FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS TOPICS ON COMPRESSIBLE NAVIER-STOKES EQUATIONS ETATS DE LA RECHERCHE SMF, CHAMB ´ ERY, MAY 21-25, 2012 RAPHA ¨ EL DANCHIN Abstract. Since the 80’s, Fourier analysis methods have known a growing impor- tance in the study of linear and nonlinear PDE’s. In particular, after the seminal paper by J.-M. Bony [4], the use of Littlewood-Paley decomposition and paradifferen- tial calculus brought a number of new results whenever those equations are considered in the whole space or the torus. We here aim at giving a survey of recent advances obtained by Fourier analysis methods in the context of compressible fluid mechanics. For simplicity, we shall focus on the barotropic compressible Navier-Stokes equations. Contents Introduction 2 1. The Fourier analysis toolbox 3 1.1. A primer on Littlewood-Paley theory 3 1.2. Functional spaces 5 1.3. Nonlinear estimates 8 1.4. Change of variables 11 1.5. Estimates for parabolic equations 12 1.6. Estimates for the linear transport equation 14 1.7. Estimates for dispersive equations 16 2. Solving the compressible Navier-Stokes equations in critical spaces 18 2.1. The incompressible Navier-Stokes equations 19 2.2. A first approach to the local existence theory 21 2.3. A Lagrangian approach for the compressible Navier-Stokes equations 25 2.4. References and remarks 32 3. Partially parabolic or dissipative linear PDEs 34 3.1. A direct analysis 34 3.2. An alternative approach 37 3.3. Applications 41 3.4. References and remarks 43 4. Global results 44 4.1. Global existence for small perturbations of a stable equilibrium state 44 4.2. On the incompressible limit 47 4.3. References and remarks 52 Appendix A. 52 References 54 Date : May 21, 2015. 1
Transcript
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS
TOPICS ON COMPRESSIBLE NAVIER-STOKES EQUATIONS
ETATS DE LA RECHERCHE SMF, CHAMBERY, MAY 21-25, 2012
RAPHAEL DANCHIN
Abstract. Since the 80’s, Fourier analysis methods have known a growing impor-tance in the study of linear and nonlinear PDE’s. In particular, after the seminalpaper by J.-M. Bony [4], the use of Littlewood-Paley decomposition and paradifferen-tial calculus brought a number of new results whenever those equations are consideredin the whole space or the torus.
We here aim at giving a survey of recent advances obtained by Fourier analysismethods in the context of compressible fluid mechanics. For simplicity, we shall focuson the barotropic compressible Navier-Stokes equations.
Contents
Introduction 21. The Fourier analysis toolbox 31.1. A primer on Littlewood-Paley theory 31.2. Functional spaces 51.3. Nonlinear estimates 81.4. Change of variables 111.5. Estimates for parabolic equations 121.6. Estimates for the linear transport equation 141.7. Estimates for dispersive equations 162. Solving the compressible Navier-Stokes equations in critical spaces 182.1. The incompressible Navier-Stokes equations 192.2. A first approach to the local existence theory 212.3. A Lagrangian approach for the compressible Navier-Stokes equations 252.4. References and remarks 323. Partially parabolic or dissipative linear PDEs 343.1. A direct analysis 343.2. An alternative approach 373.3. Applications 413.4. References and remarks 434. Global results 444.1. Global existence for small perturbations of a stable equilibrium state 444.2. On the incompressible limit 474.3. References and remarks 52Appendix A. 52References 54
Date: May 21, 2015.
1
2 R. DANCHIN
Introduction
This survey aims at presenting recent results in the theory of multi-dimensional com-pressible flows that have been obtained by means of elementary Fourier analysis. Thatapproach is relevant in any context where a good notion of Fourier transform is available,and proved to be particularly efficient and robust for investigating a number of evolutionaryfluid mechanics models. For simplicity, we shall concentrate on the following compressibleNavier-Stokes equations governing the evolution of the density ρ = ρ(t, x) ∈ R+ and of thevelocity field u = u(t, x) ∈ Rd of a barotropic viscous compressible fluid:
Above x belongs to the whole space Rd and the time variable t is nonnegative. Thenotation Du designates the Jacobian matrix of u (that is (Du)ij := ∂ju
i ) while ∇u standsfor the transposed matrix of Du (therefore Du+∇u is twice the deformation tensor). Inorder to close the system, we assume that the pressure p to be a given (smooth) functionP of the density (this is the barotropic assumption). The viscosity coefficients λ and µ aresmooth functions of the density and satisfy the conditions
(2) α := min(
infρ>0
(λ(ρ) + 2µ(ρ)), infρ>0
µ(ρ))> 0,
which ensures the second order operator in the momentum equation of (1) to be uniformlyelliptic.
We supplement the system with initial data ρ0 and u0 at time t = 0, and the conditionsat infinity that u tends to 0 and that ρ tends to some positive constant (we shall take1 for simplicity). The exact meaning of those boundary conditions will be given by thefunctional framework in which we shall solve the system.
In these notes we shall always consider the above system in the whole space Rd withd ≥ 2, although our approach is adaptable to periodic boundary conditions x ∈ Td andmore generally x ∈ Td1×Rd2 and so on. Let us also emphasize that our tools and methodsare appropriate for investigating models with more physics (e.g. nonisothermal case, fluidsendowed with internal capillarity and so on). For expository purpose, we here focus on (1).
These notes unfold as follows.
• The first section is devoted to presenting briefly the main techniques and tools thatwill be used to investigate (1). We first introduce the Littlewood-Paley decompo-sition, then define Besov spaces that are natural generalizations of the classicalSobolev spaces, and finally prove nonlinear estimates. In passing we present basicparadifferential calculus and give examples of estimates for linear equations thatmay be proved by taking advantage of Littlewood-Paley decomposition and paradif-ferential calculus. We chose to focus on the heat and Lame equations, the transportequation and linear dispersive equations, which are frequently encountered whenlinearizing systems coming from fluid mechanics or physics.• Solving (1) locally in time in critical spaces is the main purpose of the second
section. As a warm up, we combine some results of the first section with theBanach fixed point theorem (or contracting mapping argument) so as to prove thatthe incompressible Navier-Stokes equations with small data, are globally well-posed.In passing, we introduce the notion of a critical functional framework for Partial
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 3
Differential Equations. Next, we adapt this method to System (1) and get a local-in-time existence result in a critical functional framework. At first sight, it seemsthat uniqueness requires stronger assumptions than existence, a consequence of thefact that the proof does not rely on the contracting mapping argument but rather onhigh norm uniform bounds / low norm stability estimates like for solving quasilinearsymmetric hyperbolic systems. In the last part of this section, we rewrite the systemin Lagrangian coordinates so as to establish a second local-in-time existence resultby means of the Banach fixed point theorem. With this second approach, we getexistence, uniqueness and continuity of the flow map (in Lagrangian coordinates)altogether, under the same regularity hypotheses as before.• The third section is devoted to a fine analysis of the linearized system (1) about
a stable constant state. We first obtain optimal regularity or decay estimates bytaking advantage of the explicit formula for the solution. The second part of thesection is somewhat disconnected of the study of compressible fluids. Here wepresent a general method, based on energy estimates, leading to optimal decay orregularity estimates for mixed-type linear systems. Typically, the systems that weshall consider are linear first order symmetric, with additional partially parabolicor dissipative terms (in the spirit of the work by Shizuta and Kawashima [55]). Weshall show how an idea that is borrowed from Kalman controllability criterion mayhelp to prove such estimates without computing the solution explicitly.• The fourth section is devoted to the proof of global-in-time results for (1) supple-
mented with small initial data. Here the estimates of the third section play a keyrole. We first establish global well-posedness results in critical spaces and finallyinvestigate the so-called incompressible limit. This latter result requires a fineranalysis of the linearized system, pointing out its dispersive properties which arespecific to the whole space case (in contrast with the other results).
We tried to keep those notes at an elementary level so as to give a general and as lesstechnical as possible overview of how Fourier analysis techniques may be implemented.The reader may find more sophisticated results in e.g. [2], [20], [22] and in the referencestherein. For the sake of readability, most of the references and historical remarks are givenat the end of Sections 2, 3 and 4.
1. The Fourier analysis toolbox
Here we introduce the Littlewood-Paley decomposition, define Besov spaces, establishproduct and composition estimates, and finally prove estimates for different types of linearPDEs that are frequently encountered when dealing with fluid mechanics models. Moredetailed proofs may be found in e.g. [2, 20, 52, 57].
1.1. A primer on Littlewood-Paley theory. The Littlewood-Paley decomposition is adyadic localization procedure in the frequency space for tempered distributions over Rd.One of the main motivations for introducing such a localization when dealing with PDEsis that the derivatives act almost as homotheties on distributions with Fourier transformsupported in a ball or an annulus.
In the L2 framework, this noticeable property easily follows from Parseval’s formula.The Bernstein inequalities below state that it remains true in any Lebesgue space:
Proposition 1.1 (Bernstein inequalities). For all 0 < r < R, we have:
4 R. DANCHIN
• Direct Bernstein inequality: a constant C exists so that, for any k ∈ N, any cou-
ple (p, q) in [1,∞]2 with q ≥ p ≥ 1 and any function u of Lp with u supported inthe ball B(0, λR) of Rd for some λ > 0, we have
‖Dku‖Lq ≤ Ck+1λk+d( 1
p− 1q
)‖u‖Lp .• Reverse Bernstein inequality: there exists a constant C so that for any k ∈ N,p ∈ [1,∞] and any function u of Lp with Supp u ⊂ {ξ ∈ Rd / rλ ≤ |ξ| ≤ Rλ} forsome λ > 0, we have
λk‖u‖Lp ≤ Ck+1‖Dku‖Lp .
Proof. Changing variables reduces the proof to the case λ = 1. For proving the first in-equality, we fix some smooth φ with compact support and value 1 over B(0, R). One maythus write
u = φ u whence Dku = (DkF−1φ) ? u.
Therefore using convolution inequalities, one may write
‖Dku‖Lq ≤ ‖DkF−1φ‖Lr‖u‖Lpwith 1 + 1/q = 1/p+ 1/r (here we need q ≥ p), and we are done.
For proving the second inequality, we now assume that φ is compactly supported awayfrom the origin and has value 1 over the annulus C(0, r, R). We thus have
u =
(−i ξ|ξ|2
φ(ξ)
)· ∇u (ξ).
Therefore, denoting by g the inverse Fourier transform of the first term in the r.h.s.,
‖u‖Lp ≤ ‖g‖L1‖∇u‖Lp .This gives the result for k = 1. The general case follows by induction. �
As solutions to nonlinear PDE’s need not be spectrally localized in annuli (even if werestrict to initial data with this property), it is suitable to have a device which allows forsplitting any function into a sum of functions with this spectral localization. This is exactlywhat Littlewood-Paley decomposition does.
To construct it, fix some smooth radial non increasing function χ with Suppχ ⊂ B(0, 43)
and χ ≡ 1 on B(0, 34), then set ϕ(ξ) = χ(ξ/2)− χ(ξ) so that
χ+∑j∈N
ϕ(2−j ·) = 1 in Rd and∑j∈Z
ϕ(2−j ·) = 1 in Rd \ {0}.
The homogeneous dyadic blocks ∆j are defined by
∆ju := ϕ(2−jD)u := F−1(ϕ(2−jD)Fu) := 2jdh(2j ·) ? u with h := F−1ϕ.
We also introduce the low frequency cut-off operator Sj :
Sju := χ(2−jD)u := F−1(χ(2−jD)Fu) := 2jdh(2j ·) ? u with h := F−1χ.
The nonhomogeneous dyadic blocks ∆j are defined by
∆j := ∆j if j ≥ 0, ∆−1 := S0 = χ(D) and ∆j = 0 if j ≤ −2,
and we setSj :=
∑k≤j−1
∆k.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 5
The homogeneous and nonhomogeneous Littlewood-Paley decompositions for u read
(3) u =∑j
∆ju and u =∑j
∆ju.
The second equality holds true in the set S ′ of tempered distributions. This is not the caseof the first one which is true modulo polynomials only if no further assumptions on u. Away to overcome this is to restrict to the set S ′h of tempered distributions u such that
limj→−∞
‖Sju‖L∞ = 0 with Sj := χ(2−jD).
Note that loosely speaking, this condition on the low frequencies of u amounts to requiringu to tend to 0 at infinity (in the sense of distributions). Then the first equality (3) holdstrue whenever u is in S ′h.
Finally, let us emphasize that the support properties of ϕ and χ entail properties ofquasi-orthogonality for the Littlewood-Paley decomposition. With the normalization thatwe adopted here, namely Suppϕ ⊂ C(0, 3/4, 8/3) and Suppχ ⊂ B(0, 4/3), one may easilycheck that
∆j∆k = 0 if |j − k| > 1 and ∆k(Sj−1u ∆jv) ≡ 0 if |k − j| > 4.
1.2. Functional spaces. Many classical norms may be written in terms of the Littlewood-Paley decomposition. This is e.g. the case of homogeneous Sobolev or Holder norms:
Proposition 1.2. For any s ∈ R there exists a constant C ≥ 1 so that for any tempereddistribution u,
C−1‖u‖2Hs ≤
∑j
22js‖∆ju‖2L2 ≤ C‖u‖2Hs .
If s ∈ (0, 1) then we have (assuming in addition that u =∑
j ∆ju for the left inequality)
C−1‖u‖C0,s ≤ supj
2js‖∆ju‖L∞ ≤ C‖u‖C0,s .
Proof. Owing to Suppϕ(2−j ·) ∩ Supp(2−k·) = ∅ if |j − k| > 1, we have
1
2≤∑j
ϕ2(2−jξ) ≤ 1 for ξ 6= 0.
Hence, using the definition of homogeneous Sobolev norm, of ∆ju and Parseval equality1
‖u‖2Hs =
∫|ξ|2|u(ξ)|2 dξ ≈
∑j
∫|ξ|2|ϕ(2−j)u(ξ)|2 dξ ≈
∑j
22js‖∆ju‖2L2 .
As for the homogeneous Holder norm, we notice that because h has average 0,
∆ju(x) = 2jd∫h(2j(x− y)) (u(y)− u(x)) dy for all j ∈ Z.
Hence for all x ∈ Rd and j ∈ Z,
|∆ju(x)| ≤ 2−js‖u‖C0,s
(2jd∫|h(2j(x− y))|(2j |x− y|)s dy
)≤ 2−js‖u‖C0,s‖| · |sh‖L1 .
1With the convention that A ≈ B means that there exists some harmless positive constant C such thatC−1A ≤ B ≤ CA.
6 R. DANCHIN
Conversely, if Cs(u) := supj 2js‖∆ju‖L∞ <∞ then we may write for any N ∈ Z,
u(y)− u(x) =∑j<N
(∆ju(y)− ∆ju(x)) +∑j≥N
(∆ju(y)− ∆ju(x)).
Hence|u(y)− u(x)| ≤ |y − x|
∑j<N
‖∇∆ju‖L∞ + 2∑j≥N‖∆ju‖L∞ .
Therefore, taking advantage of Bernstein inequality for the terms in the first sum,
|u(y)− u(x)| ≤ Cs(u)
(|y − x|
∑j<N
2j(1−s) + 2∑j≥N
2−js).
Then taking the “best” N yields ‖u‖C0,s ≤ CCs(u). �
If looking at those two characterizations, we see that three parameters come into play:the regularity parameter s, the Lebesgue exponent that is used for ∆ju and the type ofsummation that has been performed over Z. This motivates the following definition:
Definition 1.1. For s ∈ R and 1 ≤ p, r ≤ ∞, we set
‖u‖Bsp,r :=
(∑j
2rjs‖∆ju‖rLp) 1r
if r <∞ and ‖u‖Bsp,∞ := supj
2js‖∆ju‖Lp .
We then define the homogeneous Besov space Bsp,r as the subset of distributions u ∈ S ′h
such that ‖u‖Bsp,r <∞.
Similarly we set
‖u‖Bsp,r :=
(∑j
2rjs‖∆ju‖rLp) 1r
if r <∞ and ‖u‖Bsp,∞ := supj
2js‖∆ju‖Lp ,
and define the nonhomogeneous Besov space Bsp,r as the subset of distributions u ∈ S ′ such
that ‖u‖Bsp,r <∞.
According to this definition, the space Bs2,2 coincides with the homogeneous Sobolev
space Hs and it is true that Br∞,∞ is the homogeneous Holder space C0,r if r ∈ (0, 1).
The following proposition states that, loosely speaking, having u in Bsp,r means that u
has s fractional derivatives in Lp (see the proof in e.g. [2]):
Proposition 1.3 (Characterization by finite differences). For s ∈ (0, 1) and finite p, r, wehave
‖u‖Bsp,r ≈(∫
Rd
(∫Rd
(|u(y)− u(x)||y − x|s
)p dy
|x− y|d
) rp) 1r
·
A similar result holds for p or r infinite.
The scaling properties of homogeneous Besov norms are given below:
Proposition 1.4. For any s ∈ R and (p, r) ∈ [1,+∞]2 there exists a constant C such
that for all positive λ and u ∈ Bsp,r, we have
C−1λs− d
p ‖u‖Bsp,r ≤ ‖u(λ·)‖Bsp,r ≤ Cλs− d
p ‖u‖Bsp,r .
Here are some classical embedding results:
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 7
Proposition 1.5. (1) For any p ∈ [1,∞] we have the following chain of continuous
embedding: B0p,1 ↪→ Lp ↪→ B0
p,∞.
(2) If s ∈ R, 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞, then Bsp1,r1 ↪→ B
s−d( 1p1− 1p2
)
p2,r2 .
(3) The space Bdp
p,1 is continuously embedded in the set Cb of bounded continuous func-tions. If p < ∞ then it is also embedded in the set of continuous functions goingto 0 at infinity.
Proof. The left embedding of the first property follows from the triangle inequality for theLp norm applied to
u =∑j
∆ju
whereas the right inequality is a consequence of the convolution property L1 ? Lp → Lp
which implies that
‖∆ju‖Lp ≤ ‖2jdh(2j ·)‖L1‖u‖Lp = ‖h‖L1‖u‖Lp .As for the second property, we just have to use that, owing to Bernstein inequality,
‖∆ju‖Lp2 ≤ C2j( dp1− dp2
)‖∆ju‖Lp1 .Finally, by combining the first two properties, we see that
Bdp
p,1 ↪→ B0∞,1 ↪→ L∞.
Note in particular that this implies that if u ∈ Bdp
p,1 then the series∑
j∈Z ∆ju convergesuniformly to u. As each term of the series is continuous and bounded, the same propertyholds for u. Finally, if p < ∞ then each term ∆ju goes to 0 at infinity (because it is inLp ), hence so does u. �
Here is a list of classical and important properties of Besov spaces that are of constant usein these notes (see the proof in e.g. [2]):
• The space Bsp,r is complete whenever s < d/p or s ≤ d/p and r = 1, and so does
Bsp,r without any condition on (s, p, r).
• Fatou property: if (un)n∈N is a bounded sequence of functions of Bsp,r that con-
verges in the sense of tempered distributions to some u ∈ S ′h then u ∈ Bsp,r
and ‖u‖Bsp,r ≤ C lim inf ‖un‖Bsp,r . A similar result holds in Bsp,r (where u ∈ S ′
is enough).• Duality: If u is in S ′h then we have
‖u‖Bsp,r ≤ C supφ〈u, φ〉
where the supremum is taken over those φ in S ∩ B−sp′,r′ such that ‖φ‖B−sp′,r′≤ 1.
• The following real interpolation property is satisfied for all 1 ≤ p, r1, r2, r ≤ ∞,s1 6= s2 and θ ∈ (0, 1):
[Bs1p,r1 , B
s2p,r2 ](θ,r) = Bθs2+(1−θ)s1
p,r and [Bs1p,r1 , B
s2p,r2 ](θ,r) = Bθs2+(1−θ)s1
p,r .
• For any smooth homogeneous of degree m function F on Rd \ {0} the Fourier
multiplier F (D) maps Bsp,r in Bs−m
p,r . In particular, the gradient operator maps
Bsp,r in Bs−1
p,r .
8 R. DANCHIN
The following lemma ensures that for spectrally localized series, proving that the sum ofthe series belongs to some space Bs
p,r (or Bsp,r ) amounts to getting a suitable Lp bound
for each term of the series. Consequently, the definition of Besov spaces is independent ofthe choice of (∆j)j∈Z or (∆j)j∈Z. It also turns out to be very useful for proving nonlinearestimates (see the next paragraph).
Lemma 1.1. Let 0 < R1 < R2. Let s ∈ R and 1 ≤ p, r ≤ ∞. Let (uj)j≥−1 be such thatSupp u−1 ⊂ B(0, R2) and Supp uj ⊂ 2jC(0, R1, R2) for all j ∈ N. Then∥∥∥2js‖uj‖Lp(Rd)
∥∥∥`r(N∪{−1})
<∞ =⇒ u :=∑j≥−1
uj is in Bsp,r
and we have2 ‖u‖Bsp,r .∥∥∥2js‖uj‖Lp(Rd)
∥∥∥`r(N∪{−1})
.
If s > 0 then the result is still true under the weaker assumption that Supp uj ⊂B(0, 2jR2).
A similar statement holds true in the homogeneous setting if assuming in addition that∑j<0 uj converges in S ′h.
Proof. We notice that one may find some integer N depending only on R1, R2 such thatfor all k ≥ −1,
∆ku =∑
|j−k|≤N
∆kuj .
Therefore‖∆ku‖Lp ≤
∑|j−k|≤N
‖∆kuj‖Lp ≤ C∑
|j−k|≤N
‖uj‖Lp
and we get the result.
If we only have Supp uj ⊂ B(0, 2jR2) then we just have for some integer N,
∆ku =∑
j≥k−N∆kuj .
Therefore2ks‖∆ku‖Lp ≤
∑j≥k−N
2(k−j)s 2js‖uj‖Lp
and the convolution inequality `1 ? `r → `r gives the result if s > 0. �
1.3. Nonlinear estimates. The basic questions that we address in this subsection are:let u and v belong to two (possibly different) Besov spaces:
• Does uv make sense ?• If so, where does uv lie ?• If F is a suitably smooth function, what is the regularity of F (u) ?
Formally, the product of two distributions u and v may be decomposed into
(4) uv = Tuv +R(u, v) + Tvu
withTuv :=
∑j
Sj−1u∆jv and R(u, v) :=∑j
∑|j′−j|≤1
∆ju∆j′v.
2The sign . means that the l.h.s. is bounded by the r.h.s. times a harmless multiplicative constant.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 9
The above operator T is called “paraproduct” whereas R is called “remainder”. Thedecomposition (4) has been first introduced by J.-M. Bony in [4]. However, the discreteversion of it that we shall present below is due to P. Gerard and J. Rauch in [31]. As weshall see, it is of particular efficiency inasmuch as the three terms of (4) may be treatedseparately. In particular, as Tuv is a sum of product of functions with different spectrallocalizations, it is always defined (because in Fourier variables, the sum is locally finite andLemma 1.1 applies). At the same time, it cannot be smoother than what is given by highfrequencies, namely v. As for the remainder, it may be not defined (e.g. product of Diracmasses at the same point). However, if it is defined then the regularities of u and v addup. All this is detailed in the proposition below.
Proposition 1.6. For any (s, p, r) ∈ R× [1,∞]2 and t < 0 we have
‖Tuv‖Bsp,r . ‖u‖L∞‖v‖Bsp,r and ‖Tuv‖Bs+tp,r. ‖u‖Bt∞,∞‖v‖Bsp,r .
For any (s1, p1, r1) and (s2, p2, r2) in R× [1,∞]2 we have
• if s1 + s2 > 0, 1/p := 1/p1 + 1/p2 ≤ 1 and 1/r := 1/r1 + 1/r2 ≤ 1 then
‖R(u, v)‖Bs1+s2p,r
. ‖u‖Bs1p1,r1‖v‖Bs2p2,r2
;
• if s1 + s2 = 0, 1/p := 1/p1 + 1/p2 ≤ 1 and 1/r1 + 1/r2 ≥ 1 then
‖R(u, v)‖B0p,∞. ‖u‖Bs1p1,r1‖v‖B
s2p2,r2
.
Similar results hold in homogeneous Besov spaces.
Proof. We just prove the first result of continuity for T and R. Both are consequences ofLemma 1.1. We first notice that the general term of Tuv is supported in dyadic annuliwhereas that of R(u, v) is only supported in dyadic balls. Now, we see that
and use the last part of Lemma 1.1 together with convolution inequalities. �
Putting together decomposition (4) and the above results of continuity, one may deducea number of continuity results for the product of two functions. For instance, one may getthe following tame estimate which depends linearly on the highest norm of u and v :
Corollary 1.1. Let u and v be in L∞ ∩ Bsp,r for some s > 0 and (p, r) ∈ [1,∞]2. Then
there exists a constant C depending only on d, p and s and such that
‖uv‖Bsp,r ≤ C(‖u‖L∞‖v‖Bsp,r + ‖v‖L∞‖u‖Bsp,r
).
Proof. We proceed as follows:
1. Write Bony’s decomposition uv = Tuv + Tvu+R(u, v);2. Use T : L∞ ×Bs
p,r → Bsp,r ;
3. Use R : B0∞,∞ ×Bs
p,r → Bsp,r if s > 0;
4. Take advantage of L∞ ↪→ B0∞,∞.
10 R. DANCHIN
Gluing together the inequalities obtained through the above steps completes the proof. �
Remark 1.1. Because Bd/pp,1 and B
d/pp,1 are embedded in L∞, we deduce that both spaces
Bd/pp,1 and B
d/pp,1 are (quasi) Banach algebra if p <∞.
Finally, the following composition result will be needed for handling e.g. the pressureterm when studying the compressible Navier-Stokes equations.
Proposition 1.7. Let F : R → R be a smooth function with F (0) = 0. Then for all(p, r) ∈ [1,∞]2 and all s > 0, if u ∈ Bs
p,r ∩ L∞ then F (u) ∈ Bsp,r ∩ L∞ and
(5) ‖F (u)‖Bsp,r ≤ C‖u‖Bsp,r
with C depending only on ‖u‖L∞ , F, s, p and d.
Proof. We use Meyers’s first linearization method:
F (u) =∑j
F (Sj+1u)− F (Sju) =∑j
∆ju
∫ 1
0F ′(Sju+ τ∆ju) dτ︸ ︷︷ ︸
uj
.
We note that
‖uj‖Lp ≤ C‖∆ju‖Lp and ‖∆ju‖Lp ≤ 2−js‖u‖Bsp,r ,
which ensures the convergence of the above series in Lp. However, Fuj is not localized ina ball of size 2j hence Lemma 1.1 does not apply. Nevertheless, one may show that ujbehaves as if it were as regards differentiation: we have (see e.g. [2]):
(6) ‖Dkuj‖Lp ≤ Ck2jk‖∆ju‖Lp for all k ∈ N.
Now, for any j ∈ N, one may write
(7) ∆jF (u) =∑j′≤j
∆juj′ +∑j′>j
∆juj′ .
Applying the reverse Bernstein inequality and (6) with k = [s] + 1, we discover that
‖∆juj′‖Lp . 2−kj‖Dk∆juj′‖Lp . 2−k(j−j′)‖∆j′u‖Lp .
Hence
2js∑j′≤j‖∆juj′‖Lp .
∑j′≤j
2(s−k)(j−j′)(2j′s‖∆j′u‖Lp),
and convolution inequalities for series (observe that s− k < 0) implies that the first termof the r.h.s. of (7) is in Bs
p,r and is bounded by the r.h.s. of (5).
To bound the last term of (7), we just use (6) with k = 0 so as to write that
2js∑j′>j
‖∆juj′‖Lp .∑j′>j
2−s(j′−j)(2j′s‖∆j′u‖Lp
).
Then the desired inequality follows from convolution inequalities for series. �
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 11
1.4. Change of variables. Here we establish a result of regularity concerning changesof variables in Besov spaces Bs
p,r. Let us first introduce the multiplier space M(Bsp,r) for
Bsp,r, that is the set of those distributions f such that ψf is in Bs
p,r whenever ψ is in Bsp,r,
endowed with the norm
(8) ‖f‖M(Bsp,r):= sup ‖ψf‖Bsp,r
where the supremum is taken over those functions ψ in Bsp,r with norm 1.
Proposition 1.8. Let X be a globally bi-Lipschitz diffeomorphism of Rd, and (s, p, r)with 1 ≤ p < ∞ and −d/p′ < s < d/p (or just −d/p′ < s ≤ d/p if r = 1 and just
−d/p′ ≤ s < d/p if r = ∞). Then a 7→ a ◦ X is a self-map over Bsp,r in the following
cases:
(1) s ∈ (0, 1),
(2) s ∈ (−1, 0] and JX−1 is in the multiplier space M(B−sp′,r′) defined in (8),
(3) k < s < k+1 for some integer k and DjX ∈M(Bs−1+j−kp,r ) for all j ∈ {1, · · · , k}·
Proof. First assume that s ∈ (0, 1) and that r <∞ (for shortness of presentation). Thenone may use the characterization of the norm in terms of finite differences (see Proposition1.3) so as to write:
‖u ◦X‖Bsp,r(Rd) =
(∫Rd
(∫Rd
|u(X(y))− u(X(x))|p
|y − x|d+spdy
) rp
dx
) 1r
·
Hence performing the change of variable x′ = X(x) and y′ = X(y), we get
‖u ◦X‖Bsp,r(Rd) =
(∫Rd
(∫Rd
|u(y′)− u(x′)|p
|X−1(y′)−X−1(x′)|d+spJX−1(y′) dy′
) rp
JX−1(x′) dx′) 1r
whence
‖u ◦X‖Bsp,r(Rd) ≤ ‖JX−1‖1p
+ 1r
L∞(Rd)‖DX‖
s+ dp
L∞(Rd)‖u‖Bsp,r(Rd).
The condition that s < d/p ensures in addition that u belongs to some Lebesgue spacewith finite index, or goes uniformly to 0 at infinity (case r = 1 and s = d/p). Hence the
same property holds for u ◦X, which suffices to conclude that u ◦X ∈ Bsp,r(Rd).
The result for negative s may be achieved by duality: we write that
‖u ◦X‖Bsp,r(Rd) ≤ C sup‖v‖
B−sp′,r′
(Rd)≤1
∫Rdv(z)u(X(z)) dz.
Now, setting x = X(z), we have∫Rdv(z)u(X(z)) dz =
∫Rdu(x)v(X−1(x))JX−1(x) dx
≤ ‖u‖Bsp,r(Rd)‖v ◦X−1JX−1‖B−s
p′,r′ (Rd).
So the definition of the multiplier space and the first part of the lemma allows to conclude.
The larger values of s may be treated by induction, using the chain rule
D(u ◦X) = (Du ◦X) ·DX,and the definition of multiplier spaces. �
12 R. DANCHIN
1.5. Estimates for parabolic equations. Consider the heat equation
(9) ∂tu−∆u = f, u|t=0 = u0
or, more generally3
(10) ∂tv + |D|σv = g, v|t=0 = v0.
We want to establish estimates of the form
‖u‖L∞(X) + ‖∂tu,D2u‖L1(X) ≤ C(‖u0‖X + ‖f‖L1(X)
)(11)
‖v‖L∞(X) + ‖∂tv, |D|σv‖L1(X) ≤ C(‖v0‖X + ‖g‖L1(X)
).(12)
In the case of the heat equation, the gain of two derivatives compared to the source termwhen performing a L1 -in-time integration is the key to a number of well-posedness resultsin a critical functional framework for models arising in fluid mechanics (see e.g. the nextsections).
It is well known that if r ∈ (1,∞) and X = Lq or W s,q for some s ∈ R and q ∈ (1,∞)then, for the heat equation we have
‖∂tu,D2u‖Lr(X) ≤ C‖f‖Lr(X).
On the one hand, those inequalities fail for the endpoint case r = 1 and, more generally,whenever X is a reflexive Banach space. On the other hand, as noticed by J.-Y. Cheminin [8], Inequality (11) is true for Besov spaces with third index 1. This is stated in thefollowing theorem.
Theorem 1.1. Estimates (11) and (12) hold true for any p ∈ [1,∞], σ ∈ R and s ∈ R if
X = Bsp,1.
Proving the theorem relies on the following:
Lemma 1.2. There exist two positive constants c and C such that for any j ∈ Z, p ∈[1,∞] and λ ∈ R+, we have
‖e−λ|D|σ∆j‖L(Lp;Lp) ≤ Ce−cλ2σj .
Proof. If p = 2 this is a mere consequence of Parseval’s formula. In the general case,one may first reduce the proof to j = 0 (just perform a suitable change of variable) thenconsider a function φ in D(Rd \ {0}) with value 1 on a neighborhood of the support of ϕso as to write
eλ∆∆0u = F−1(φe−λ|·|
σ ∆0u)
= gλ ? ∆0u with gλ(x) := (2π)−d∫ei(x|ξ)φ(ξ)e−λ|ξ|
σdξ.
If it is true that
(13) ‖gλ‖L1 ≤ Ce−cλ
then using the convolution inequality L1 ? Lp → Lp implies that
‖eλ∆∆0u‖Lp ≤ ‖gλ‖L1‖∆0u‖Lp ≤ Ce−cλ‖∆0u‖Lp .
3With the notation F(|D|σu)(ξ) := |ξ|σFu(ξ).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 13
Proving (13) follows from integration by parts: we have
gλ(x) = (1 + |x|2)−d∫Rdei(x|ξ)(Id −∆ξ)
d(φ(ξ)e−λ|ξ|
σ)dξ.
Therefore, combining Leibniz and Faa-di-Bruno’s formulae, we may conclude that
|gλ(x))| ≤ C(1 + |x|2)−de−cλ,
which obviously implies (13). �
Proof of Theorem 1.1 We focus on the case σ = 2 (heat equation). If u satisfies (9)then for any j ∈ Z,
∂t∆ju−∆∆ju = ∆jf.
Hence, according to Duhamel’s formula
∆ju(t) = et∆∆ju0 +
∫ t
0e(t−τ)∆∆jf(τ) dτ.
Taking advantage of Lemma 1.2, we thus have
(14) ‖∆ju(t)‖Lp . e−c22jt‖∆ju0‖Lp +
∫ t
0e−c2
2j(t−τ)‖∆jf(τ)‖Lp dτ.
Multiplying by 2js and summing up over j yields∑j
2js‖∆ju(t)‖Lp .∑j
e−c22jt2js‖∆ju0‖Lp +
∫ t
0e−c2
2j(t−τ)∑j
‖∆jf(τ)‖Lp dτ
whence
‖u‖L∞t (Bsp,1) . ‖u0‖Bsp,1 + ‖f‖L1t (B
sp,1).
Note that integrating (14) with respect to time also yields
22j‖∆ju‖L1t (L
p) .(
1− e−c22jt)(‖∆ju0‖Lp + ‖∆jf‖L1
t (Lp)
).
Therefore, multiplying by 2js and summing up over j yields
(15) ‖u‖L1t (B
s+2p,1 ) .
∑j
(1− e−c22jt
)2js(‖∆ju0‖Lp + ‖∆jf‖L1
t (Lp)
),
which is slightly better than what we wanted to prove.4 �
Remark 1.2. Let us point out that starting from (14) and using other convolution inequal-ities gives a whole family of estimates for the heat equation. However, as time integrationhas been performed before summation over j, the norms that naturally appear are
‖ · ‖Lat (Bσb,c)
:=∥∥∥2jσ‖ · ‖Lat (Lb)
∥∥∥`c
where ‖ · ‖Lat (Lb) := ‖ · ‖La((0,t);Lb(Rd)).
4As obviously (1− e−c22jt) is bounded by 1 and tends to 0 when t goes to 0+.
14 R. DANCHIN
Therefore (14) implies that
‖u‖Lρ1t (B
s+ 2ρ1
p,r ). ‖u0‖Bsp,1 + ‖f‖
Lρ2t (B
s−2+ 2ρ2
p,r )for 1 ≤ ρ2 ≤ ρ1 ≤ ∞.
Of course those quantities may be compared with the norm in La(0, t; Bσb,c) according to
Minkowski inequality: we have
‖ · ‖Lat (Bσb,c)
≤ ‖ · ‖Lat (Bσb,c)if a ≤ c
‖ · ‖Lat (Bσb,c)≤ ‖ · ‖
Lat (Bσb,c)if a ≥ c.
As a corollary, we get similar estimates for the following Lame system which is thelinearization of the velocity equation of (1) about the constant solution (ρ, u) = (1, 0), ifone neglects the pressure term:
(16) ∂tu− µ∆u− (λ+ µ)∇divu = f, u|t=0 = u0
with µ > 0 and ν := λ+ 2µ > 0.
Corollary 1.2. If u0 ∈ Bsp,1 and f ∈ L1(R+; Bs
p,1) then (16) has a unique solution u in
Cb(R+; Bsp,1)∩L1(R+; Bs
p,1). Furthermore, there exists a constant C depending only on µ/νsuch that
‖u‖L∞t (Bsp,1)
+ ν‖u‖L1t (B
s+2p,1 ) ≤ C
(‖u0‖Bsp,1 + ‖f‖L1
t (Bsp,1)
).
Proof. Let P and Q denote the projector over divergence-free and potential vector fields,respectively (that is FQu(ξ) = |ξ|−2ξ (ξ · Fu(ξ)) in Fourier variables, and P = Id − Q).Then we have
∂tPu− µ∆Pu = Pf and ∂tQu− ν∆Qu = Qf.Given that P and Q are homogeneous multipliers of degree 0, applying Theorem 1.1completes the proof. �
1.6. Estimates for the linear transport equation. In this paragraph, we focus on theproof of a priori estimates for the following transport equation that plays a fundamentalrole in fluid mechanics:
(Tλ)
{∂ta+ v · ∇a+ λa = f
a|t=0 = a0
where λ is a given nonnegative parameter.
Roughly, if v is a Lipschitz time-dependent vector-field, and if a0 ∈ X and f ∈L1(0, T ;X), with X a Banach space then we expect (T ) to have a unique solution a ∈C([0, T );X) satisfying (if λ = 0):
‖a(t)‖X ≤ eCV (t)
(‖a0‖X +
∫ t
0e−CV (τ)‖f(τ)‖X dτ
)with V (t) :=
∫ t
0‖∇v(τ)‖L∞ dτ.(17)
This is obvious if X is the Holder space C0,ε with ε ∈ (0, 1) as (in the case f ≡ 0 tosimplify) the solution to (T0) is given by
a(t, x) = a0(ψ−1t (x))
where ψt stands for the flow of v at time t.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 15
Therefore,|a(t, x)− a(t, y)| = |a0(ψ−1
t (x))− a0(ψ−1t (y))|
≤ ‖a0‖C0,ε |ψ−1t (x)− ψ−1
t (y)|ε
≤ ‖a0‖C0,ε‖∇ψ−1t ‖εL∞ |x− y|ε.
As ‖∇ψ−1t ‖L∞ ≤ exp(V (t)), we get the result in this particular case.
Littlewood-Paley decomposition will enable us to prove a similar result in a much moregeneral framework.
Theorem 1.2. Assume that
1 ≤ p ≤ p1 ≤ ∞, 1 ≤ r ≤ ∞, −min( dp1,d
p′
)< s < 1 +
d
p1·
Then any smooth enough solution to (17) satisfies the a priori estimate
‖a‖L∞t (Bsp,r)
+ λ‖a‖L1t (B
sp,r)≤ eCV (t)
(‖a0‖Bsp,r + ‖f‖
L1t (B
sp,r)
)with
V (t) :=
∫ t
0‖∇v(τ)‖
Bdp1p1,1
dτ.
If r = 1 (resp. r =∞) then the case s = 1+d/p1 (resp. s = −min(dp1, dp′)
) also works.
Proof. Applying ∆j to (T ) gives
(18) ∂t∆ja+ v · ∇∆ja+ λ∆ja = ∆jf + Rj with Rj := [v · ∇, ∆j ]a.
In the case p ∈ (1,∞), multiplying both sides by |∆ja|p−2∆ja and integrating over Rdyields
1
p
d
dt‖∆ja‖pLp + λ‖∆ja‖pLp +
1
p
∫v · ∇|∆ja|p dx =
∫ (∆jf + Rj
)|∆ja|p−2∆ja dx.
Therefore
‖∆ja(t)‖Lp + λ‖∆ja‖L1t (L
p) ≤ ‖∆ja0‖Lp(19)
+
∫ t
0
(‖∆jf‖Lp + ‖Rj‖Lp +
‖divv‖L∞p
‖∆ja‖Lp)dτ.
Having p tend to 1 or ∞ implies that (19) also holds if p = 1 or p =∞.Now, under the above conditions over s, p , the remainder term Rj satisfies
(20) ‖Rj(t)‖Lp ≤ Ccj(t)2−js‖∇v(t)‖B
dp1p1,1
‖a(t)‖Bsp,r with ‖(cj(t))‖`r = 1.
This may be proved by taking advantage of Bony’s decomposition. Indeed we have (withthe summation convention over repeated indices):
Rj = [Tvk , ∆j ]∂ka+ T∂k∆javk − ∆jT∂kav
k +R(vk, ∂k∆ja)− ∆jR(vk, ∂ka).
Let us just explain how to bound the first term which is the only one where having acommutator improves the estimate (bounding the other terms mainly stems from Lemma1.1 or Proposition 1.6). Owing to the properties of spectral localization, we have
[Tvk , ∆j ]∂ka =∑|j−j′|≤4
[Sj′−1vk, ∆j ]∂k∆j′a.
16 R. DANCHIN
Let h := F−1ϕ. Remark that
[Sj′−1vk, ∆j ]∂k∆j′a(x) = 2jd
∫Rdh(2j(x− y))
(Sj′−1v
k(x)− Sj′−1vk(y)
)∂k∆j′a(y) dy.
Hence, according to the mean value formula,
[Sj′−1vk, ∆j ]∂k∆j′a(x)
= 2jd∫Rd
∫ 1
0h(2j(x− y))
((x− y) · ∇Sj′−1v
k(y + τ(x− y)))∂k∆j′a(y) dτ dy.
So finally,
‖[Tvk , ∆j ]∂ka‖Lp . 2−j‖∇v‖L∞∑|j′−j|≤4
‖∂k∆j′a‖Lp . ‖∇v‖L∞∑|j′−j|≤4
‖∆j′a‖Lp .
Hence, given that Bdp1p1,1
↪→ L∞, this term may be bounded according to (20).
Let us resume to (18). Using (19) and (20), multiplying by 2js then summing up overj yields
‖a‖L∞t (Bsp,r)
+ λ‖a‖L1t (B
sp,r)≤ ‖a0‖Bsp,r +
∫ t
0‖f‖Bsp,r dτ + C
∫ t
0V ′‖a‖Bsp,r dτ
with ‖a‖L∞t (Bsp,r)
:=∥∥2js‖∆ja‖L∞t (Lp)
∥∥`r
and ‖a‖L1t (B
sp,r)
:=∥∥2js‖∆ja‖L1
t (Lp)
∥∥`r.
Then applying Gronwall’s lemma yields the desired inequality for a. �
1.7. Estimates for dispersive equations. Let (U(t))t∈R be a group of unitary operatorson L2(Rd) satisfying the dispersion inequality :
(21) ‖U(t)f‖L∞ ≤C
|t|σ‖f‖L1 for some σ > 0.
Interpolating between L2 7→ L2 and L1 7→ L∞, we deduce that
‖U(t)f‖Lp ≤(C
|t|σ
) 1p′−
1p
‖f‖Lp′ for all 2 ≤ p ≤ ∞.
The basic examples are the groups generated by the Schrodinger equation
i∂tu+ ∆u = 0 in Rd
for which σ = d/2, or by the wave equation
∂2ttu−∆u = 0 in Rd
for which σ = (d− 1)/2.
Definition 1.2. A couple (r, p) ∈ [2,∞]2 is admissible if 1/r+ σ/p = σ/2 and (r, p, σ) 6=(2,∞, 1).
Theorem 1.3 (Strichartz estimates). Let (U(t))t∈R satisfy the above hypotheses. Then
(1) For any admissible couple (r, p) we have ‖U(t)u0‖Lr(Lp) . ‖u0‖L2 ;(2) For any admissible couples (r1, p1) and (r2, p2) we have∥∥∥∫ t
0U(t− τ)f(τ) dτ
∥∥∥Lr1 (Lp1 )
. ‖f‖Lr′2 (Lp
′2 )·
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 17
Note that compared to Sobolev embedding Hd( 1
2− 1p
)↪→ Lp, the first Strichartz estimate
provides a gain of d(12 −
1p) = d
rσ derivative when taking Lr norm in time. As for the
relation 1r + σ
p = σ2 , it may be guessed from a dimensional analysis.
Following the approach of Ginibre and Velo in [35], the proof of Strichartz estimatesrelies mainly on two ingredients:
(1) The TT ? argument (see below);(2) The Hardy-Littlewood-Sobolev inequality.
Lemma 1.3 (TT ? argument). Let T : H → B be a bounded operator from the Hilbertspace H to the Banach space B and T ? : B′ → H be the adjoint operator defined by
∀(x, y) ∈ B′ ×H, (T ?x | y)H = 〈x, Ty〉B′,B.
Then we have
‖TT ?‖L(B′;B) = ‖T‖2L(H;B) = ‖T ?‖2L(B′;H).
Proof. We just prove the homogeneous Strichartz inequality away from the endpoints.For that purpose, let us introduce the operator
T : u0 7−→ U(t)u0.
For smooth functions φ, we have
T ? : φ 7−→∫RU(−t′)φ(t′) dt′ and TT ? : φ 7−→
[t 7→
∫RU(t− t′)φ(t′) dt′
].
Hence applying the TT ? argument with
H = L2(Rd), B = Lr(R;Lp(Rd)), B′ = Lr′(R;Lp
′(Rd),
we see that proving ‖Tu0‖Lr(Lp) ≤ C‖u0‖L2 is equivalent to
(22) ‖TT ?φ‖Lr(Lp) ≤ C‖φ‖Lr′ (Lp′ ).
Now, we have
‖TT ?φ(t)‖Lp ≤∫R‖U(t− t′)φ(t′)‖Lp dt.
So taking advantage of the dispersion inequality Lp′ → Lp and of the relation σ( 1
p′−1p) = 2
r ,we get
‖TT ?φ(t)‖Lp ≤∫R
1
|t− t′|2r
‖φ(t′)‖Lp′ dt.
Applying the Hardy-Littlewood-Sobolev inequality gives (22) if 2 < r <∞. �
Remarks:
(1) The endpoint (r, p) = (∞, 2) is given by the fact that (U(t))t∈R is unitary on L2.The endpoint (r, p) = (2, 2σ/(σ−1)) if σ > 1 is more involved (Keel & Tao [43]).
(2) The nonhomogeneous Strichartz inequality follows from similar arguments (see e.g.[2, 35]).
(3) In the case of the linear wave or Schrodinger equation, using (∆j)j∈Z allows to getStrichartz estimates involving Besov norms.
18 R. DANCHIN
As an example we consider the following acoustic wave equation that will play a funda-mental role in the study of the incompressible limit (see the last section):
(23)
{∂tb+ |D|v = F,
∂tv − |D|b = G.
It is well known that (23) is associated to a group U(t) of unitary operators on L2(Rd)which satisfies the dispersion inequality (see e.g. [2], Chap. 8)
‖U(t)(b0, v0)‖L∞ ≤ Ct−d−12 ‖(b0, v0)‖L1 .
Hence Strichartz estimates are available for this system if d ≥ 2. We claim that
Proposition 1.9. Let (b, v) satisfy (23). Then we have for any s ∈ R
‖(b, v)‖LrT (B
s+d( 1p−12 )+1
rp,1 )
. ‖(b0, v0)‖Bs2,1 + ‖(F,G)‖L1T (Bs2,1)
whenever p ≥ 2,2
r≤ min
(1, (d− 1)
(1
2− 1
p
))and (r, p, d) 6= (2,∞, 3).
Proof. Localizing (23) by means of (∆j)j∈Z, we get for all j ∈ Z,{∂t∆jb+ |D|∆jv = ∆jF,
∂t∆jv − |D|∆jb = ∆jG.
Let us first concentrate on the case j = 0. Then applying Theorem 1.3 yields
Finally, multiplying both sides by 2js and performing a summation over j yields the desiredinequality. �
2. Solving the compressible Navier-Stokes equations in critical spaces
This section is devoted to proving local well-posedness results for the compressibleNavier-Stokes equations (1) in critical spaces. As a warm up, we first present the critical(or scaling invariant) spaces approach for the simpler case of incompressible Navier-Stokesequations and establish a global well-posedness result for small data by means of the Banachfixed point theorem (or contracting mapping argument). The rest of this section concerns(1). We first solve it by means of the “standard” approach for nonlinear hyperbolic sys-tems, which amounts to proving first uniform a priori estimates for the “high norm” of thesolution and next stability estimates for a lower norm (this approach may equivalently bereformulated in terms of the Schauder-Tikhonoff fixed point theorem, see e.g. [20]). Next,we recast (1) in Lagrangian coordinates and show that it is possible to solve the obtainedsystem by means of the Banach fixed point theorem.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 19
2.1. The incompressible Navier-Stokes equations. The incompressible Navier-Stokesequations read:
(NS)
{∂tu+ div(u⊗ u)− µ∆u+∇P = 0,
divu = 0.
Here u : [0, T ) × Rd → Rd stands for the velocity field, and P : [0, T ) × Rd → R, forthe pressure. The viscosity µ is a given positive number. If we want to solve the Cauchyproblem for (NS) then we have to prescribe some initial divergence-free velocity field u0.
Introducing the Leray projector over divergence-free vector fields, namely P := Id +∇(−∆)−1div , System (NS) recasts in
∂tu+ Pdiv(u⊗ u)− µ∆u = 0.
This latter equation enters in the class of generalized Navier-Stokes equations:
(GNS) ∂tu+Q(u, u)− µ∆u = 0
with
FQj(u, v)(ξ) :=∑
αj,m,n,pk,`
ξn ξp ξm|ξ|2
F(uk v`)(ξ).
All the coefficients are supposed to be constant. Hence the entries of Q(u, v) are first orderhomogeneous Fourier multipliers applied to bilinear expressions. From the point of viewof homogeneous Besov spaces, the action of such multipliers is exactly the same as that ofthe gradient operator (see subsection 1.2).
We claim that one may find some functional framework in which System (GNS) maybe solved by means of the following abstract lemma:
Lemma 2.1. Let X be a Banach space and B : X ×X → X a continuous bilinear mapwith norm M. Then there exists a unique solution v in B(0, 2‖v0‖X) to
(E) v = v0 + B(v, v)
whenever
(24) 4M‖v0‖X < 1.
Proof. Denoting F : v 7→ v0 + B(v, v), we see that
‖F (v)− v0‖X ≤M‖v‖2X .
Hence if (24) is satisfied and ‖v‖X ≤ 2‖v0‖X then F maps the closed ball B(0, 2‖v0‖X)into itself.
Next, considering v1 and v2 in this closed ball, we see that
Hence Condition (24) ensures that F is strictly contracting, and the existence of a uniquefixed point in B(0, 2‖v0‖X) thus follows from the standard fixed point theorem in completemetric spaces. �
Assume in addition that there exists a one-parameter family (Tλ)λ>0 acting on X andwhich leaves (E) invariant that is:
v = v0 + B(v, v) ⇐⇒ Tλv = Tλv0 + B(Tλv, Tλv) for all λ > 0.
20 R. DANCHIN
Then the smallness condition (24) recasts in
4M‖Tλv0‖X < 1 for all λ > 0.
In other words, either the problem may be solved for any data in X, or the norm in X hasto be invariant (up to an irrelevant constant) by Tλ for all λ. If this latter occurs then Xis called a scaling invariant space for (E).
In the applications, a dimensional analysis often allows to find such a family (Tλ)λ>0.This is the case for instance if considering evolutionary equations such as the nonlinearSchrodinger, wave or heat equations with a power-like nonlinearity. As regards the (gener-alized) Navier-Stokes equations, one may take
(25) Tλv(t, x) := λv(λ2t, λx).
Hence one may tempt to solve (GNS) in spaces X with norm invariant by the abovetransformation. As an example, we prove below an existence statement in scaling invarianthomogeneous Besov spaces5.
Theorem 2.1. Let u0 ∈ Bdp−1
p,r with divu0 = 0. Assume that p is finite. There exist twopositive constants c and C such that if
‖u0‖Bdp−1
p,r
≤ cµ
then (GNS) has a unique global solution u in the space6
X := L∞(R+; Bdp−1
p,r ) ∩ L1(R+; Bdp
+1p,r )
satisfying‖u‖X := ‖u‖
L∞(Bdp−1
p,r )+ µ‖u‖
L1(Bdp+1
p,r )≤ C‖u0‖
Bdp−1
p,r
.
Proof. We want to apply the abstract lemma to X = L∞(R+; Bdp−1
p,r ) ∩ L1(R+; Bdp
+1p,r ),
v0(t) := eµt∆u0 and B(u, v)(t) := −∫ t
0eµ(t−τ)∆Q(u, v) dτ.
Heat estimates (see Theorem 1.1 and Remark 1.2) imply that
‖v0‖X ≤ C‖u0‖Bdp−1
p,r
.
That B : X ×X → X is a consequence of embedding properties in Besov spaces, and ofcontinuity results for paraproduct and remainder (see Proposition 1.6). Indeed we have
‖Q(u, v)‖L1(B
dp−1
p,r )≤ C‖u⊗ v‖
L1(Bdpp,r).
Hence, using the fact that7
‖R(u, v)‖L1(B
dpp,r)
. ‖u‖L∞(B
dp−1
p,r )‖v‖
L∞(Bdp+1
p,r )
‖Tuv‖L1(B
dpp,r)
. ‖u‖L∞(B−1
∞,∞)‖v‖
L∞(Bdp+1
p,r )
5Proposition 1.4 ensures that the norms in Theorem 2.1 are indeed invariant by (25).6see the definition of tilde norms in Remark 1.2.7The product laws for Lρ(Bσp,r) work the same as the usual ones, the time Lebesgue exponent just
behaves according to Holder inequality. Note that for the remainder we need that (d/p− 1) + (d/p+ 1) > 0hence p <∞.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 21
and a similar inequality for Tvu, and because Bdp−1
p,r ↪→ B−1∞,∞, we eventually find that, for
some C = C(d, p,Q):
‖Q(u, v)‖L1(B
dp−1
p,r )≤ Cµ−1‖u‖X‖v‖X .
Hence
‖B(u, v)‖X ≤ C ′µ−1‖u‖X‖v‖X .Therefore B satisfies (24) provided ‖u0‖
Bdp−1
p,r
≤ cµ with c small enough. �
2.2. A first approach to the local existence theory. This paragraph is dedicated tosolving (1) locally in time, in critical spaces. To simplify the presentation, we focus on smallperturbations of the constant density state ρ = 1. Hence, setting ρ = 1 + a, λ = λ(1),µ = µ(1) and A = µ∆ + (λ+ µ)∇div , System (1) rewrites
Note that up to a change of the pressure law, the barotropic compressible Navier-Stokesequations are invariant by the rescaling
a(t, x)→ a(λ2t, λx), u(t, x)→ λu(λ2t, λx).
In the homogeneous Besov spaces scale, this induces to take
a0 ∈ Bdp1p1,r1 and u0 ∈ B
dp2−1
p2,r2 .
It is not clear that one may solve (26) in such spaces for any choice of p1, p2, r1, r2.Indeed, first, in order to preclude vacuum and keep ellipticity of the second order operator
in the velocity equation, an L∞ control for a is needed. In the scale of Besov spaces Bdp1p1,r1 ,
having r1 = 1 is the only choice which ensures (continuous) inclusion in L∞. As for thevelocity equation, a gain of two derivatives is required to handle the term J(a)Au (as Ais second order). According to Corollary 1.2, we thus have to take r2 = 1. This is all the
more appropriate that this will ensure ∇u to be in L1T (B
dp2p2,1
), a property that is needed
to transport the initial Besov regularity of a (see Theorem 1.2). Finally, owing to thecoupling between the mass and velocity equations, we take p1 = p2 = p for simplicity.
So, in short, we want to investigate the well-posedness issue of (26) for data
a0 ∈ Bdp
p,1 and u0 ∈ Bdp−1
p,1 .
According to Corollary 1.2 and to Theorem 1.2, we expect a to be in C([0, T ]; Bdp
p,1), andu to be in the space
Ep(T ) :={u ∈ C([0, T ]; B
dp−1
p,1 ), ∂tu,∇2u ∈ L1(0, T ; Bdp−1
p,1 )}·
22 R. DANCHIN
We shall endow Ep(T ) with the norm
‖u‖Ep(T ) := ‖u‖L∞T (B
dp−1
p,1 )+ ‖∂tu,∇2u‖
L1T (B
dp−1
p,1 ).
Let us now state our first local well-posedness existence result in critical spaces.
Theorem 2.2. Assume that a0 ∈ Bdp
p,1 and that u0 ∈ Bdp−1
p,1 with 1 ≤ p < 2d. There exists
a constant c = c(p, d,G) such that if
(27) ‖a0‖Bdpp,1
≤ c
then (26) has a local-in-time solution (a, u) with a in C([0, T ]; Bdp
p,1) and u in Ep(T ).Uniqueness holds true if p ≤ d.
The proof mostly relies on the estimates for the transport equation and for the Lamesystem that have been presented in the first section. The important fact is that, if onerestricts oneself to local-in-time results for the Cauchy problem, then one may just considerthe density and velocity equations independently.
Before going further into the proof of existence however, let us emphasize that onecannot expect to reduce System (26) to the model problem of Lemma 2.1. This is due tothe hyperbolic nature of the transport equation which entails a loss of one derivative in theLipschitz-type stability estimates. Hence, existence will rather stem from bounds in highnorm for the solution and stability in low norms or, alternately, from Schauder-Tikhonovtype fixed point arguments.
2.2.1. Uniform estimates in large norm. We assume that (26) with data a0 ∈ Bdp
p,1 and
u0 ∈ Bdp−1
p,1 has a solution (a, u) such that
a ∈ C([0, T ]; Bdp
p,1) and u ∈ C([0, T ]; Bdp−1
p,1 ) ∩ L1([0, T ]; Bdp
+1
p,1 ).
We claim that if a0 satisfies (27) then this solution may be bounded in terms of the data.
Let U(T ) := ‖∇u‖L1T (B
dpp,1). Estimates for the transport equation imply that
‖a‖L∞T (B
dpp,1)≤ eCU(T )
(‖a0‖
Bdpp,1
+
∫ T
0e−CU‖(1 + a)divu‖
Bdpp,1
dτ
).
From product laws in Besov spaces, we have:
‖(1 + a)divu‖Bdpp,1
. (1 + ‖a‖Bdpp,1
)‖∇u‖Bdpp,1
.
Inserting this in the above inequality and applying Gronwall’s lemma, we thus get
‖a‖L∞T (B
dpp,1)≤ eCU(T )‖a0‖
Bdpp,1
+ eCU(T ) − 1.
Hence, if eCU(T ) − 1 ≤ η for some η ∈ (0, 1] then
‖a‖L∞T (B
dpp,1)≤ 2‖a0‖
Bdpp,1
+ η.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 23
Let us now prove estimates for the velocity. From Corollary 1.2, we get
‖u‖L∞T (B
dp−1
p,1 )∩L1T (B
dp+1
p,1 ). ‖u0‖
Bdp−1
p,1
+
∫ T
0‖u · ∇u+∇(G(a))‖
Bdp−1
p,1
dt
+
∫ T
0‖I(a)(Du+∇u) · ∇a+ J(a)divu∇a+K(a)∆u+ L(a)∇divu‖
Bdp−1
p,1
dt.
Product and composition laws in Besov spaces yield if d > 1 and 1 ≤ p < 2d,
‖u · ∇u‖Bdp−1
p,1
. ‖u‖Bdp−1
p,1
‖∇u‖Bdpp,1
,
‖K(a)∆u‖Bdp−1
p,1
+ ‖L(a)∇divu‖Bdp−1
p,1
. ‖a‖Bdpp,1
‖∇2u‖Bdp−1
p,1
,
‖I(a)(Du+∇u) · ∇a+ J(a)divu∇a‖Bdp−1
p,1
. (1 + ‖a‖Bdpp,1
)‖∇u‖Bdpp,1
‖∇a‖Bdp−1
p,1
,
‖∇(G(a))‖Bdp−1
p,1
. ‖a‖Bdpp,1
.
Hence
‖u‖L∞T (B
dp−1
p,1 )+ ‖u‖
L1T (B
dp+1
p,1 ). ‖u0‖
Bdp−1
p,1
+
∫ T
0‖u‖
Bdp−1
p,1
‖u‖Bdp+1
p,1
dt+ ‖a‖L∞T (B
dpp,1)‖u‖
L1T (B
dp+1
p,1 )+ T‖a‖
L∞T (Bdpp,1).
The last-but-one term may be absorbed by the left hand-side if ‖a‖L∞T (B
dpp,1)
is small. Ac-
cording to (27), this may be ensured if ‖a0‖Bdpp,1
and η are small enough. Note also that
applying Gronwall lemma shows that the term with the integral may be eliminated if U(T )is small enough. From this, we conclude that
‖u‖L∞T (B
dp−1
p,1 )+ ‖u‖
L1T (B
dp+1
p,1 )≤ C
(‖u0‖
Bdp−1
p,1
+ T (‖a0‖Bdpp,1
+ η)).
It is clear that U(T ) tends to 0 for T going to 0. However, in order to implement aniterative scheme for solving (26) and get a fixed time interval on which all the terms of thesequences satisfy the above estimates, it is suitable to exhibit a lower bound for T in termsof the data. To achieve it, one may split u into uL + u with uL solution to
∂tuL −AuL = 0, uL|t=0 = u0.
We have
U(T ) ≤ ‖uL‖L1T (B
dp+1
p,1 )+ ‖u‖
L1T (B
dp+1
p,1 ).
The first term goes to 0 for T tending to 0 with a decay that may be described accordingto (15). We expect the second term to be small for T small as u(0) = 0. In order to get amore accurate information, one may use the fact that u satisfies
∂tu−Au = −u · ∇u− uL · ∇u− uL · ∇uL −∇(G(a))
+I(a)(Du+∇u) · ∇a+ J(a)divu∇a+K(a)∆u+ L(a)∇divu.
24 R. DANCHIN
By combining Corollary 1.2 and the product laws in Besov spaces, we get
(28) ‖u‖L1T (B
dp+1
p,1 )+ ‖u‖
L∞T (Bdp−1
p,1 ).∫ T
0‖u‖
Bdp−1
p,1
‖u‖Bdp+1
p,1
dτ
+ ‖uL‖L2T (B
dpp,1)‖u‖
L2T (B
dpp,1)
+ ‖uL‖L1T (B
dp+1
p,1 )‖uL‖
L∞T (Bdp−1
p,1 )
+ ‖a‖L∞T (B
dpp,1)‖u‖
L1T (B
dp−1
p,1 )+ T‖a‖
L∞T (Bdpp,1).
Arguing by interpolation yields for any β > 0,
‖uL‖L2T (B
dpp,1)‖u‖
L2T (B
dpp,1)≤ β‖uL‖
L∞T (Bdp−1
p,1 )‖u‖
L1T (B
dp+1
p,1 )+ Cβ−1‖uL‖
L1T (B
dp+1
p,1 )‖u‖
L∞T (Bdp−1
p,1 ).
Note that ‖uL‖L∞T (B
dp−1
p,1 )≤ C‖u0‖
Bdp−1
p,1
. Therefore, taking β small enough, using Gronwall
lemma and (27), we conclude by a standard bootstrap argument that the l.h.s. of (28) maybe made smaller than any given ε for all t ∈ [0, T ] if, for some α (depending on ε and‖u0‖
Bdp−1
p,1
) we have
max(T, ‖uL‖L1T (B
dp+1
p,1 )) ≤ α.
Hence, according to (15), it suffices to choose T ∈]0, α] so that∑j
(1− e−c22jT
)2j( dp−1)‖∆ju0‖Lp . α.
This gives a (non so) explicit lower bound for the time interval on which the norm of thesolution (a, u) may be bounded in terms of the initial data.
2.2.2. Stability estimates in small norm. Consider two solutions (a1, u1) and (a2, u2) of(26) with the above regularity. The difference (δa, δu) := (a2−a1, u2−u1) satisfies
Owing to the hyperbolic nature of the mass equation, one loses one derivative in the stabilityestimates: indeed, δF1 has at most the same regularity as ∇a1 . This induces also a lossof one derivative for δu (look at δG1 for instance). Hence, we expect to be able to provestability estimates only in
FT := C([0, T ]; Bdp−1
p,1 )×(C([0, T ]; B
dp−2
p,1 ) ∩ L1T (B
dp
p,1))d.
The most unpleasant effect of this loss of one derivative is that when applying compositionand product laws in Besov spaces for bounding the norms of δFi and δGj , one has to imposestronger conditions on p and on d. Indeed, Proposition 1.6 applied to our situation forcesus to assume that
d > 2 and 1 ≤ p < d
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 25
in which case, after a few computations, we conclude that if T and ‖a1‖L∞T (B
dpp,1)
are small
enough then we have the following stability estimate:
‖δa‖L∞T (B
dp−1
p,1 )+ ‖δu‖
L∞T (Bdp−2
p,1 )+ ‖δu‖
L1T (B
dpp,1). ‖δa(0)‖
Bdp−1
p,1
+ ‖δu(0)‖Bdp−2
p,1
,
which implies uniqueness.
The limit case d = 2 or p = d is more involved as, for instance,
(30)(δa ∈ B0
d,1 and Au2 ∈ B0d,1
)=⇒ δaAu2 ∈ B−1
d,∞ only.
Hence estimates have to be performed in a Besov space with third index ∞ . On the onehand, this is not a trouble for δa as applying Theorem 1.2 and product laws implies
‖δa‖L∞T (B0d,∞) ≤
(‖δa(0)‖B0
d,∞+ (1 + ‖a1‖L∞T (B1
d,1))‖δu‖L1T (B1
d,1)
)e‖u2‖
L1T(B2d,1
).
On the other hand, owing to (30), we need to generalize Corollary 1.2 to Besov spaces withthird index ∞, similarly as in Remark 1.2. We can thus expect to get some control on thefollowing quantity:
‖δu‖L1T (B1
d,∞):= sup
j2j‖∆jδu‖L1
T (Ld),
which is sligthly weaker than ‖δu‖L1T (B1
d,1) . At this point, one has to resort to the following
logarithmic interpolation inequality (see [19]):
(31) ‖δu‖L1T (B1
d,1) . ‖δu‖L1T (B1
d,∞)log
(e+‖δu‖
L1T (B0
d,∞)+ ‖δu‖
L1T (B2
d,∞)
‖δu‖L1T (B1
d,∞)
),
which plugged in the estimate for δa and combined with Osgood lemma (see e.g. [2], Chap.3) eventually yields
‖δa‖L∞t (B0d,∞)+‖δu‖
L∞t (B−1d,∞)∩L1
t (B1d,∞).(‖δa(0)‖B0
d,∞+‖δu(0)‖B−1
d,∞
)exp(−∫ t0 αdτ)
where α is in L1(0, T ) and depends only on the high norms of the two solutions.
From those a priori estimates, it is not difficult to construct an iterative scheme forproving rigorously the local-in-time existence of a solution. We get a sequence of smoothsolutions which satisfies (uniformly) the bounds in high norm of the first step, on somefixed time interval. Then resorting to compactness arguments and to the Fatou propertyfor Besov spaces allows to prove that this sequence satisfies (26) and belongs to the requiredspace. In the case p ∈ [1, d], uniqueness follows from the above stability estimates in smallnorm.
We do not give more details here for we shall see another method in the next paragraphto avoid this loss of regularity, and get uniqueness for the full range p ∈ [1, 2d[.
2.3. A Lagrangian approach for the compressible Navier-Stokes equations. Wehere aim at solving System (1) in the Lagrangian coordinates. The main motivation is thatthe mass is constant along the flow hence only the (parabolic type) equation for the velocityhas to be considered. Consequently, we expect to be able to solve the well-posedness issueby means of the Banach fixed point theorem and the flow map will thus be Lipschitzcontinuous.
26 R. DANCHIN
To start with, let us derive the Lagrangian equations corresponding to (1). To simplifythe presentation we assume λ and µ to be constant.
We need first to recall a few algebraic relations involving changes of coordinates. Forthe time being, fix some C1 -diffeomorphism X over Rd. For H : Rd → Rm, we setH(y) = H(x) with x = X(y). With this convention, the chain rule writes
(32) DyH(y) = DxH(X(y)) ·DyX(y) with (DyX)ij = ∂yjXi.
The following lemma proposes an alternate formula for the divergence and gradient oper-ators in Lagrangian coordinates.
Lemma 2.2. Let H be a vector-field over Rd, and K be a scalar function over Rd. Thenthe following relations hold true8:
∇xK = J−1 divy (adj (DyX)K),(34)
divxH = J−1 divy (adj (DyX)H)(35)
where J(y) := | det(DyX(y))|, and adj (DyX) stands for the adjugate of DyX, that is thetranspose of the cofactor matrix of DyX.
Proof. Proving the first item relies on the following series of computations (based on inte-grations by parts and (33)) which hold for any vector-field φ with coefficients in9 C∞c (Rd):∫
∇xK(x) · φ(x) dx = −∫K(x)divxφ(x) dx
= −∫K(X(y))divyφ(y)J(y) dy
= −∫J(y)K(y)Dyφ(y) : A(y) dy
=
∫φ(y) · divy(adj (DX)K)(y) dy
=
∫φ(x)J−1(X−1(x))divy(adj (DyX)K)(X−1(x)) dx.
Proving the second item is similar. �
Let (ρ, u) be a solution of (1). In order to derive the compressible Navier-Stokes equa-tions in Lagrangian coordinates, we introduce the flow X associated to the vector-field u,that is the solution to
(36) X(t, y) = y +
∫ t
0u(τ,X(τ, y)) dτ.
In Lagrangian coordinates, the density and velocity field are defined by
The two fundamental observations are that Jρ = ρ0 and that we may forget any referenceto the initial Eulerian vector-field u in the equations because (36) implies that
(37) X(t, y) = y +
∫ t
0u(τ, y) dτ.
So finally the Lagrangian Navier-Stokes equations read
ρ0∂tu− µdiv(adj (DX)TA∇u
)− (λ+ µ)div
(adj (DX)(TA : ∇u)
)(38)
+div(adj (DX)P (ρ)
)= 0
with ρ := J−1ρ0, J := |detDX|, A := (DX)−1 and X defined by (37).
Theorem 2.3. Let p ∈ [1,∞) and u0 be a vector-field in Bdp−1
p,1 . Assume that the initial
density ρ0 satisfies a0 := (ρ0− 1) ∈ Bdp
p,1 ∩M(Bdp−1
p,1 ). There exists a constant c dependingonly on p and on d such that if
(39) ‖a0‖M(B
dp−1
p,1 )≤ c
then System (37),(38) has a unique local solution (ρ, u) with (a, u) ∈ C([0, T ]; Bdp
p,1)×Ep(T ).
Moreover, the flow map (a0, u0) 7−→ (a, u) is Lipschitz continuous from Bdp
p,1 × Bdp−1
p,1 to
C([0, T ]; Bdp
p,1)× Ep(T ).
In Eulerian coordinates, this yields the following improvement of Theorem 2.2:
Theorem 2.4. Under the above assumptions with 1 ≤ p < 2d, System (1) has a unique
local solution (ρ, u) with a ∈ C([0, T ]; Bdp−1
p,1 ) and u ∈ Ep(T ).
In order to reformulate the system in terms of a fixed point problem, we write Equation(38) as follows10:
)with J = detDX, A = (DX)−1 and X defined in (37).
10In the rest of this paragraph we drop the “bars” over ρ and u .
28 R. DANCHIN
The left-hand side of the above equation is the linear Lame system with constant co-efficients that has been considered in Corollary 1.2. This motivates our defining the mapΦ : v 7→ u where u stands for the solution to
and where Xv is the flow of v, Av = (DXv)−1 and Jv = detDXv.
Note that any fixed point of Φ is a solution to (40). The existence of a fixed point forΦ seen as a self-map of Ep(T ) will be guaranteed by the Banach fixed point theorem, aswe shall see below.
First step: estimates for I1, I2, I3 and I4 . Assume that v is in Ep(T ) and that for asmall enough constant c,
(42)
∫ T
0‖Dv‖
Bdpp,1
dt ≤ c.
We claim that all the terms of the right-hand side of (41) belong to L1(0, T ; Bdp−1
p,1 ).
Regarding I1(w), this readily stems from the definition of multiplier spaces which yields
(43) ‖I1(w)‖L1T (B
dp−1
p,1 )≤ ‖a0‖
M(Bdp−1
p,1 )‖∂tw‖
L1T (B
dp−1
p,1 ).
Next, thanks to product laws, to Lemma A.1 and to (42), we have
(44) ‖I2(v, w)‖L1T (B
dpp,1)
+ ‖I3(v, w)‖L1T (B
dpp,1)≤ C‖Dv‖
L1T (B
dpp,1)‖Dw‖
L1T (B
dpp,1).
For the pressure term (that is I4(v)), we use the fact that under assumption (42), we have,by virtue of Proposition 1.7 and of flow estimates (see (92) and (95)),
(45) ‖I4(v)‖L∞T (B
dpp,1)≤ C
(1 + ‖Dv‖
L1T (B
dpp,1)
)(1 + ‖a0‖
Bdpp,1
).
Second step: Φ maps a suitable closed ball in itself. At this stage, one may assert that if
v ∈ Ep(T ) satisfies (42) then the right-hand side of (41) belongs to L1(0, T ; Bdp−1
p,1 ). Hence
Corollary 1.2 implies that Φ(v) is well defined and maps Ep(T ) to itself. However it is notclear that it is contractive over the whole set Ep(T ). So, as in the previous paragraph, weintroduce the solution uL to
∂tuL − µ∆uL − (λ+ µ)∇divuL = 0, uL|t=0 = u0.
Of course, Corollary 1.2 guarantees that uL belongs to Ep(T ) for all T > 0. In particular,if T and R are small enough then any vector-field in BEp(T )(uL, R) satisfies (42).
We claim that if T is small enough (a condition that will be expressed in terms of the freesolution uL ) and if R is small enough (a condition that will depend only on the viscositycoefficients and on p, d and P ) then
v ∈ BEp(T )(uL, R) =⇒ u ∈ BEp(T )(uL, R).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 29
Indeed u := u− uL satisfies{∂tu− µ∆u− (λ+ µ)∇div u = I1(v) + µdivI2(v, v) + λdivI3(v, v)− divI4(v),
u|t=0 = 0.
So Corollary 1.2 yields11
‖u‖Ep(T ) . ‖I1(v)‖L1T (B
dp−1
p,1 )+ ‖I2(v, v)‖
L1T (B
dpp,1)
+ ‖I3(v, v)‖L1T (B
dpp,1)
+ T‖I4(v)‖L∞T (B
dpp,1).
Inserting inequalities (43), (44) and (45), we thus get:
‖u‖Ep(T ) . ‖Dv‖2L1T (B
dpp,1)
+ ‖a0‖M(B
dp−1
p,1 )‖∂tv‖
L1T (B
dp−1
p,1 )+ T (1 + ‖a0‖
Bdpp,1
).
That is, keeping in mind that v is in BEp(T )(uL, R),
‖u‖Ep(T ) ≤ C(‖a0‖
M(Bdp−1
p,1 )(R+ ‖∂tuL‖
L1T (B
dp−1
p,1 ))
+‖DuL‖2L1T (B
dpp,1)
+R2 + T (1 + ‖a0‖Bdpp,1
)).
So we see that if T satisfies
(46) CT (1 + ‖a0‖Bdpp,1
) ≤ R/2 and ‖DuL‖L1T (B
dpp,1)
+ ‖∂tuL‖L1T (B
dp−1
p,1 )≤ R
then we have
‖u‖Ep(T ) ≤ 2C‖a0‖M(B
dp−1
p,1 )R+ 2CR2 +R/2.
Hence there exists a small constant η = η(d, p) such that if
(47) ‖a0‖M(B
dp−1
p,1 )≤ η,
if R has been chosen small enough and if T satisfies (46) then u is in BEp(T )(uL, R) and
(42) is fulfilled for all vector-field of BEp(T )(uL, R).
Third step: contraction properties. We claim that under Conditions (46) and (47) with asmaller R if needed, the map Φ is 1/2-Lipschitz over BEp(T )(uL, R). Indeed, let v1 and
v2 be in BEp(T )(uL, R) and denote
u1 := Φ(v1) and u2 := Φ(v2).
Let X1 and X2 be the flows associated to v1 and v2. Set Ai = (DXi)−1 and Ji :=
detDXi for i = 1, 2. The equation satisfied by δu := u2 − u1 reads
Bounding δf3 works exactly the same. So we see that if Conditions (46) and (47) aresatisfied (with smaller η and larger C if need be) then we have
‖δu‖Ep(T ) ≤1
2‖δv‖Ep(T ).
Hence, the map Φ : BEp(T )(uL, R) 7→ BEp(T )(uL, R) is 1/2-Lipschitz and Banach fixed
point theorem thus ensures that it admits a unique fixed point in BEp(T )(uL, R). Thiscompletes the proof of existence of a solution in Ep(T ) for System (38).
Fourth step: Stability estimates in Ep(T ). We aim at proving stability estimates in Ep(T )for the solutions to (38). This will give both uniqueness in Ep(T ) and that the flow mapis Lipschitz.
So we consider two initial velocity fields u0,1 and u0,2 in Bdp−1
p,1 , and densities ρ0,1 and
ρ0,2 such that a0,1 and a0,2 are in Bdp
p,1 and that (39) is fulfilled.
We want to compare the solutions u1 and u2 in Ep(T ) of System (38), correspondingto data (ρ0,1, u0,1) and (ρ0,2, u0,2).
The proof being similar to that of the contractivity of Φ, we skip some details. Onceagain, we have to bound ‖δu‖Ep(t) in terms of ‖δu0‖
Bdp−1
p,1
and ‖δρ0‖M(B
dp−1
p,1 )where δu :=
u2 − u1, δu0 := u0,2 − u0,1 and δρ0 := ρ0,2 − ρ0,1. The equation satisfied by δu now reads
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 31
Using Corollary 1.2, we gather that for all t ∈ [0, T )
‖δu‖Ep(t) . ‖δu0‖Bdp−1p,1
+1∑i=0
‖δfi‖L1t (B
dp−1p,1 )
+ t‖δf2‖L∞t (B
dpp,1)
+4∑i=3
‖δfi‖L1t (B
dpp,1).
Fix some T0 ∈ (0, T ] so that (42) is satisfied by both solutions on [0, T0]. From the definition
of the multiplier space M(Bdp−1
p,1 ), we readily have
‖δf0‖L1T0
(Bdp−1
p,1 )≤ ‖δρ0‖
M(Bdp−1
p,1 )‖∂tu1‖
L1T0
(Bdp−1
p,1 ).
The terms δf1, δf3 and δf4 may be bounded exactly as in the previous paragraph. As forδf2, we may write
δf2 =(adj (DX1)P (ρ0,1J
−11 )−adj (DX2)P (ρ0,1J
−12 ))+adj (DX2)
(P (ρ0,1J
−12 )−P (ρ0,2J
−12 ))
and argue as above.
We eventually conclude that for all t ∈ [0, T0], we have
‖δu‖Ep(t) . ‖δu0‖Bdp−1
p,1
+ ‖δρ0‖M(B
dp−1
p,1 )‖∂tu1‖
L1t (B
dp−1
p,1 )+ ‖(Du1, Du2)‖
L1t (B
dpp,1)‖Dδu‖
L1t (B
dpp,1)
+t‖δa0‖Bdpp,1
+(t(1 + ‖a0‖
Bdpp,1
) + ‖Du2‖L1t (B
dpp,1)
)‖Dδu‖
L1t (B
dpp,1).
Therefore
‖δu‖Ep(t) ≤ C(‖δa0‖
Bdpp,1∩M(B
dp−1
p,1 )+ ‖δu0‖
Bdp−1
p,1
)for all t ∈ [0, T0]
if T0(1 + ‖a0‖Bdpp,1
) is small enough and for i = 1, 2, and a small constant η
supt∈[0,T0]
‖ui(t)‖Bdp−1
p,1
+
∫ T0
0
(‖∂tui‖
Bdp−1
p,1
+ ‖Dui‖Bdpp,1
)dt ≤ η.
This completes the proof of stability estimates for the velocity on a small enough timeinterval. Extending them to the whole initial interval [0, T ] may be done by repeating theargument.
In order to get the Lipschitz continuity of the (Lagrangian) density with respect to theinitial data, we just have to notice that
a2 − a1 = (a0,2 − a0,1)J−12 + a0,1(J−1
2 − J−11 ).
So using (95) and (101) ensures that for all t ∈ [0, T ],
‖(a2 − a1)(t)‖Bdpp,1
. ‖δa0‖Bdpp,1
+ ‖a0,1‖Bdpp,1
‖δu‖L1t (B
dpp,1).
Proof of Theorem 2.4. For data (ρ0, u0) satisfying the assumptions of Theorem 2.4, one
may construct a local solution (u,∇P ) to System (38) in C([0, T ]; Bdp
p,1) × Ep(T ). If Xu
denotes the “flow” to u which is defined according to (89) then the results of the appendixensure that, for all t ∈ [0, T ], Xu(t, ·) is a C1 diffeomorphism of Rd. In particular, onemay set
u (t, ·),and the algebraic relations that are derived in the appendix show that (ρ, u) satisfies
System (1). In addition, given that DXu(t)− Id belongs to Bdp
p,1, the map z 7→ z ◦X±1u (t)
32 R. DANCHIN
is continuous on Bdp
p,1 and on Bdp−1
p,1 (see Proposition 1.8). Therefore the Eulerian velocity
u is in Ep(T ) (we have to use that the chain rule also ensures that ∇u = (TAu ·∇u)◦X−1u )
provided that 1 ≤ p < 2d, and the Eulerian density ρ is in C([0, T ]; Bdp
p,1).
In order to prove uniqueness, we consider two solutions (ρ1, u1) and (ρ2, u2) correspond-ing to the same data (ρ0, u0), and perform the Lagrangian change of variable pertainingto the flow of u1 and u2 respectively. The obtained vector-fields u1 and u2 are in Ep(T )and both satisfy (38) with the same ρ0 and u0. Hence they coincide, as a consequence ofthe uniqueness part of Theorem 2.3.
2.4. References and remarks. For the incompressible Navier-Stokes equations (NS) inbounded domains the idea of combining the abstract lemma (based on the contractingmapping argument) with dimensional analysis and semi-group theory has been initiatedby H. Fujita and T. Kato in [30]. It has been adapted to the whole space setting by J.-Y.Chemin in [7]. There
X = C(R+; Hd2−1) ∩ L2(R+; H
d2 )
and the initial data is in the homogeneous Sobolev space Hd2−1
There are a number of critical functional spaces in which (NS) may be globally solvedfor small data, for instance:
• the space C(R+;Ld) (see Giga [34], Kato [42], Furioli-Lemarie-Terraneo [32]);
• the space C(R+; Bdp−1
p,1 )∩L1(R+; Bdp
+1
p,1 ) with p <∞ and more general Besov spaces
(see the works by Cannone-Meyer-Planchon in [5] and by H. Kozono and M. Ya-mazaki in [44]). Theorem 2.1 follows from the work by J.-Y. Chemin in [8].
Let us emphasize that the method that we presented here (that combines the abstractlemma, scaling considerations and product laws in Besov spaces) does not use the verystructure of the nonlinearity: it applies indistinctly to any Generalized Navier-Stokes Equa-tions (GNS), even to those which have no physical meaning and that do not possess anyconserved quantity. There are examples of such equations for which global well-posednesshold true for any data (this is the case of the viscous multi-dimensional Burgers equationbecause the maximum principle applies), and other examples for which finite time blow-upoccurs for some large data (see e.g. [49]). Therefore one cannot expect from this methodmuch more than global existence for small data, or local existence for large data.
The mathematical study of viscous compressible flows has a long history. The existenceof local-in-time classical solutions for the full Navier-Stokes equations has been establishedby J. Nash in 1962 [51]. Uniqueness was proved a few years before by J. Serrin in [54] (seealso [41]), and the bounded or unbounded domain case has been considered later by A.Tani in [56].
A completely different approach, based on the (formal) conservation of energy and onthe ideas of the pioneering work by J. Leray [46] for (NS) has been proposed by P.-L.Lions in the nineties [47] (see also the work by E. Feireisl in [28]). The fact that theconstructed solutions are the same as Nash’ in the case of smooth data with no vacuumhas been established recently by E. Feireisl and A. Novotny in [29]. In between these twoapproaches, D. Hoff established the existence of global solutions with discontinuous densityand velocity with (low) Sobolev regularity (see [38]). The construction has been improvedin [39] so as to offer a description of the evolution of the discontinuity curves of the density.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 33
In this section and, more generally, in these notes we concentrated on the well-posednessissue for critical regularity data, a study that has been initiated by the author in [15]. Thelocal existence result of Theorem 2.2 has been first established therein (together with asimilar result for the full Navier-Stokes system), under some smallness condition over a0 .That condition may be replaced by the nonvacuum assumption 1+a0 > 0 (see the paper [21]by the author in the case p = 2 and the work by Chen-Miao-Zhang in [9] for more generalp and density-dependent viscosity coefficients). Finally, it has been observed recently byB. Haspot [36] that the Lebesgue exponents for a and u may be taken different. Writingout the equation for the effective velocity (see the next section) is the key to this latterresult. The nonvacuum assumption may be weakened for smooth enough data satisfyingsome compatibility conditions (see [13]). Whether data with critical regularity may beconsidered in this context is an open question, though.
Using the Lagrangian coordinates in order to solve the Cauchy problem for (26) hasbeen used in the seminal work by J. Nash [51] to establish the local-in-time existence ofclassical solutions. It has been used again more recently in a series of papers by A. Valliand V. Zajaczkowski (see e.g. [58, 59]) for rougher data. Adapting this approach to thecritical regularity framework has been done recently by the author in [23], and extendedto the full Navier-Stokes equations in [12]. As already pointed out, back to the Eulerianframework, this allows to get uniqueness for the full range 1 ≤ p < 2d (the conditionp < 2d being a consequence of the product laws in Besov spaces). Hence the requiredregularity for uniqueness is somewhat weaker than that of P. Germain in [33], B. Haspotin [37] and D. Hoff in [40]. Even though the flow map is Lipschitz continuous for theLagrangian equations, we do not expect this to be true for the Eulerian formulation, as themass equation is of hyperbolic type. Results in that direction have been proved recentlyby Chen-Miao-Zhang in [11], if p > 2d.
The smallness condition over a0 in Theorem 2.3 is weaker than that of Theorem 2.2.In these notes we assumed (39) for simplicity but similar results hold true whenever theinitial density is bounded away from 0. In addition, the viscosity coefficients may dependsmoothly on the density (see [23])). As in [37], introducing the effective velocity shouldallow to extend Theorem 2.3 or 2.4 to different Lebesgue exponents for the density and thevelocity.
Finally, let us point out that using Lagrangian coordinates proved to be also adapted tothe study of inhomogeneous incompressible Navier-Stokes equations. In fact, for this latter
system, it suffices to assume that a0 is in the multiplier space M(Bdp−1
p,1 ) which contents
piecewise constant functions across C1 surfaces (see [26] for more details). We do not knowhow to consider such data with our method for the compressible Navier-Stokes equations,the main obstacle being the explicit dependency of the pressure on the density.
We end this section with a few comments concerning the global well-posedness issue.Regardless of the approach presented in this section (Lagrangian or Eulerian coordinates),proving global well-posedness just by combining estimates for the transport equation andfor the Lame system is hopeless. This may be seen at the level of the a priori estimatesfor (26) that we obtained hitherto. The problem is that estimates for the transport equa-tion naturally involve quantities such as ‖a‖
L∞T (Bdpp,1)
whereas ‖a‖L1T (B
dpp,1)
is needed when
bounding u by means of parabolic estimates. So far, we used that
‖∇(G(a))‖L1T (B
dp−1
p,1 ). T‖a‖
L∞T (Bdpp,1),
34 R. DANCHIN
which is not so clever when T goes to infinity. In particular, by such rough an estimate onecannot take advantage of any monotonicity property of the pressure law. The diagnosticis clear : while the pressure term may be omitted in the linear analysis leading to local-in-time existence results, it has to be included in the linear analysis for the global existencetheory. Performing this linear analysis is one of the main purposes of the next section.
3. Partially parabolic or dissipative linear PDEs
This section is devoted to proving global-in-time a priori estimates for constant coeffi-cients systems having both first order symmetric terms and parabolic or dissipative termsacting on some components of the solution. In the context of compressible barotropic flu-ids, the main motivation for this study is that the linearized system about (0, 0) has thisstructure. For this particular system, in the first subsection, we shall obtain the desiredestimates by means of an explicit computation.
In the applications however (e.g. linearizations of more elaborated compressible models),it is not always possible to compute the solution explicitly. This motivates our presenting amore robust method which is mostly based on introducing a suitable Lyapunov functionalwhich captures the decay properties of the system.
3.1. A direct analysis. The linearized barotropic compressible Navier-Stokes equationsabout the state (a, u) = (0, 0) read
(52)
{∂ta+ divu = 0,
∂tu− µ∆u− (λ+ µ)∇divu+ α∇a = 0with α := P ′(1).
Applying operators P and Q to the second equation and setting ν := λ + 2µ , System(52) translates into
(53)
∂ta+ divQu = 0,
∂tQu− ν∆Qu+ α∇a = 0,
∂tPu− µ∆Pu = 0.
In the homogeneous Besov spaces setting, it is equivalent to bound Qu or v := |D|−1divu ,the advantage of the latter function being that it is real valued. So we are led to considering
∂ta+ |D|v = 0,
∂tv − ν∆v − α|D|a = 0,
∂tPu− µ∆Pu = 0.
Note that the vorticity part of the velocity field Pu satisfies a mere heat equation withconstant diffusion, the property of which is well described by Theorem 1.1. So we have toconcentrate on the first two equations, namely
(54)
{∂ta+ |D|v = 0,
∂tv − ν∆v − α|D|a = 0.
Taking the Fourier transform with respect to the space variable yields
d
dt
(av
)= A(ξ)
(av
)with A(ξ) :=
(0 −|ξ|α|ξ| −ν|ξ|2
).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 35
The characteristic polynomial of A(ξ) is X2 + ν|ξ|2X +α|ξ|2 , the discriminant of which is
δ(ξ) := |ξ|2(ν2|ξ|2 − 4α).
If α < 0 then there is one positive eigenvalue hence the linear system is unstable.Therefore we assume from now on that α > 0 (i.e. P ′(1) > 0), that is we focus on the casewhere the pressure law is increasing in some neighborhood of the reference density. Notealso that changing v into v/
√α reduces the study to the case α = 1, an assumption that
we shall always make in what follows.
The low frequency regime ν|ξ| < 2. There are two distinct complex conjugated eigenvalues:
λ±(ξ) = −ν|ξ|2
2(1± iS(ξ)) with S(ξ) :=
√4
ν2|ξ|2− 1 .
After some computations we find that
a(t, ξ) = etλ−(ξ)
(1
2
(1 +
i
S(ξ)
)a0(ξ)− i
ν|ξ|S(ξ)v0(ξ)
)+etλ+(ξ)
(1
2
(1− i
S(ξ)
)a0(ξ) +
i
ν|ξ|S(ξ)v0(ξ)
),
v(t, ξ) = etλ−(ξ)
(i
ν|ξ|S(ξ)a0(ξ) +
1
2
(1− i
S(ξ)
)v0(ξ)
)+etλ+(ξ)
(− i
ν|ξ|S(ξ)a0(ξ) +
1
2
(1 +
i
S(ξ)
)v0(ξ)
).
For ξ → 0, we have
a(t, ξ) ∼ 12etλ−(ξ) (a0(ξ)− iv0(ξ)) + 1
2etλ+(ξ) (a0(ξ) + iv0(ξ)) ,
v(t, ξ) ∼ 12etλ−(ξ) (ia0(ξ) + v0(ξ)) + 1
2etλ+(ξ) (−ia0(ξ) + v0(ξ)) .
Hence, the low frequencies of a and v have a similar behavior. As |etλ±(ξ)| = e−νt|ξ|2/2
In other words, in the L2 framework, the low frequencies of (a, u) behave as if satisfyingthe heat equation with diffusion ν. This is no longer true in Lp with p 6= 2 however, forthe eigenvalues have a nonzero imaginary part.
The high frequency regime ν|ξ| > 2. There are two distinct real eigenvalues:
λ±(ξ) := −ν|ξ|2
2(1±R(ξ)) with R(ξ) :=
√1− 4
ν2|ξ|2
and we find that
a(t, ξ) = etλ−(ξ)
(1
2
(1 +
1
R(ξ)
)a0(ξ)− 1
ν|ξ|R(ξ)v0(ξ)
)+etλ+(ξ)
(1
2
(1− 1
R(ξ)
)a0(ξ) +
1
ν|ξ|R(ξ)v0(ξ)
),
36 R. DANCHIN
v(t, ξ) = etλ−(ξ)
(1
ν|ξ|R(ξ)a0(ξ) +
1
2
(1− 1
R(ξ)
)v0(ξ)
)+etλ+(ξ)
(− 1
ν|ξ|R(ξ)a0(ξ) +
1
2
(1 +
1
R(ξ)
)v0(ξ)
).
For |ξ| → ∞, we have R(ξ) → 1 and 1 − R(ξ) ∼ 2/(νξ)2. Hence λ+(ξ) ∼ −ν|ξ|2 andλ−(ξ) ∼ − 1
ν · In other words, a parabolic and a damped mode coexist and the asymptoticbehavior of (a, v) for |ξ| → ∞ is given by
a(t, ξ) ∼ e−tν(a0(ξ)− (ν|ξ|)−1v0(ξ)
)+ e−νt|ξ|
2 (−(ν|ξ|)−2a0(ξ) + (ν|ξ|)−1v0(ξ)),
v(t, ξ) ∼ e−tν((ν|ξ|)−1a0(ξ)− (ν|ξ|)−2v0(ξ)
)+ e−νt|ξ|
2 (−(ν|ξ|)−1a0(ξ) + v0(ξ)).
At first, one would expect the damped mode to dominate as e−νt|ξ|2
is negligible com-
pared to e−tν for ξ going to infinity. This is true as far as a is concerned. This is not quite
the case for v however owing to the negative powers of ν|ξ| in the formula. More precisely,by taking advantage of Parseval formula, we get
Lemma 3.1. There exist two positive constants c and C such that for any j ∈ Z satisfying2jν ≥ 1 and t ∈ R+, we have
‖∆ja(t)‖L2 ≤ Ce−t2ν
(‖∆ja0‖L2 + (2jν)−1‖∆jv0‖L2
),
‖∆jv(t)‖L2 ≤ C(
(2jν)−1e−t2ν ‖∆ja0‖L2 +
(e−cνt2
2j+(ν2j)−2e−
t2ν
)‖∆jv0‖L2
).
The same inequalities hold true for any p ∈ [1,∞]. Indeed, arguing as in the proof ofLemma 1.2 yields
In other words, even at the linear level, we see that it is suitable to work with the sameregularity for ∇a and v. For low frequencies however, one has to work with a and v, afact which does not follow from our scaling considerations for (26).
Putting together all the estimates for the dyadic blocks and using Duhamel’s formula,we get the following proposition:
Proposition 3.1. Let (s, s′) ∈ R2 and p ∈ [1,+∞]. Assume that (a, u) satisfies
(LPH)
{∂ta+ divu = F,
∂tu−Au+∇a = G.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 37
Then we have for the low frequencies:
‖(a, u)‖`L∞t (Bs
′2,1)
+ ν‖(a, u)‖`L1t (B
s′+22,1 )
. ‖(a0, u0)‖`Bs′
2,1
+ ‖(F,G)‖`L1t (B
s′2,1)
and for the high frequencies:
ν‖a‖hL∞t (Bs+1
p,1 )+ ‖a‖h
L1t (B
s+1p,1 )
+ ‖u‖hL∞t (Bsp,1)
+ ν‖u‖hL1t (B
s+2p,1 )
. ν‖a0‖hBs+1p,1
+ ‖u0‖hBsp,1 + ν‖F‖hL1t (B
s+1p,1 )
+ ‖G‖hL1t (B
sp,1),
where the index ` (resp. h) means that only low (resp. high) frequencies have been takeninto account when computing the norm12.
3.2. An alternative approach. For more cumbersome constant coefficients linear sys-tems (corresponding to e.g. the full Navier-Stokes-Fourier system or MHD equations),computing explicitly the solutions may be very complicated if not impossible. In addition,note that in the present case, Proposition 3.1 does not enable us to tackle the global well-posedness issue for the compressible Navier-Stokes equations. Indeed, assuming with noloss of generality that P ′(1) = 1, System (26) rewrites for some function K with K(0) = 0,{
∂ta+ divu = −u · ∇a− adivu,
∂tu−Au+∇a = −u · ∇u− a1+aAu−∇(aK(a)).
Obviously, Proposition 3.1 does not enable us to estimate (a, u) as putting the convectionterm −u · ∇a in the r.h.s. will cause a loss of one derivative (there is no gain of regularityby time integration in the first equation).
In this paragraph, we present quite a robust method that may used to prove L2 (or evenLp−Lq ) decay or regularity estimates for linear systems having an antisymmetric part anda partially dissipative (or parabolic) part acting on some components of the solution. Wefocus on systems of linear partial (or pseudo) differential equations of the type
(56) ∂tw +A(D)w +B(D)w = 0
with w : R+ × Rd → Rn, and
• A(D) = (Aij(D))1≤i,j≤n with Aij(D) homogeneous Fourier multiplier of degree α,• B(D) = (Bij(D))1≤i,j≤n with Bij(D) homogeneous Fourier multiplier of degree β.
We assume in addition that A(D) is antisymmetric, namely13
(57) Re ((A(ξ)η) · η) = 0 for all (ξ, η) ∈ Rd × Cn,
and that B(D) satisfies the following partial ellipticity property :
(58) |ξ|β Re ((B(ξ)η) · η) ≥ κ|B(ξ)η|2 for all (ξ, η) ∈ Rd × Cn
where κ is a positive real number.
Classical examples of such systems are :
• Partially dissipative symmetric linear systems such as (in the simplest case) :
(59)
{∂tu+ ∂xv = 0
∂tv + ∂xu+ λv = 0with λ > 0.
12We mean that ‖z‖`Bσp,1
=∑
2jν≤1 2jσ‖∆jz‖Lp and that ‖z‖hBσp,1
=∑
2jν>1 2jσ‖∆jz‖Lp .13In this paragraph, we agree that η1 · η2 denotes the Hermitian inner product in Cn.
38 R. DANCHIN
• The linearized barotropic Navier-Stokes equations :
Indeed, System (59) satisfies the above assumptions with d = 1, n = 2, α = 1, β = 0 andκ = λ−1, while one has to take n = d+ 1, α = 1, β = 2 and κ = cν−1 (with c dependingonly on µ/ν ) for System (60).
In what follows we set for all ω ∈ Sd−1 and ρ > 0,
A(ξ) = ραAω and B(ξ) = κ−1ρβBω with ξ = ρω.
Keeping (57) and (58) in mind, we thus have
(61) Re ((Aωη) · η) = 0 and Re ((Bωη) · η) ≥ |Bωη|2 for all (ω, η) ∈ Sd−1 × Cn.The partial Fourier transform (with respect to x) of the solution w to (56) thus satisfiesthe following first order linear ODE:
Of course, the decay properties of w are closely related to the value of the eigenvaluesassociated to E(ξ). In the sequel, we propose a method that allows to avoid computingthose eigenvalues. Before going further, let us notice that as w is explicitly given by
(63) w(t, ξ) = w0(ξ) exp
(− tρ
β
κ
(κρα−βAω +Bω
)),
one may restrict our attention to the case α = 1, β = 0 and κ = 1. Indeed, settingτ := (tρβ)/κ, and % := κρα−β, we see that any decay estimate for the function
z(τ) := z0 exp(−τ(%Aω +Bω)
)will imply a decay estimate for w(t, ξ).
In order to exhibit the decay properties of z, we introduce the following Lyapunovfunctional :
(64) L(τ) := |z(τ)|2 + min(%, %−1)n−1∑k=1
εk Re((BωA
k−1ω z) · (BωAkωz)
)where ε1, · · · , εn−1 are small positive parameters depending only on A and B, the valueof which will be given below.
Let us notice that z satisfies
(65) z′ + (%Aω +Bω)z = 0.
Hence, taking advantage of (61) we easily gather that
L′(τ) + 2 Re((Bωz) · z
)+ min(1, %2)
n−1∑k=1
εk|BωAkωz|2
= −min(%, %−1)
{n−1∑k=1
εk
(Re((BωA
k−1ω Bωz)· (BωAkωz)
)+ Re
((BωA
k−1ω z) · (BωAkωBωz)
))}
−min(1, %2)
n−1∑k=1
εk Re((BωA
k−1ω z) · (BωAk+1
ω z)).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 39
We claim that a suitable choice of ε1, · · · , εn−1 enables us to absorb the r.h.s by the l.h.sand thus to get decay estimates. There are several types of terms that have to be handled:
• Terms (BωAk−1ω Bωz)· (BωAkωz) with 1 ≤ k ≤ n − 1: Cauchy-Schwarz inequality
implies that∣∣(BωAk−1ω Bωz)· (BωAkωz)
∣∣ ≤ ‖BωAk−1ω ‖|Bωz| |BωAkωz|.
Hence, taking advantage of (61) and of the boundedness of BωAk−1ω for ω in the
compact set Sd−1,∣∣(BωAk−1ω Bωz)· (BωAkωz)
∣∣ ≤ Ck√Re ((Bωz) · z) |BωAkωz|.
Therefore,
min(%, %−1)εk∣∣(BωAk−1
ω Bωz)· (BωAkωz)∣∣≤ 1
20nRe ((Bωz)·z)+5nε2
kC2k min(%2, %−2)|BωAkωz|2.
Hence this term may be absorbed by the left-hand side if we take εk so thatnεkC
2k � 1.
• Terms (BωAk−1ω z)· (BωAkωBωz) with 2 ≤ k ≤ n− 1:
They may be treated exactly as the previous ones.• Term (Bωz)· (BωAωBωz): We have∣∣(Bωz)· (BωAωBωz)∣∣ ≤ ‖BωAω‖ |Bωz|2 ≤ ‖BωAω‖ Re ((Bωz) · z).
Hence it suffices to take ε1 small enough with respect to 1/ supω ‖BωAω‖.• Terms (BωA
k−1ω z) · (BωAk+1
ω z) with 1 ≤ k ≤ n− 2: We write that
εk∣∣(BωAk−1
ω z) · (BωAk+1ω z)
∣∣ ≤ εk−1
4|BωAk−1
ω z|2 +ε2k
εk−1|BωAk+1
ω z|2.
Therefore those terms may be absorbed if, say, 4ε2k ≤ εk−1εk+1.
• Term (BωAn−2ω z)·(BωAnωz): Here we have to use Cayley-Hamilton’s theorem which
ensures that there exist some complex numbers c0ω, · · · , cn−1
ω so that
Anω = c0ωIn + c1
ωAω + · · ·+ cn−1ω An−1
ω .
Note that the coefficients cjω depend continuously on the coefficients of the matrixAω. Because ω belongs to the compact set Sd−1, we thus deduce that they may beuniformly bounded by some positive real number M. Hence it suffices to absorb:
εn−1M∣∣(BωAn−2
ω z) · (BωAjωz)∣∣ , j = 0, · · · , n− 1.
Once again, we write that∣∣(BωAn−2ω z) · (BωAjωz)
∣∣ ≤ |BωAn−2ω z| |BωAjωz|.
Hence
εn−1M∣∣(BωAn−2
ω z) · (BωAjωz)∣∣ ≤ (Mεn−1)2
εj|BωAn−2
ω z|2 +εj4|BωAjωz|2.
Hence we just have to impose in addition that, say, 4(Mεn−1)2 ≤ εjεn−2 for j =0, · · · , n− 1.
40 R. DANCHIN
Finally, putting together the above bounds, we deduce that there exist14 ε0, · · · , εn−1 sothat for all ω ∈ Sd−1 and τ ∈ R+, we have
(66) L′(τ) +min(1, %2)
2
n−1∑k=0
εk|BωAkωz(τ)|2 ≤ 0.
Note that taking ε0, · · · , εn−1 smaller if needed, one may ensure in addition that
(67)1
2L ≤ |z|2 ≤ 2L
Now, setting
Nω := minη∈Sn−1
n−1∑k=0
εk|BωAkωη|2,
and combining (64) and (67), we end up with
L(t) ≤ e−14
min(1,%2)NωτL(0).
Then coming back to the initial variables and function w, we conclude that
Let us emphasize that, unless β = α, this inequality means that the decay properties ofthe solution are different in low and high frequency, the threshold between the two regimes
being at |ξ| = κ1
β−α . Depending on the properties of Nω (that, in general, may vanishfor some values of ω ), this inequality provides a score of optimal algebraic or exponentialdecay estimates for w without computing it explicitly.
Let us assume from now on that
(69) minω∈Sd−1
Nω > 0.
Keeping in mind that Sd−1 is compact, Condition (69) is equivalent to the fact that Nω
is positive for any ω ∈ Sd−1. It is well known (see e.g. [3]) that this former condition isequivalent to the following Kalman rank condition: the n2 × n matrix
BωBωAω· · ·
BAn−1ω
has rank n,
or to the Shizuta-Kawashima condition:
(SK) kerBω ∩ {eigenvectors of Aω} = {0}.
Assuming that (69) is satisfied, one may conclude that for low frequencies |ξ| < κ1
β−α (resp.
high frequencies |ξ| > κ1
β−α ), the solutions to (56) behave, from the point of view of decayestimates or smoothing properties, as those of
14One may take for instance εj = ε√j+1 with ε small enough.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 41
At the L2 level, this may be seen more accurately after localizing the equation for w bymeans of the Littlewood-Paley decomposition (∆j)j∈Z. Then combining (68) and Fourier-Plancherel theorem, we deduce that
(70) ‖∆jw(t)‖L2 ≤ 2‖∆jw0‖L2e−min(2jβ ,κ22(2α−β)j)κ−1t for all t ∈ R+ and j ∈ Z.
Note that, by taking advantage of Duhamel’s formula, we may afford to have a right-handside f in (56). Then Inequality (70) implies that
(71) ‖∆jw‖L∞t (L2) + min(κ−12jβ, κ2j(2α−β)
)‖∆jw‖L1
t (L2) . ‖∆jw0‖L2 + ‖∆jf‖L1
t (L2).
This means that there is a gain of max(β, 2α − β) (resp. min(β, 2α − β)) derivatives inlow frequencies (resp. high frequencies) when performing a L1 -in-time integration.
If we assume for example that β ≤ α then by multiplying both sides of the aboveinequality by 2js and summing up over Z, it is easy to deduce estimates for w in any“hybrid” Besov space built on L2. Setting the threshold between low and high frequencies
at κ1
β−α , we get
‖w‖`L∞t (Bs2,r)
+ κ ‖w‖`L1t (B
s+2α−β2,r )
. ‖w0‖`Bs2,r + ‖f‖`L1t (B
s2,r)
(72)
‖w‖hL∞t (Bs2,r)
+ κ−1‖w‖hL1t (B
s+β2,r ). ‖w0‖hBs2,r + ‖f‖h
L1t (B
s2,r).(73)
Above we used the notation
‖w‖`Lqt (B
σ2,r)
:=
( ∑2j(β−α)<κ
(2jσ‖∆jw‖Lqt (L2)
)r) 1r
‖w‖hLqt (B
σ2,r)
:=
( ∑2j(β−α)≥κ
(2jσ‖∆jw‖Lqt (L2)
)r) 1r
·
3.3. Applications. Let us first look at the partially dissipative system (59). Then thecorresponding matrices Aω and Bω read
Aω =
(0 i sgnω
i sgnω 0
)and Bω =
(0 00 1
)·
As pointed out before, (57) is fulfilled, and B satisfies (58) with κ = λ−1 and β = 0. Inaddition,
BωAω =
(0 0
i sgnω 0
)·
Therefore the Kalman rank condition is satisfied, and thus the above analysis carries out tothis example. The threshold between low and high frequencies is at λ. The correspondingLyapunov functional reads (for small enough ε):
‖(u, v)‖2L2 + ελ−1
∫Rv∂xu dx in low frequencies
‖(u, v)‖2L2 + ελ
∫Rv |D|−2∂xu dx in high frequencies.
As we have α = 1 and β = 0, we conclude that there is parabolic smoothing with diffusionλ−1 on the whole solution (u, v) in low frequency, and exponential decay with parameterλ for high frequencies.
42 R. DANCHIN
The above method allows to exhibit exactly the same qualitative behavior for the lin-earized Navier-Stokes system (60) which is of hyperbolic-parabolic type : indeed here wehave α = 1 and β = 2 so that we still have parabolic smoothing for low frequencies andexponential decay for high frequencies. We may easily check that (58) is fulfilled withκ = cν−1 (c depending only on µ/ν ) and β = 2, and that the associated matrices read
(with λ = λ/ν and µ = µ/ν ):
Aω =
(0 i~ωiT~ω 0
)and Bω =
(0 0
0 µId + (λ+ µ)~ω ⊗ ~ω
)·
We notice that A2ω = −Id hence we may stop at k = 1 in the definition of the Lyapunov
Therefore, in low frequencies 2jν ≤ 1, we have parabolic smoothing (with parameter ν )for a and u. The corresponding Lyapunov functional reads
‖(∆ja, ∆ju)‖2L2 + εν
∫Rd
∆ju · ∇∆ja dx
with ε small enough.
This method gives exponential decay in high frequencies for (a, u). Hence in particular
(74) ‖(a, u)‖hL∞t (Bs2,r)
+ ν−1‖(a, u)‖hL1t (B
s2,r)≤ ‖(a0, u0)‖h
Bs2,r.
However, our scaling considerations from Section 2 suggest that it is suitable to work withthe same level of regularity for ∇a and u . Besides, the explicit computations that havebeen performed in the previous subsection point out a parabolic type decay for u which isabsent from (74).
In Fourier variables, using |D|a rather than a amounts to making the change of unknown
z(t, ξ) = P (ξ)w(t, ξ) with P (ξ) =
(|ξ| 00 Id
).
Hence
z(t, ξ) =
(P (ξ)e−tE(ξ)P−1(ξ)
)z0(ξ).
This shows that, for any fixed ξ, the function z(·, ξ) (hence (∇a, u)(·, t)) has the same decayproperties as (a, u)(·, ξ). However, the matrix P (ξ) is uniformly bounded only on boundedsets, hence we cannot get any handy information over (∇a, u) (in terms of (∇a0, u0)) forvery high frequencies. For the very case of (60) however, it is possible to recover thismissing information in high frequency by introducing the modified velocity w := Qu +ν−1(−∆)−1∇a.
Indeed, on the one hand the divergence-free part Pu of the velocity satisfies
∂tPu− µ∆Pu = 0.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 43
On the other hand, the system for (∇a,w) reads{∂t∇a+ ν−1∇a = −∆w
∂tw − ν∆w = ν−1w − ν−2(−∆)−1∇a.
Therefore for any j ∈ Z and p ∈ [1,+∞],
‖ν∆j∇a‖L∞t (Lp) + ‖∇∆ja‖L1t (L
p) . ‖ν∆j∇a0‖Lp + ν22j‖∆jw‖L1t (L
p)
‖∆jw‖L∞t (Lp)+ν22j‖∆jw‖L1t (L
p).‖∆jw0‖Lp+(ν2j)−2(ν22j‖∆jw‖L1
t (Lp)+‖∇∆ja‖L1
t (Lp)
).
Hence plugging the second inequality in the first one, and assuming that ν2j is largeenough, we conclude that
‖ν∆j∇a‖L∞t (Lp) + ‖∇∆ja‖L1t (L
p) + ‖∆jw‖L∞t (Lp)+ν22j‖∆jw‖L1t (L
p)
≤ C(ν‖∆j∇a0‖Lp + ‖∆jw0‖Lp
).
We thus recover the decay estimates of the previous subsection.
3.4. References and remarks. The linear analysis based on explicit computations hasbeen performed in [6] and [9]. The approach described in the second subsection is mainlyborrowed from a recent paper by K. Beauchard and E. Zuazua [3]. Proving optimal Lpt (L
qx)
estimates in the case where (69) is not satisfied was the main motivation of that work. Thecelebrated (SK) condition has been first introduced by S. Kawashima and Y. Shizuta in[55]. Effective velocity has been used by many authors. Here we adapted to the linearsetting the recent work by B. Haspot in [37].
Besides its generality, one of the main interest of the approach that is presented in Sub-section 3.2 is its robustness: to some extent, it may be extended to nonconstant coefficients.For instance, let us add a “convection term” to the partially dissipative toy model (59).We get {
∂tu+A∂xu+ ∂xv = 0∂tv +B∂xv + ∂xu+ λv = 0
where A and B are given functions.
Applying ∆j to the system, we get{∂t∆ju+ Sj−1A∂x∆ju+ ∂x∆jv = Rj(A, u)
∂t∆jv + Sj−1B∂x∆jv + ∂x∆ju+ λ∆jv = Rj(B, v)
where Rj(A, u) and Rj(B, v) are “remainder” terms that may computed explicitly and
estimated as in (20). Note that, owing to the low frequency cut-off, the functions Sj−1A and
Sj−1B are smooth. Writing the evolution equation for the spectrally localized Lyapunovfunctionals:
‖(∆ju, ∆jv)‖2L2 + ελ−1
∫R
∆jv∂x∆ju dx if 2j ≤ λ
‖(∆ju, ∆jv)‖2L2 + ελ
∫R
∆jv |D|−2∂x∆ju dx if 2j > λ
44 R. DANCHIN
and taking ε small enough (independently of A and B ) allows to get the following inequal-ity for all j ∈ Z :
So using Gronwall lemma and estimating the remainder terms as in (20) (here more preciseregularity assumptions over A and B have to be made) allows to recover the same estimatesas in the case A ≡ B ≡ 0, up to some exponential term due to the use of Gronwall lemma.
A similar approach does for the linearized compressible Navier-Stokes equations (see[2, 14]) or more generally for the linearization of the full Navier-Stokes equation (see [16]).
The main drawback of this approach is that the imaginary part of eigenvalues of thesystem is not taken into account. In other words, we cannot keep track of the possibledispersive properties of the system by such rough a device. We shall see in the next sectionhow this deficiency may be overcome for the compressible Navier-Stokes equations.
4. Global results
This section is devoted to the proof of global-in-time results for the barotropic Navier-Stokes equations. First we establish the global existence for small perturbations of a stableconstant equilibrium (ρ, 0) (that is we assume that the reference positive density ρ is suchthat P ′(ρ) > 0) and next we investigate the so-called incompressible limit, that is theconvergence of (1) to the incompressible Navier-Stokes equations when the Mach numbertends to 0. The estimates that have been proved in the previous section will play a crucialrole. In particular we have to keep in mind that, in low frequency, some eigenvalues havenonzero imaginary part, hence we are stuck to the L2 framework. At the same time, atthe linear level there is no obstacle to use a Lp type framework for high frequencies.
For simplicity, we assume from now that the Lame coefficients λ and µ are constant.
4.1. Global existence for small perturbations of a stable equilibrium state. Thissubsection is devoted to proving the following global well-posedness result.
Theorem 4.1. Let p ∈ [2, 2d) ∩ [2,min(4, 2dd−2 ]. Assume that α := P ′(1) > 0, a0 ∈ B
dp
p,1
and u0 ∈ Bdp−1
p,1 and that in addition a`0 and u`0 are in Bd2−1
2,1 . There exist two constants cand M depending only on d, and on the parameters of the system such that if
‖(a0, u0)‖`Bd2−1
2,1
+ ν‖a0‖hBdpp,1
+ ‖u0‖hBdp−1
p,1
≤ cν
then (26) has a unique global-in-time solution (a, u) with
(a, u)` ∈ Cb(R+; Bd2−1
2,1 ) ∩ L1(R+; Bd2
+1
2,1 ), ah ∈ Cb(R+; Bdp
p,1) ∩ L1(R+; Bdp
p,1),
uh ∈ Cb(R+; Bdp−1
p,1 ) ∩ L1(R+; Bdp
+1
p,1 ).
Remark 4.1. The smallness condition is satisfied for small densities and large highlyoscillating velocities: take uε0 : x 7→ φ(x) sin(ε−1x · ω)n with ω and n in Sd−1 and φ ∈
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 45
S(Rd). Then one may check that
‖uε0‖hBdp−1
p,1
≤ Cε1− dp if p > d,
and that the low frequencies of uε0 go to 0 in Bd2−1
2,1 faster than any power of ε. Hence suchan initial velocity with small enough ε and a0 generates a global unique solution.
As uniqueness for the full range p < 2d follows from Lagrangian approach presented inSubsection 2.3, we concentrate on the proof of existence. This is mainly a matter of provingglobal a priori estimates for small enough data. The quantity that we have to bound reads
X(t) := ‖(a, u)‖`L∞t (B
d2−1
2,1 )+ ν‖(a, u)‖`
L1t (B
d2+1
2,1 )
+ν‖a‖hL∞t (B
dpp,1)
+ ‖a‖hL1t (B
dpp,1)
+ ‖u‖hL∞t (B
dp−1
p,1 )
+ ν‖u‖hL1t (B
dp+1
p,1 )
.
In order to bound the low frequency part of the norm, we shall take advantage of Proposition3.1 (where parabolic smoothing in the L2 framework are available for both a and u). Aspointed out in the previous section, in high frequency, the fundamental observations arethat at the linear level:
• Pu satisfies a heat equation (hence parabolic smoothing in any Besov space);• The compressible parabolic mode tends to be collinear to Qu+ ν−1α(−∆)−1∇a ;• The density a is damped.
In order to make those observations more accurate, let us rewrite the nonlinear system (26)as follows:
(75)
∂ta+ u · ∇a+ (1 + a)divQu = 0,
∂tQu+Q(u · ∇u)− ν∆Qu+∇(G(a)) = −Q(J(a)Au),
∂tPu+ P(u · ∇u)− µ∆Pu = −P(J(a)Au).
The last equation is a heat equation with quadratic terms. Hence one may expect thatparabolic smoothing for Pu holds in any (critical) Besov space. To handle the first twoequations, our linear analysis motivates us to introducing the effective velocity
w := Qu+ ν−1α(−∆)−1∇a
which is expected to satisfy also parabolic type estimates in the high frequency regime.
First step: Bounds for the effective velocity. The effective velocity w satisfies the heatequation:
∂tw − ν∆w = −Q(u · ∇u)−Q(J(a)Au)
+(G′(0)−G′(a))∇a− ν−1α(−∆)−1∇div((1 + a)u).
All the terms of the right-hand side (but the last one) are at least quadratic hence expectedto be small if we start with small data. The last term has a linear part that is lower order.One can make this heuristics more accurate by combining the regularity estimates for theheat equation and product and composition estimates in Besov spaces. Applying a highfrequency cut-off to the equation for w, we readily get for any p ∈ [2, 2d) (omitting the
46 R. DANCHIN
dependency with respect to α):
‖w‖hL∞t (B
dp−1
p,1 )
+ ν‖w‖hL1t (B
dp+1
p,1 )
. ‖w0‖hBdp−1
p,1
+ ν−1‖w − ν−1α(−∆)−1∇a‖hL1t (B
dp−1
p,1 )
+(‖u‖
L∞t (Bdp−1
p,1 )+ ‖a‖
L∞t (Bdpp,1)
)‖∇u‖
L1t (B
dpp,1)
+ ‖a‖2L2t (B
dpp,1)
+ ν−1‖au‖hL1t (B
dp−1
p,1 )
.
The terms with Qu and au do not have the right scaling. However, they be made arbitrarilysmall if setting the threshold between low and high frequencies high enough. More precisely,if this threshold is at |ξ| = 2j0 with j0 s.t. 1� 2j0ν then
(76) ν−1‖w‖hL1t (B
dp−1
p,1 )
≤ ν−12−2j0‖w‖hL1t (B
dp+1
p,1 )
� ν‖w‖hL1t (B
dp+1
p,1 )
.
A similar inequality may be proved for ‖au‖hL1t (B
dp−1
p,1 )
, and we thus end up with
(77) ‖w‖hL∞t (B
dp−1
p,1 )
+ ν‖w‖hL1t (B
dp+1
p,1 )
. ‖w0‖hBdp−1
p,1
+ ν−2α‖a‖hL1t (B
dp−2
p,1 )
+ ν−1X2(t).
Arguing as in (76), we see that the term involving a is very small compared to ‖a‖hL1(B
dpp,1)
.
Second step: Parabolic estimates for Pu. Applying Theorem 1.1 and product estimates tothe last equation of (75), we readily get
‖Pu‖L∞t (B
dp−1
p,1 )+ ‖Pu‖
L1t (B
dp+1
p,1 ). ‖Pu0‖
Bdp−1
p,1
+ ν−1X2(t).
Third step: Decay estimates for a. We notice that
∂ta+ u · ∇a+ ν−1αa = −adivu− divw.
Because α > 0, Theorem 1.2 implies (if ‖∇u‖L1t (B
dpp,1)
is small enough) that
(78) ‖a‖hL∞t (B
dpp,1)
+ ν−1‖a‖hL1t (B
dpp,1)
≤ ‖a0‖hBdpp,1
+ ‖divw‖hL1t (B
dpp,1)
+ ν−1X2(t).
Now, according to (77),
(79) ν‖divw‖hL1t (B
dpp,1)
. ‖w0‖hBdp−1
p,1
+ (ν2j0)−2‖a‖hL1t (B
dpp,1)
+ ν−1X2(t).
Hence plugging (78) in (79) and taking j0 large enough, we deduce that
‖w‖hL∞t (B
dp−1
p,1 )
+ν‖w‖hL1t (B
dp+1
p,1 )
+ν‖a‖hL∞t (B
dpp,1)
+‖a‖hL1t (B
dpp,1)
. ‖w0‖hBdp−1
p,1
+ν‖a0‖hBdpp,1
+ν−1X2(t).
Of course, as Qu = w−ν−1α(−∆)−1∇a, one may replace w by Qu in the above inequality.
Fourth step: Low frequency estimates. As explained before, we have to restrict ourselves toBesov spaces of type Bs
2,1 . Now, by taking advantage of Proposition 3.1, we get (for some
function k vanishing at 0):
‖(a, u)‖`L∞t (B
d2−1
2,1 )+ ν‖(a, u)‖`
L1t (B
d2+1
2,1 ). ‖(a0, u0)‖`
Bd2−1
2,1
+ ‖div(au)‖`L1t (B
d2−1
2,1 )
+‖u · ∇u‖`L1t (B
d2−1
2,1 )+ ‖J(a)Au‖`
L1t (B
d2−1
2,1 )+ ‖k(a)∇a‖`
L1t (B
d2−1
2,1 ).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 47
At this point, in order to bound the nonlinear terms by ν−1X2(t), in addition to 1 ≤ p < 2d,the assumptions that p ≤ 4 and p ≤ 2d
d−2 are needed. This is due to the fact that productbetween high frequencies generate low frequencies. The latter have to be bounded in aspace related to L2 whereas only Lp type estimates are available for high frequencies.Heuristically, the condition that p ≤ 4 corresponds to the fact that for the product of twoLp functions to be in Lq for some q ≤ 2, we need p ≤ 4. To simplify the presentation, weskip the details. The reader may refer to the paper by B. Haspot [37], or to [24].
Last step: Global estimate. Putting all the previous estimates together, we get for anyp ∈ [2, 4] ∩ [2, 2d), the inequality
(80) X(t) ≤ C(X(0) + ν−1X2(t)).
Now it is clear that as long as
(81) 2CX(t) ≤ ν,Inequality (80) ensures that
(82) X(t) ≤ 2CX(0).
Using a bootstrap argument, one may conclude that if X(0) is small enough with respectto ν then (81) is satisfied as long as the solution exists. Hence (82) holds globally in time.
4.2. On the incompressible limit. It is common sense that slightly compressible flowsshould not differ much from incompressible flows. In this subsection, we justify rigorouslythis heuristics in the asymptotics ε going to 0 (where ε stands for the Mach number thatis the ratio of the typical velocity over the sound speed) if we look at times scales of order1/ε. More precisely, we establish the convergence of the solutions of the barotropic Navier-Stokes equations to those of the incompressible Navier-Stokes equations when the Machnumber ε tends to 0.
Because we expect the relevant time scale to be 1/ε, it is natural to set
(ρ, u)(t, x) = (ρε, εuε)(εt, x).
With these new variables, the original system (1) recasts in ∂tρε + div(ρεuε) = 0,
∂t(ρεuε) + div(ρεuε ⊗ uε)− µ∆uε−(λ+µ)∇div uε +
∇P ε
ε2= 0.
with P ε = P (ρε).
In the case of well-prepared data:
ρε0 = 1 +O(ε2) and uε0 with div uε0 = O(ε),
the time derivatives may be bounded independently of ε for ε going to 0. Hence noacoustic waves have to be taken into account and one may prove that the solution tendsto the incompressible Navier-Stokes equations when ε goes to 0 by a standard approach(see e.g. [45]). Besides, asymptotic expansions may be derived for the solution if suchexpansions are available for the data.
Here, we shall consider ill-prepared data, namely
ρε0 = 1 + εbε0 and uε0
with bε0 and uε0 bounded in a suitable space when ε goes to 0. Note that it is not supposedthat divuε0 = 0.
48 R. DANCHIN
Initially, for such data, the time derivative of the solution is of order ε−1 and highlyoscillating acoustic waves that are likely to interact do have to be considered. Whetherconvergence to the incompressible Navier-Stokes equations holds true anyway is the mainissue. This is the question that we want to address now in the whole space setting and ifP ′(1) = 1 (with no loss of generality).
Denoting ρε = 1 + εbε , it is found that (bε, uε) satisfies
(NSCε)
∂tb
ε +div uε
ε= −div(bεuε),
∂tuε + uε · ∇uε − Auε
1 + εbε+ (1+k(εbε))
∇bε
ε= 0,
(bε, uε)|t=0 = (b0, u0),
with A := µ∆ + (λ+µ)∇div and k a smooth function satisfying k(0) = 0.
According to Theorem 2.2, System (NSCε) is locally well-posed for data having criticalregularity. We claim that:
(1) lim infε→0 Tε ≥ T where Tε stands for the lifespan of (bε, uε) and T stands for thelifespan of the solution v to the incompressible Navier-Stokes equation:
(NS)
{∂tv + P(v · ∇v)− µ∆v = 0,
v|t=0 = Pu0.
(2) Tε = +∞ for small ε if T = +∞,(3) uε tends to v and bε converges to 0.
To simplify the presentation, in these notes, we only consider the case where the data b0and u0 are so small that the solution to (NSCε) as well as that of (NS) are global.
Decomposing the velocity into its divergence-free and potential parts, the linearizedcompressible Navier-Stokes equations about (bε, uε) = (0, 0) read
∂tbε +
divQuε
ε= 0,
∂tQuε − ν∆Quε +∇bε
ε= 0,
∂tPuε − µ∆Puε = 0.
As pointed out in the previous section, the last equation is the heat equation whereasdenoting vε := |D|−1divQuε, the first two equations are equivalent to
(BMε)
∂tb
ε +|D|vε
ε= 0,
∂tvε − ν∆vε − |D|v
ε
ε= 0.
This latter system may be solved explicitly by using the Fourier transform:
d
dt
(bε(ξ)vε(ξ)
)=
(0 −ε−1|ξ|ε−1|ξ| −ν|ξ|2
)(bε(ξ)vε(ξ)
).
As in the previous section, there are two regimes: if νε|ξ| > 2 then the eigenvalues read
λ±(ξ) = −ν|ξ|2
2
(1±
√1− 4
ε2ν2|ξ|2
)
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 49
whereas in the low frequency regime νε|ξ| < 2, one has
λ±(ξ) = −ν|ξ|2
2
(1± i
√4
ε2ν2|ξ|2− 1
).
Therefore
λ+(ξ) ∼ −ν|ξ|2 and λ−(ξ) ∼ − 1
ε2νfor ξ →∞,
and
λ± (ξ) ∼ −ν |ξ|2
2∓ i |ξ|
εfor ξ → 0.
Hence, in high frequency we expect to have
• one parabolic mode with diffusion ν,• one damped mode with coefficient 1
ε2ν,
whereas, in low frequency (BMε) behaves like
d
dtz − ν
2∆z ∓ i |D|
εz = 0.
Because the low frequency regime tends to invade the whole Rd when ε → 0 (note thatthe threshold between the two regimes is at |ξ| = 2(νε)−1 ), it has to be studied with thegreatest care. In Rd, taking advantage of the large imaginary part of the eigenvalues forlow frequencies (a property that has not been used so far) turns out to be the key to provingconvergence for ε tending to 0 as it supplies dispersion, hence dispersive estimates throughProposition 1.9.
Let us now state our result of convergence for global small solutions.
Theorem 4.2. Assume that P ′(1) > 0. There exist two positive constants η and Mdepending only on d, λ/µ and P, such that if
(83) Cεν0 := ‖bε0‖Bd2−1
2,1
+ εν‖bε0‖Bd22,1
+ ‖uε0‖Bd22,1
≤ η
then the following results hold:
(1) System (NSCε) has a unique global solution (bε, uε) with
‖bε‖L∞(B
d2−1
2,1 )∩L2(Bd22,1)
+ εν‖bε‖L∞(B
d22,1)
+ ‖uε‖L∞(B
d2−1
2,1 )+ ν‖uε‖
L1(Bd2+1
2,1 )≤MCεν0 .
(2) If Puε0 ⇀ v0 then the incompressible Navier-Stokes equations (NS) with data v0
have a unique solution v with
(84) ‖v‖L∞(B
d2−1
2,1 )+ µ‖v‖
L1(Bd2+1
2,1 )≤M‖v0‖
Bd2−1
2,1
.
(3) (bε,Quε) tends to 0 in Lr(Bdp−1+ 1
r
p,1 ) with the decay rate ε1r , where
• p = 2(d− 1)/(d− 3) and r = 2 if d ≥ 4,• p ∈ [2,∞) and r = 2p/(p−2) if d = 3,• p ∈ [2,∞] and r = 4p/(p−2) if d = 2.
(4) Puε tends to v in L∞(Bdp− 3
2
p,1 ) ∩ L1(Bdp
+ 12
p,1 ) with the decay rate ε12 if d ≥ 4. If
d = 3 (resp. d = 2) convergence holds true in C(R+; B−1−α∞,1 ) with the decay rate
εα for any α ∈]0, 1/2[ (resp. α ∈]0, 1/6]).
The proof comprises four steps:
50 R. DANCHIN
(1) Global existence and uniform estimates for (NSCε);(2) Global existence for the corresponding limit system (NS);(3) Convergence to 0 for the “compressible part” of the solution, namely (bε,Quε);(4) Convergence of Puε to the solution v of (NS).
Step 1. Global existence and uniform estimates. Making the change of unknowns
we notice that (bε, uε) solves (NSCε) if and only if (b, u) solves (26) with rescaled datab0 := εbε0(ε·) and u0 := εuε0(ε·). Hence, if η is small enough in (83), Theorem 4.1 withp = 2 ensures that the solution (b, u) is global. Then resuming to the original variables, weget the first part of the above statement: there exists a global solution (bε, uε) such that
‖bε‖`L∞(B
d2−1
2,1 )+ ν‖bε‖`
L1(Bd2+1
2,1 )+ εν‖bε‖h
L∞(Bd22,1)
+ (εν)−1‖bε‖hL1(B
d22,1)
+‖uε‖L∞(B
d2−1
2,1 )+ ν‖uε‖
L1(Bd2+1
2,1 )≤M
(‖bε0‖`
Bd2−1
2,1
+ εν‖bε0‖hBd22,1
+ ‖uε0‖Bd2−1
2,1
).
Let us emphasize that in the above inequality the threshold between low and high frequen-cies is at 1/(εν).
Step 2. Global existence for (NS). Assuming that η is small enough in (83), Theorem 2.1
with p = 2 and r = 1 ensures that (NS) has a unique global solution v ∈ Cb(R+; Bd2−1
2,1 )∩
L1(R+; Bd2
+1
2,1 ).
Step 3. Convergence to 0 for (bε,Quε). The functions b and u defined above satisfy
(86)
∂tb+ divQu = F := −div(bu),
∂tQu+∇b = G := −Q(u · ∇u+ 1
1+bAu+K(b)∇b).
As pointed out before, the acoustic wave equation (that is the l.h.s of (86)) is associatedto a group of unitary operators on L2(Rd) satisfying the dispersion inequality (21) withσ = (d− 1)/2. More precisely, applying Proposition 1.9 with s = d/2− 1, we get
‖(b,Qu)‖Lr(B
dp−1+1
rp,1 )
. ‖(b0,Qu0)‖Bd2−1
2,1
+ ‖(F,G)‖L1(B
d2−1
2,1 )
with (p, r) satisfying the hypotheses of Theorem 4.2.
Now, according to product laws in Besov spaces (see Proposition 1.6, Corollary 1.1 andProposition 1.7), we may write
‖div(bu)‖L1(B
d2−1
2,1 ). ‖b‖
L2(Bd22,1)‖u‖
L2(Bd22,1)
‖Q(u · ∇u)‖L1(B
d2−1
2,1 ). ‖u‖2
L2(Bd22,1)
‖Q((1 + b)−1Au)‖L1(B
d2−1
2,1 ). ‖u‖
L1(Bd2+1
2,1 )
‖K(b)∇b‖L1(B
d2−1
2,1 ). ‖b‖2
L2(Bd22,1).
Therefore, using the global a priori estimate for (b, u), we end up with
‖(F,G)‖L1(B
d2−1
2,1 )≤ CCεν0 .
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 51
Resuming to the solution (bε, uε) to (NSCε) through the change of variables (85), onemay conclude that
‖(bε,Quε)‖Lr(B
dp−1+1
rp,1 )
≤ CCεν0 ε1r .
Step 4. Convergence of the incompressible part. Assuming Puε0 = v0 for simplicity, thevector-field wε := Puε − v satisfies
There are three types of (quadratic) terms in Hε :
• The first two terms are linear in wε, and their coefficient is small as uε and v aresmall. Hence one expect them to be negligible.• Owing to Quε, the next two terms decay like some power of ε (previous step).• The last term is small because εbε
1+εbε is of order εbε.
In order to make all this rigorous, one has to use appropriate norms. To simplify thepresentation, let us just consider the (nonphysical) case d ≥ 4. Setting p := 2(d−1)/(d−3)Theorem 1.1 ensures that
(88) ‖wε‖L1(B
dp+1
2p,1 )
+ ‖wε‖L∞(B
dp−
32
p,1 ). ‖Hε‖
L1(Bdp−
32
p,1 ).
As uε and v are small, the first two terms of the r.h.s may be absorbed by the l.h.s: indeedwe have
‖P(wε ·∇v + uε ·∇wε)‖L1(B
dp−
32
p,1 ). ‖wε‖
L2(Bdp−
12
p,1 )
(‖uε‖
L2(Bd22,1)
+ ‖v‖L2(B
d22,1)
).
Now, the term involving wε may be bounded by the left-hand side of (88) (use interpola-tion), while the other two terms are small according to the first two steps of the proof.
Next, we use the fact that, by virtue of the previous step,
‖P(Quε ·∇v+uε ·∇Quε)‖L1(B
dp−
32
p,1 ). ‖Quε‖
L2(Bdp−
12
p,1 )
(‖v‖
L2(Bd22,1)
+‖uε‖L2(B
d22,1)
). Cεν0 ηε
12 .
Finally, because Bd2− 3
22,1 ↪→ B
dp− 3
2
p,1 , we have∥∥ εbε
1 + εbεAuε
∥∥L1(B
dp−
32
p,1 ). ‖εbε‖
L∞(Bd2−
12
2,1 )‖D2uε‖
L1(Bd2−1
2,1 ).
By interpolation, it is not difficult to check that
‖εbε‖Bd2−
12
2,1
. ε12(‖bε‖`
Bd2−1
2,1
+ ε‖bε‖hBd22,1
).
So putting the above inequalities together, we conclude that
‖wε‖L1(B
dp+1
2p,1 )
+ ‖wε‖L∞(B
dp−
32
p,1 )≤ Cη2ε
12 with p = 2(d− 1)/(d− 3).
The cases d = 2 and d = 3 may be treated along the same lines. �
52 R. DANCHIN
4.3. References and remarks. In the whole space case, the global existence of strongsolutions for small perturbations of a stable constant state has been first established by A.Matsumura and T. Nishida in [48] for data with high Sobolev regularity. Data with Lp
type regularity have been considered by P.B. Mucha in [50].Global results in the critical regularity framework (in the spirit of Theorem 4.1) have
been first proved by the author in [14] in the case p = 2 (see also [16] for a similar resultpertaining to the full Navier-Stokes equations). The case of exponents p 6= 2 (with slightlydifferent hypotheses) has been considered independently in a joint work with F. Charve [6]and by Q. Chen, C. Miao and Z. Zhang [10] in 2009. More recently, Haspot proposed asimpler method in [37], which is essentially the one that we presented here.
The approach that has been proposed here for the incompressible limit has been initiatedin [17]. There, the case of large data has also been examined: it has been proved by abootstrap argument that if the solution to the limit system exists until time T then itis also true for the compressible solution if ε is small enough. In particular the solutionbecomes global if it is so for the solution to the limit system. Hence in dimension two, anyslightly compressible solution is global in time.
The relevancy of Strichartz estimates for establishing the incompressible limit in thewhole space has been first noticed in [27] in the framework of weak solutions with finiteenergy. Global results also hold in the torus. As regards the incompressible limit however,the approach is quite different (for an approach based on Schochet’s filtering method, see[18] and the references therein).
Finally, let us mention that results related to the low Mach number limit for the fullNavier-Stokes equations have been established recently by T. Alazard in [1]. The limitsystem in the case of large entropy variations is no longer the incompressible Navier-Stokesequations, though (see [25] for a recent study of that limit system). The case of smalldata with small entropy variations has been investigated very recently in the Lp criticalregularity framework in [24].
Appendix A
Here we establish various estimates for the flow that have been used repeatedly in Sub-section 2.3 and justify that having a flow Xv defined by (37) allows to reconstruct theEulerian velocity field.
Recall that if v : [0, T ) × Rd → Rd is measurable, such that t 7→ v(t, x) is in L1(0, T )for all x ∈ Rd with, in addition, ∇v ∈ L1(0, T ;L∞) then it has, by virtue of the Cauchy-Lipschitz theorem, a unique C1 flow Xv satisfying
Xv(t, y) = y +
∫ t
0v(τ,Xv(τ, y)) dτ for all t ∈ [0, T ).
Furthermore, for all t ∈ [0, T ), the map Xv(t, ·) is a C1 -diffeomorphism over Rd.The difficulty that has to be faced is that we are given the velocity field v in Lagrangian
coordinates. Therefore, we first have to check whether the “flow” Xv(t, ·) defined by
(89) Xv(t, y) := y +
∫ t
0v(τ, y) dτ
is a C1 diffeomorphism over Rd. This property is required for constructing the Eulerianvector-field v by setting v(t, ·) := v ◦X−1
v (t, ·).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 53
So let us assume that we are given some vector field v over [0, T ]× Rd with
v ∈ C([0, T ]; Bdp−1
p,1 ), ∂tv ∈ L1(0, T ; Bdp−1
p,1 ) and Dv ∈ L1(0, T ; Bdp
p,1).
Differentiating (89) with respect to the space variable yields
(90) DXv(t, y) = Id +
∫ t
0Dv(τ, y) dτ.
As Bdp
p,1(Rd) is embedded in the set C0(Rd) of continuous functions going to 0 at infinity,
we deduce that Xv is a C1 function over [0, T ]×Rd. If assuming in addition that Condition(42) is satisfied with c small enough then using embedding we see that
(91) |DXv(t, y)− Id | ≤ 1/2 for all (t, y) ∈ [0, T ]× Rd.
Hence, for any t ∈ [0, T ], the map Xv(t, ·) is a local diffeomorphism. In order to show thatit is a global diffeomorphism, it suffices to establish that for all (t, x) ∈ [0, T ] × Rd, theequation
Xv(t, y) = x
has a unique solution. As Xv(0, x) = x and (91) is fulfilled, this is a mere consequence ofthe implicit function theorem, considering x as a parameter.
The following estimates have been used extensively in Subsection 2.3.
Lemma A.1. Let p ∈ [1,∞). Under Assumption (42), we have for all t ∈ [0, T ],
‖Id − adj (DXv(t))‖Bdpp,1
. ‖Dv‖L1t (B
dpp,1),(92)
‖Id −Av(t)‖Bdpp,1
. ‖Dv‖L1t (B
dpp,1),(93)
‖adj (DXv(t))TAv(t)− Id ‖
Bdpp,1
. ‖Dv‖L1t (B
dpp,1),(94)
‖J±1v (t)− 1‖
Bdpp,1
. ‖Dv‖L1t (B
dpp,1).(95)
Proof. Recall that (see e.g. the appendix of [26]) for any d× d matrix C we have
(96) Id − adj (Id + C) =(C − (TrC)Id
)+ P2(C),
where the entries of the matrix P2(C) are polynomials with respect to the coefficients ofC, with all terms of degree at least 2.
Applying this relation to the matrix DXv(t), and keeping (90) in mind, we deduce that
Id − adj (DXv(t)) =
∫ t
0
(Dv − div v Id
)dτ + P2
((∫ t
0Dv dτ
)).
Given that Bdp
p,1 is a Banach algebra and that (42) holds, we readily get (92).
Proving (93) relies on the following identity:
(97) Av(t) = (Id + Cv(t))−1 =
∑k∈N
(−1)k(Cv(t))k with Cv(t) =
∫ t
0Dv dτ,
and (94) stems from (92) and (93).
54 R. DANCHIN
For proving the last item, we use the fact that
(98) Jv(t, y) = 1 +
∫ t
0divv(τ,Xv(τ, y)) Jv(τ, y) dτ = 1 +
∫ t
0(Dv : adj (DXv))(τ, y) dτ.
Hence, if Condition (42) holds then we have (95) for Jv. In order to get the inequalityfor J−1
v , it suffices to use the fact that
J−1v (t, y)− 1 = (1 + (Jv(t, y)− 1))−1 − 1 =
∑k≥1
(−1)k∫ t
0Dv : adj (DXv) dτ.
This completes the proof of the lemma. �
Lemma A.2. Let v1 and v2 be two vector-fields satisfying (42), and δv := v2 − v1. Thenwe have for all p ∈ [1,∞) and t ∈ [0, T ] (with obvious notation):
(99) ‖A2 −A1‖L∞t (B
dpp,1). ‖Dδv‖
L1t (B
dpp,1),
(100) ‖adj (DX2)− adj (DX1)‖L∞t (B
dpp,1). ‖Dδv‖
L1t (B
dpp,1),
(101) ‖J2 − J1‖L∞t (B
dpp,1). ‖Dδv‖
L1t (B
dpp,1).
Proof. In order to prove the first inequality, we use the fact that, for i = 1, 2, we have
Ai = (Id + Ci)−1 =
∑k≥0
(−1)kCki with Ci(t) =
∫ t
0Dvi dτ.
Hence
A2 −A1 =∑k≥1
(Ck2 − Ck1
)=
(∫ t
0Dδv dτ
)∑k≥1
k−1∑j=0
Cj1Ck−1−j2 .
So using the fact that Bdp
p,1 is a Banach algebra, it is easy to conclude to (99).
The second inequality is a consequence of the decomposition (96) and of the Taylorformula which ensures that, denoting δC := C2 − C1,
Proving the third inequality relies on similar arguments. It is only a matter of using (98).The details are left to the reader. �
References
[1] T. Alazard: Low Mach number limit of the full Navier-Stokes equations, Archive for Rational Mechanicsand Analysis, 180(1), pages 1–73 (2006).
[2] H. Bahouri, J.-Y. Chemin and R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equa-tions, Grundlehren der mathematischen Wissenschaften, 343, Springer (2011).
[3] K. Beauchard and E. Zuazua: Large time asymptotics for partially dissipative hyperbolic systems,Arch. Ration. Mech. Anal., 199(1), pages 177–227 (2011).
[4] J.-M. Bony: Calcul symbolique et propagation des singularites pour les equations aux derivees partielles
non lineaires, Annales Scientifiques de l’Ecole Normale Superieure, 14(4), 209–246 (1981).[5] M. Cannone, Y. Meyer and F. Planchon: Solutions autosimilaires des equations de Navier-Stokes,
Seminaire Equations aux Derivees Partielles de l’Ecole Polytechnique, 1993–1994.
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 55
[6] F. Charve and R. Danchin: A global existence result for the compressible Navier-Stokes equationsin the critical Lp framework, Archive for Rational Mechanics and Analysis, 198(1), pages 233–271(2010).
[7] J.-Y. Chemin: Remarques sur l’existence pour le systeme de Navier-Stokes incompressible, SIAMJournal of Mathematical Analysis, 23, pages 20–28 (1992).
[8] J.-Y. Chemin: Theoremes d’unicite pour le systeme de Navier-Stokes tridimensionnel, Journald’Analyse Mathematique, 77, pages 27–50 (1999).
[9] Q. Chen, C. Miao and Z. Zhang: Well-posedness in critical spaces for the compressible Navier-Stokesequations with density dependent viscosities, Revista Matematica Iberoamericana, 26(3), pages 915–946(2010).
[10] Q. Chen, C. Miao and Z. Zhang: Global well-posedness for the compressible Navier-Stokes equationswith the highly oscillating initial velocity, Communications on Pure and Applied Mathematics, 63(9),pages 1173–1224 (2010).
[11] Q. Chen, C. Miao and Z. Zhang: On the ill-posedness of the compressible Navier-Stokes equations inthe critical Besov spaces, arXiv:1109.6092.
[12] N. Chikami and R. Danchin: On the well-posedness of the full compressible Navier-Stokes system incritical Besov spaces, Journal of Differential Equations, 258, pages 3435–3467 (2015).
[13] Y. Cho, H.J. Choe and H. Kim: Unique solvability of the initial boundary value problems for com-pressible viscous fluids, Journal de Mathematiques Pures et Appliquees, 83(2), pages 243–275 (2004).
[14] R. Danchin: Global existence in critical spaces for compressible Navier-Stokes equations, InventionesMathematicae, 141(3), pages 579–614 (2000).
[15] R. Danchin: Local theory in critical spaces for compressible viscous and heat-conductive gases, Com-munications in Partial Differential Equations, 26, 1183–1233 (2001).
[16] R. Danchin: Global existence in critical spaces for flows of compressible viscous and heat-conductivegases, Archive for Rational Mechanics and Analysis, 160(1), 2001, pages 1–39.
[17] R. Danchin: Zero Mach number limit in critical spaces for compressible Navier-Stokes equations,
Annales Scientifiques de l’Ecole Normale Superieure, 35(1), pages 27–75 (2002).[18] R. Danchin: Zero Mach number limit for compressible flows with periodic boundary conditions, Amer-
ican Journal of Mathematics, 124(6), pages 1153–1219 (2002).[19] R. Danchin: On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA
Nonlinear Differential Equations Appl., 12(1), 111–128 (2005).[20] R. Danchin : Fourier Analysis Methods for PDEs, 2006, available on http://perso-math.univ-mlv.fr
/users/danchin.raphael/recherche.html/.[21] R. Danchin: Well-posedness in critical spaces for barotropic viscous fluids with truly nonconstant
density, Communications in Partial Differential Equations, 32, 1373–1397 (2007).[22] R. Danchin: A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations,
Science China, Mathematics, 55(2), 245–275 (2012).[23] R. Danchin: A Lagrangian approach for the compressible Navier-Stokes equations, Annales de l’Institut
Fourier, 64(2), pages 753–791 (2014).[24] R. Danchin and L. He: The incompressible limit in Lp type critical spaces, hal-01067758 (2014).[25] R. Danchin and X. Liao: On the well-posedness of the full low-Mach number limit system in general
critical Besov spaces, Communications in Contemporary Mathematics, 14(3) 47 pages (2012).[26] R. Danchin and P. B. Mucha: A Lagrangian approach for the incompressible Navier-Stokes equations
with variable density, Communications on Pure and Applied Mathematics, 65(10), 1458–1480 (2012).[27] B. Desjardins and E. Grenier: Low Mach number limit of viscous compressible flows in the whole space,
Royal Society of London Proceedings, Series A, 455, pages 2271–2279 (1999).[28] E. Feireisl: Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2003[29] E. Feireisl and A. Novotny: Weak–Strong Uniqueness Property for the Full Navier-Stokes-Fourier
system, Archive for Rational Mechanics and Analysis, 204(2), pages 683–706 (2012).[30] H. Fujita and T. Kato: On the Navier-Stokes initial value problem I, Archive for Rational Mechanics
and Analysis, 16, 269-315 (1964).
[31] P. Gerard and J. Rauch: Propagation de la regularite locale de solutions d’´quations aux derivdespartielles non lineaires, Annales de l’Institut Fourier, 37, pages 65–84 (1987).
[32] G. Furioli, P.-G. Lemarie-Rieusset and E. Terraneo: Unicite des solutions mild des equations de Navier-Stokes dans L3(R3) et d’autres espaces limites, Revista Matematica Iberoamericana, 16(3), pages 605–667 (2000).
56 R. DANCHIN
[33] P. Germain: Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math.Fluid Mech., 13(1), pages 137–146 (2011).
[34] Y. Giga: Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of theNavier-Stokes system. J. Differential Equations, 62(2), pages 186–212 (1986).
[35] J. Ginibre and G. Velo: Generalized Strichartz inequalities for the wave equations, Journal of FunctionalAnalysis, 133, pages 50–68 (1995).
[36] B. Haspot: Well-posedness in critical spaces for the system of compressible Navier-Stokes in largerspaces, Journal of Differential Equations, 251, pages 2262–2295 (2011).
[37] B. Haspot: Existence of global strong solutions in critical spaces for barotropic viscous fluids, Archivefor Rational Mechanics and Analysis, 202(2), pages 427–460 (2011).
[38] D. Hoff: Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropicflow with discontinuous initial data, Journal of Differential Equations, 95, pages 33–74 (1992).
[39] D. Hoff: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions,Comm. Pure Appl. Math., 55(11), pages 1365–1407 (2002).
[40] D. Hoff: Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressibleflow, SIAM Journal on Mathematical Analysis, 37(6), pages 1742–1760 (2006).
[41] N. Itaya: On the initial value problem of the motion of compressible viscous fluid, especially on theproblem of uniqueness, Journal of Mathematics of Kyoto University, 16, pages 413–427 (1976).
[42] T. Kato: Strong Lp -solutions of the Navier-Stokes equation in Rm , with applications to weak solutions.Math. Z. 187(4) 471–480 (1984).
[43] M. Keel and T. Tao: Endpoint Strichartz estimates, American Journal of Mathematics, 120, 1998,pages 995–980.
[44] H. Kozono and M. Yamazaki: Semilinear heat equations and the Navier-Stokes equations with distri-butions in new function spaces as initial data, Communications in Partial Differential Equations, 19,pages 959–1014 (1994).
[45] H.-0. Kreiss, J. Lorenz and M. Naughton: Convergence of the solutions of the compressible to thesolutions of the incompressible Navier-Stokes equations, Advances in Pure and Applied Mathematics,12, pages 187–214 (1991).
[46] J. Leray: Sur le mouvement d’un liquide visqueux remplissant l’espace, Acta Mathematica, 63, pages193–248 (1934).
[48] A. Matsumura and T. Nishida: The initial value problem for the equations of motion of viscous andheat-conductive gases, Journal of Mathematics of Kyoto University, 20, pages 67–104 (1980).
[49] S. Montgomery-Smith: Finite time blow up for a Navier-Stokes like equation. Proceedings of the Amer-ican Mathematical Society, 129(10), 2001, pages 3025–3029.
[50] The Cauchy problem for the compressible Navier-Stokes equations in the Lp -framework, NonlinearAnal., 52(4), pages 1379–1392 (2003).
[51] J. Nash: Le probleme de Cauchy pour les equations differentielles d’un fluide general, Bulletin de laSociete Mathematique de France, 90, pages 487–497 (1962).
[52] T. Runst and W. Sickel: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partialdifferential equations, Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin (1996).
[53] S. Schochet: Fast singular limits of hyperbolic PDEs. Journal of Differential Equations, 114(2), pages476–512 (1994).
[54] J. Serrin: On the uniqueness of compressible fluid motions, Archive for Rational Mechanics and Anal-ysis, 3, pages 271–288 (1959).
[55] Y. Shizuta and S. Kawashima: Systems of equations of hyperbolic-parabolic type with applications tothe discrete Boltzmann equation, Hokkaido Math. J., 14(2), pages 249–275 (1985).
[56] A. Tani: On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res.Inst. Math. Sci., 13, pages 193–253 (1977).
[57] H. Triebel: Interpolation theory, function spaces, differential operators. North-Holland MathematicalLibrary, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.
[58] A. Valli: An existence theorem for compressible viscous fluids, Annali di Matematica Pura ed Applicata,130 (4), pages 197–213 (1982).
FOURIER ANALYSIS METHODS FOR COMPRESSIBLE FLOWS 57
[59] A. Valli and W. Zajaczkowski: Navier-Stokes equations for compressible fluids: global existence andqualitative properties of the solutions in the general case, Communications in Mathematical Physics,103(2), pages 259–296 (1986).
(R. Danchin) Universite Paris-est, LAMA UMR 8050 and Institut Universitaire de France,61, avenue du General de Gaulle, 94010 Creteil Cedex, France
