = dw(t), t > 0

u(0) = vx.

{w(t)}t0 is a Wiener process, defined on a complete probability space(, F,P) with filtration {Ft}t0, which is cylindric in the space of finite energy

H0

2, i.e.w(t) =

kZ20

wk(t)ek

where {wk}kZ2+ is a sequence of standard independent complex valued Wienerprocesses and wk = wk for k Z2+ (for k Z

2+: wk(t) = ak(t) + ibk(t) and

{ak}, {bk} i.i.d. with Eakaj = Ebkbj = kj ). This is a process with continuouspaths taking values in H2 for any < 1 (see, e.g., [DPZ92]). In other terms,dw(t) is a Gaussian space-time white noise.

We shall denote by E the expectation with respect to the measure P.

Remark 3.2. For noise which is more regular in space(coloured Gaussiannoise) the techniques to analyse equation (4) are very different from ours.

E.g. solutions with finite energy have been discussed with global existencein space dimensions 2 and 3, uniqueness being known only for d=2 (as for the deterministic case). Further typical results include existence anduniqueness of invariant measures and ergodicity (mostly for many Fouriermodes but some also for few Fourier modes) See, e.g., [Cru89a], [Cru89b],[BDPD04], [BG96], [BL04], [Car03], [Cha96], [CK04b], [CE06], [Cut03],[DPD03], [Fer99], [Fer01], [Fer03], [Fer06], [FG95], [FG98], [Fla], [Fla94],[Fla02], [Fla03], [FGGT05], [FR01], [LJS97], [Mel00], [MS02], [Pes85],[QY98], [Rob91], [Rob03], [EMS01], [ES00b], [BT73], [Cho78], [VF88],[BCF92], [FY92], [Kot95], [CG94], [FM95], [Fer97a], [ES00a], [KS01],[BKL01], [MR04], [BDS04], [MR05], [HM06].

For d = 3 there is an existence result on invariant measures and ergodicityin [DPD03] (for further results see the lectures by F. Flandoli).

Equation (4) is equivalent to the following equations for the Fourier com-ponents

d uk(t) + [|k|2uk(t) + Bk(u(t))] dt = dwk(t), t > 0

where Bk(u) =i2

hZ20,h=k ch,kuhukh , with ch,k =

(hk)|k|2|h||k|

(hk)(hk)|h||kh||k| .


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3.2 The Gaussian Invariant Measure Given by the Enstrophy(and the Viscosity Parameter)

We shall consider the centered Gaussian measure on the space of complexvalued sequences {uk}kZ2+ , heuristically defined as the infinite product ofcentered Gaussian measures k on C

= R R

(5) (du) =kZ2+

k(duk)

where

k(duk) =|k|2

e|k|

2|uk|2duk

It is identifiable with the standard centered Gaussian measure N(0, ||H12).Let us denote by E the expectation with respect to this measure: EF U F(u) (du).

In particular

E [ukuj ] =

1

|k|2 if k = j

0 if k = j

Heuristically we can write (5) as Z1 exp(S(u))du, S being the enstro-phy associated to the velocity field u: S(u) =

kZ2+ |k|

2|uk|2. Thus, cor-

responds to the Gaussian measure S of Section 2.4.For later use we shall need more information on the support of the measure

. We have, for any integer n

E

u2n

H2n

= E

T2

|k uk|k|

ek(x)|2n

d x

=

T2

E

|k uk|k|

ek(x)|2n

d x

= cn

k |k|2E |uk|

2n

(6)

for some constant cn > 0. In these calculations we have used that, for anyk C:

(7) E |

kukk|2

n

=

(2n)!

2n n! k|k|2 E (|uk|

2)

n

and the fact that |ek(x)| =12

for any x T2.Since E (|uk|

2) = 1|k|2 , the above calculation implies that there exists apositive constant cn such that

E

u2

H2n

Eu

2n

H2n

1n

1

cn

kZ20

1

|k|2+2


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The latter series converges as soon as > 0. Hence (H2n ) = 1 for any > 0and integer n. Since we are in a bounded domain, we have the embeddingH2(n+1) H

q H

2n for 2n < q < 2(n + 1). Therefore

(Hq ) = 1 > 0, 1 q <

and by interpolation (see the details in [AF04b], based on [BL76]) we getBesov spaces of full measure :

Proposition 3.1. For any > 0

(Bpq ) = 1 > 0, 1 p q < .

Remark 3.3. It was already known from [ARdFHK79] that the space H02 offinite energy velocity vectors has not full measure with respect to ; in factone has even (H02) = 0.

With calculations similar to (6) we can obtain that, P-a.s., the paths w ofthe Wiener process w(t) belong to H1p for > 0 and 1 p < .

We collect the main properties of the Fourier components Bk. We look foruniform estimates for the sequence of finite approximations BNk . The extensionto the infinite dimensional dynamics has to be checked carefully. In fact, wecannot deal directly with B(u) for any u in the support of the measure ,

since such u are too irregular for the quadratic term

B(u) to be defined. Butall the components Bk are well defined.

Proposition 3.2. For any k Z20

kBk = 0(8)

Bk = Bk(9)

Bk Lp() for any 1 p < (10)

Indeed each component Bk is the Lp()-limit (as N ) of the Galerkin

approximations

B

N

k (u) =

h0


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Remarks 3.1.

(i). Let B(u) =k Bk(u)ek. B, resp Bk are the corresponding quantities

B, resp Bk discussed in Section 2.2 written for the variables uk in-stead of the variables k

(ii). The coefficients chk are naturally related with the coefficients chk ofSection 2.4.

Notice that the dynamics (4) is a combination of the Euler dynam-ics and of the stochastic Stokes dynamics. Therefore, if the stochastic lineardynamics dz(t) + Az(t)dt = dw(t) has as a unique invariant measure ex-actly , then the whole dynamics (4) has as infinitesimal invariant mea-sure. We have that the stochastic Stokes equation corresponds to a system

of uncoupled linear equations dzk(t) + |k|2

zk(t)dt = dwk(t). Each compo-nent has invariant measure corresponding to the law of the stationary processzk(t) =

t e

(ts)|k|2dwk(s); this is a stationary centered Gaussian processwhose covariance is 1/(|k|2). Hence the infinite product (k Z2+) of theseGaussian measures is an invariant measure for the stochastic Stokes equation.We point out that the proper choice of the noise and of the viscosity coefficientgives this expression for the invariant measure (for each > 0 there exists aunique invariant measure ).

We can see this infinitesimal invariance of also by introducing theKolmogorov operator K associated to the stochastic equation (4) (see, e.g.,[AF02b])

K =kZ20

uk

uk +

k2

uk Bk

uk

kZ20

uk

uk kZ20

Bk

uk

where ( uk ) is the dual of the operator uk in the space L

2().As done above for the Liouville operator L, we have that the Kolmogorov

operator K is a linear operator in L2(), well-defined on D(K) = F Cb . In-

deed, K = Q+L with Q =k(uk

) uk , the positive symmetric OrnsteinUlenbeck operator defined on F Cb . The operator K, defined on FC

b , is

dissipative and closable. The measure is infinitesimal invariant also for Q;hence for the sum Q + L. (see [AF02b]).

Remark 3.4. For other results on Kolmogorovs equation for stochasticNavierStokes, see e.g. [FG98, BDPD04, Sta07]. For particular results of

uniqueness of closed extensions of K see [AF02b] and [ABF07]. For problemof uniqueness of other, somewhat related, generators in infinite dimensionalspaces, see e.g. [AKR92, AKR95, AR95, BDPD04, DPD99, DPD07, DPT00,DV87, KR07, LR98, Ebe99, Sta99, Sta03, DP04, RS06].

We now provide an estimate of the quadratic term B with respect to themeasure . This will be useful later on.


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Proposition 3.3. For any viscosity > 0 and any 1 < , we have

(11)

U

B(u)H12d(u) < > 0.

Proof. See [AF04b]. We only point out that [AC90] showed BH2 L2()

for any > 32

. In [AF04b] that result was improved by showing thatE |Bk|

2 c22 log |k| for any |k| 2.

Remark 3.5. According to the latter result, the nonlinear term B(u) is defined for-a.e. u. Since (H02) = 0 but (H

q ) = 1 ( > 0, 1 < q < ),

the elements u for which the nonlinear term B(u) exists are (non regular)distributions. In [DPD02] it is explained that B(u) L( ; H

12

) for1 < , > 0, as follows. Denote by : u u : the renormalized square(Wick square), defined as : u u := u u E (u u) (see, e.g., [Sim74]).Consider the finite dimensional approximations uN :=

|k|Nukek; one has

that supNE : uN uN :

H2< . Notice that (: uN uN :) =

(uNuNE (uNuN)) = (uNuN). Hence B(uN)P[(: uNuN :)] and

in the limitB(u) = P[(: uu :)] is well defined, i.e. B(u) L( ; H12 ).

Moreover, in [Deb02] there is a useful proposition providing this result inthe Besov spaces, i.e. B(u) L( ; B1p q ) for any > 0 and ,p,q 1.

3.3 Existence of Strong Solutions

The results in this section are from [DPD02, Deb02].Set

z(t) =

t

e(ts)Adw(s), t R

which is a stationary solution of the stochastic Stokes equation

(12) dz(t) + Az (t)dt = dw(t)

This is a linear stochastic equation. We know from Section 3.2 that the invari-ant law L(z(t)) is exactly the Gaussian measure given by the enstrophy andthe viscosity parameter. We have z C(R; Bp q ) P-a.s. for any > 0, p , q 1.

This is shown by means of Kolmogorovs criterium, as in [DPZ92].Then we define v = u z. Taking the difference between (4) and (12) the

additive noise disappears; hence v satisfies the random equation

dv(t)

dt+ Av(t) + B(v(t) + z(t)) = 0

Using the bilinearity of B, we can write this equation in the integral form


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(13) v(t) = etA(xz(0))t0

e(ts)A [B(v)+ B(v, z)+B(z, v)+ B(z)] ds

The term B(z) is defined according to (11). Indeed, B(z) L(0, T; H12 )P-a.s., because

E

T0

B(z(t))H12dt = T

U

B(u)H12d(u)

Since the initial data are not smooth, it is not expected that the equation forv has a solution with paths in C([0, T]; Hs2) for s 0. On the other hand, thisis a parabolic equation for which it will be proved in Proposition 3.4 that thesolution v exists in C([0, T]; Bpq ) L

(0, T; Bpq) for some < 0 and > 0;

this is enough to define all the terms in (13).To have short notations, it is convenient to introduce the space

E = C([0, T]; Bpq ) L(0, T; Bpq)

for , > 0 and p,q, 1.This is a Banach space with norm vE = vC([0,T];Bpq ) + vL(0,T;Bpq)We have a local existence result.

Proposition 3.4. Let the real parameters ,p,q,,,, satisfy 2 p,q 0, 1 , < and

0 < < 0.


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In the above proof and in the following, we denote by c different constants.When needed, we shall specify them by a subindex.

Lemma 3.2. If f L(0, T; H12 ), thent0

e(ts)Af(s)ds E and

t0

e(ts)Af(s)dsE c2fL(0,T;H12 )

for some constantc2.

Proof. We use

e(ts)Af(s)H+1 2

p

2

c

(t s)1+2+

2

1p

f(s)

H12

and the embedding (see [BL76] Th. 6.5.1)

H+1 2

p

2 Bpq for p, q 2

We deduce

t0

e(ts)Af(s)dsBpq t0

c

(t s)1+2 +

2 1p

f(s)H12 ds

By Youngs inequality, the convolution integral is in L(0, T) if

1

+

2

+

2

1

+

1

p


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Theorem 3.1. Let the real parameters ,p,q,, satisfy 2 p, q < ,1 < and

0 < < 0, > 0, 1 <


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Let ux be any other process defined on the same probability space(, F, {Ft},P), with the same properties given above for u

x and solving equa-tion (4) with the same {Ft}-Wiener process as for u

x. Define the differenceUx = uxux; then Ux C([0, T]; Bpq ). From now on we drop the dependenceon x and work pathwise (P-a.s.). U satisfies the equation

(23)

ddtU(t) + AU(t) = B(u(t)) + B(u(t)), t > 0

U(0) = 0

Bearing in mind the regularizing effect of the Stokes operator A, somethingmore can be proven. More precisely, (22) grants that the right hand sideof the first equation in (23) belongs to the space L(0, T; H12 ) for any

1 < , > 0. By Proposition 3.5 in the Appendix one has that

(24) U L(0, T; H12 ) C([0, T]; B1 22 )

This holds for any > 0, 1 < . Hence we have proven that any solutionU to equation (23) must belong to the functional space := 10, ,

where , := L(0, T; H12 ) C([0, T]; B

1 2

2 ). Let us point out that for

2 p q we have B1 2

2 B 2

+ 2

pp Bp B

pq and for we

have Bpq Bpq ; therefore B

1 22 B

pq . Thus the regularity specified in

(24) is stronger than the regularity U C([0, T]; Bpq ) given by the definitionof U itself.

Remark 3.7. The regularizing effect of the Stokes operator is not enough toobtain more regularity in the stochastic equation (4), because of the presenceof the cylindric noise dw. As soon as the noise disappears in (23), the solutionis more regular. This is enough to get uniqueness.

Bearing in mind the bilinearity of the operator B, the equation for U can bewritten in the following form

(25)

ddtU(t) + AU(t) + B(u(t), U(t)) + B(U(t), u(t)) = 0, t > 0

U(0) = 0

The function U 0 is a solution to (25). We are going to prove that this isthe only solution of (25) in the class .

To prove this, we first show that, given u, u C([0, T]; Bpq ), under theassumptions below there exists a unique solution U to the problem (25) be-longing to a class less regular than . This is proved in Theorem 3.2 below.From this, uniqueness in the smaller class immediately follows. This con-cludes our proof that the unique solution for (23) is U 0. What remains tobe proven is therefore the following


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Theorem 3.2. Let real numbers , a be given as well as 2 p, q < , 1 b < satisfying the following conditions

0 < < a

2+

1

p.

Moreover

(29)

t0

e(t)AB(u(), U()) dL([0,T];Bpq )

c5

T12 1b+a2 1p u

C([0,T];B

pq )U

Lb

(0,T;Ba

pq)

if1

2+

a

2>

1

b+

1

p.

Hence, if U D and the conditions on the parameters hold, thent0

e(t)AB(u(), U()) d D.

We perform the same computations for B(U, u).


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The mapping

U

t0

e(t)A [B(u(), U()) + B(U(), u())]d

is then a contraction in DT with T T and such that

(30) T < min

(c4 NT)1/( 122 1p ) , (c5 NT)

1/( 12 1b+a2 1p )

where NT = uC([0,T];Bpq ) + uC([0,T];Bpq ). Hence on the interval [0, T]

there exists a unique solution U with the regularity specified in D. One hasU(t) = 0 for 0 t < T. Notice that the amplitude of the time interval for

local existence depends only on the C([0, T]; Bpq )-norms ofu and u; thereforewe can continue in such a way as to cover the time interval [0, T] with a finitenumber of intervals of amplitude 3

4T.

Since this holds for any finite T, the proof is achieved.

Choose now the parameters of Proposition 3.2 to be p = q = b = 3, = 1

6, a = 1

2. In this way, bearing in mind Proposition 3.1, we have fixed

a set Bpq of initial data such that (Bpq ) = 1 (but many other choices

are possible); moreover the assumptions of Theorem 3.2 are satisfied. Choosealso the parameters = 3, = 16 for the regularity of (24). Finally, by anembedding theorem (see [BL76] Theorem 6.5.1) we have

B1 22

Bpq

H12 Bapq

Hence D . And the uniqueness in D implies the uniqueness in .We have therefore proven the following

Theorem 3.3. Pathwise uniqueness of the solutions to the stochastic NavierStokes equation with space-time Gaussian white noise (4), for which isan invariant measure, holds in the following precise sense: there exists a setS U with (S) = 1 such that for -a.e. x S the C([0, T]; S)-valuedpaths of any two solutions of (4), defined on the same probability space withthe same Wiener process and having invariant measure , coincide P-a.s.

A further result concerns the ergodicity of the stochastic NavierStokesflow given by the above equations. It is contained in [Deb02]:

Theorem 3.4. The solution process ux of Theorem 3.2 is exponentiallyL2() ergodic in the sense that

(ux(t))

d


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as t , exponentially quickly, for all measurable L2().I.e. > 0 such that as t :

(31) (ux(t))) L2() et L2()

The proof uses the fact that is Gaussian and the classical Dirichlet operatorL associated with (cf. [Alb00]) has a spectral gap (of some length ).

Remark 3.8. Ergodicity for stochastic equations for fluids has been an inten-sively discussed topic in recent years. One of the main points has been to proveergodicity also in the presence of noise concentrated only in a few Fouriermodes, see [HM06].

Remark 3.9. Theproblemofextendingtheaboveconsiderationsto3dimensionshaving a flow for Euler deterministic resp. stochastic NavierStokes equationswith an explicit (physically relevant) invariant measure is open. For the 3dimensional space or the 3 dimensional torus the only explicit positive invariant functional for the deterministic Euler equation is the energy. But the corre-sponding Gaussian measure seems to have too singular support to be related insome way to the equation itself.

3.5 Some Additional Remarks and Complements

In the whole presentation we have restricted our attention to incompressiblefluids. We shall continue to do so, except for the remark 8 below.

1. Stochastic Euler equations have been studied e.g. in [Bes99], [BF99],[CFM07], [BP01], [CC99], [Kim02], [MV00], both with additive and mul-tiplicative noise of the Gaussian type. Deterministic Euler equations arestudied e.g. in [Lio98, Bre99]. Dissipative Euler equations are studied e.g.[BF00]. See also [CC06].

2. Deterministic Burgers equations with initial random conditions have beenstudied e.g. [AMS94], [LOR94a, LOR94b] with interesting connectionswith the study of large scale structures in astrophysics [ASZ82], [SZ89],see also e.g. [Sin91]. A useful tool in the study of the Burgers equationis the Cole-Hopf transformation to a heat equation see [ABHK85] for anearly proposal for its use. The stochastic Burgers equation in one resp.higher dimensions has been studied with Gaussian white noise e.g. in

[DPDT05], [DPDT94], with spatial periodic and white Gaussian in time[LNP00], [BCJL94], [LW01], [TRW03], [E01], see also [EKMS00]. An ana-lytic approach (uniqueness, infinitesimal invariance) to stochastic Burgersequation has been developed in [RS06], [BR01], [DPDT94] Propagation ofchaos results for Burgers equation has been obtained e.g., [Szn87], [Osa87].Burgers equation with other types of noise (Levy noise) has been discussed[TW06], [TW03], [TZ03].


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3. Stochastic NavierStokes equations with other types of Gaussian noisehave been studied, see the references in Remark 3.2. Recently also additiveand multiplicative noises of Levy type have been considered, see [ABW].Ergodicity of finite dimensional approximations of the stochastic, NavierStokes equation is discussed by [Rom04]. For other studies on ergodicityof 2D NavierStokes equations with random forcing, see e.g., [Fer97a],[FM95], [HM06], [KS01], [Kuk06], [Mat99].

4. The study of the deterministic limit of stochastic NavierStokes (or Euler)equations when there is a small parameter > 0 in front of the noise and is sent to zero is largely open, see, however [ABHK85], [Cru89b], [Hab91]for some initial considerations.

5. The study of the behaviour of solution of stochastic (and deterministic)

NavierStokes equations as the viscosity coefficient tends to zero (i.e.the passage NavierStokes Euler) has been performed in several pub-lications. It was suggested in [ABHK85] and developed e.g. in [TRW03],some remarks are also in [Cru89b] [Hab91]. In [Kuk04] on a 2D-torus andfor an additive noise of the Gaussian white type in time and smooth inspace it is shown that for a subsequence (i) of viscosity parameters thedouble limit limj0 limT uj(T+ t) yields a stationary solution of thedeterministic Euler equation. See also, e.g., [Che96a], [CMR98], [Fre97],[Swa71].

6. There exists a probabilistic approach to deterministic hydrodynamicalequations. In particular the solution of deterministic NavierStokes equa-tions can be expressed by the solution process associated with a backwardKolmogorov equation. This goes back to E. Nelson and has been devel-

oped by Belopolskaya and Daletski (78-90), Busnello [Bus99], [BFR05],[AB02], [AB06], [Oss05], [BRV01], see also [BDS04], [FR02], [Rap02b].

7. Other problems connected with Euler and NavierStokes equation relatetoa) coupling with heat equation (Benard problem) see e.g. [Fer97b].b) approximations, see, e.g. [DG95]c) physical properties like turbulence see, e.g. [Cho94]d) optimal control [Bar98], [CFMT83], [GSS02]

8. The phenomena associated with compressible fluids are quite differentfrom those associated with incompressible fluids. In particular blow upphenomena for good initial data have been intensively studied. Thereis indeed a large literature on compressible Burgers and Burgers-type

equations, with resp. without viscosity term, as well as on determinis-tic and stochastic compressible Euler and NavierStokes equations see,e.g., [FY92], [Mas00], [Roz03], [Roz04].

9. There are interesting basic connections between singularity phenomenaoccuring in (stochastic) equation for fluids and those occuring in thestudy of wave propagation and in certain quantum fields. E.g. the invari-ant measure constructed in Section 2 have close similarities with those


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constructed for wave propagation in [MV94], [CH97] and with the sto-chastic quantization equations studied in the theory of quantum fields, see,e.g. [AR95], [DPT05], [ABR], [ALZ06], [AR96], [BCM88], [JLM85], [MR],[Sim74]. Analogies are already apparent in the non linearities and thefact of renormalization needed, an instance of which we saw in Section 2,where we had to introduce the renormalized energy : E: (this is similar torenormalizing the non linear term in the stochastic quantization equation,see, e.g., [JLM85], [AR95], [AR96]). Also the form of invariant measuresI is similar to the one occuring in the theory of quantum fields. Otherrelations concern the analogy of the classical limit of quantum fields andthe 0 in hydrodynamics, see, e.g., [ABHK85]. Also quantum fields de-scribed by stochastic differential equations with Poisson noise have been

considered (see, e.g., [AGY05]), they have analogies with the invariantmeasures for the Euler-NavierStokes equations discussed in [AF04a]. Itis well known from quantum field theory that problems get worse with thedimension of space increasing and this is also so in stochastic hydrody-namics. A renormalization group approach, see, e.g. [Sin05b], [Sin05a],[Fri95], (see also [Sin08]), has been considered to be helpful in handlingthese problems. In any case it is likely that progress in one of these areas,fluids resp. quantum fields, will be beneficial to the other area. These aremost challenging areas for future mathematical research and, in partic-ular, further development of the type of infinite dimensional stochasticanalysis we tried to present in form of an introduction in these lectures.

3.6 AppendixWe give a result of regularity for parabolic equations.

Proposition 3.5. LetT (0, ], 1 < < and R. Let A be the Stokesoperator described in Section 3.1.For any f L(0, T; H2 ), the Cauchy problem

ddtX(t) + AX(t) = f(t), t (0, T]

X(0) = 0

has a unique solution X W1,(0, T) {X L(0, T; H+22 ) :ddtX

L(0, T; H2 )}. Moreover, the solution depends continuously on the data inthe sense that there exists a constant c

,such that

T0

[X(t)H+22+ ddtX(t)

H2 ] dt

1/

c,T0

f(t)H2 dt1/

Finally, X Cb([0, T]; B+2 2

2 ).

Proof. The Stokes operator A is a positive self adjoint operator in H2 withdomain H+22 and it generates an analytic semigroup in H

2 . Then the first


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part of the proposition is obtained applying Theorem 3.2 in [DV87]. Moreover,by interpolation we get that the space W1,(0, T) is continuously embedded

in the space Cb([0, T]; B+2 22 ), that is there exists a positive constant c such

that

(32) XCb([0,T];B

+2 2

2 ) c X

W1, (0,T)

Acknowledgements: The first author is very grateful to G. Da Prato,M. Rockner and L. Tubaro and all organizers of the Cetraro School for avery kind invitation to lecture at the School, which provided much stimula-tion for further studies. The help and kind understanding of the editors are

also gratefully acknowledged.

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Florea-Some Methods of Infinite Dimensional Analysisi in Hydrodynamics_An Introduction

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