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SCIENCE CHINA Mathematics

. ARTICLES . February 2012 Vol. 55 No. 2: 245–275 doi: 10.1007/s11425-011-4357-8

A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations

Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010

DANCHIN Rapha¨el

LAMA UMR 8050, Universit´eParis-Est, 61, avenue du G´en´eral de Gaulle, 94010 Cr´eteilCedex, France Email: [email protected]

Received March 11, 2011; accepted May 17, 2011

Abstract Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey paper, we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space. We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics.

Keywords compressible fluids, critical regularity, Besov spaces, Fourier analysis, Mach number

MSC(2010) 35B25, 35B40, 76N10

Citation: Danchin R. A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations. Sci China Math, 2012, 55(2): 245–275, doi: 10.1007/s11425-011-4357-8

1 Introduction

This survey aims at presenting a few recent results for the compressible Navier-Stokes equations that have been proved by means of modern Fourier analysis methods. These techniques are relevant in any context where a good notion of Fourier transform is available. Here, for simplicity, we shall concentrate on the whole space case Rd (that is, the space variable of the solutions to the Navier-Stokes equations will be in Rd). However, our approach may be easily adapted to periodic boundary conditions x ∈ Td and more generally to x ∈ Td1 × Rd2 and so on (see, for example, [16]). Let us also stress that the methods that we introduce here are quite robust. As a matter of fact, we expect this survey to be a guidebook for solving more elaborate evolutionary nonlinear partial differential equations not necessarily related to fluid mechanics. In addition to the Fourier analysis techniques, adopting a critical functional framework will be another thread of these notes. Let us present the idea on an abstract model problem. Let X be a Banach space of functions v = v(t, x) with t ∈ R+ and x ∈ Rd, and B : X × X → X, a continuous bilinear map with norm

M. Then the standard fixed point theorem ensures that there exists a unique solution v in B(0, 2∥v0∥X ) to

v = v0 + B(v, v), (E) whenever

4M∥v0∥X < 1. (1.1)

⃝c Science China Press and Springer-Verlag Berlin Heidelberg 2012 math.scichina.com www.springerlink.com 246 Danchin R Sci China Math February 2012 Vol. 55 No. 2

Assume that (E) has the following scaling invariance property: there exists a triplet (α, β, γ) such that for all λ > 0, the transform α β γ v 7→ vλ with vλ(t, x) = λ v(λ t, λ x)

preserves Equation (E) if v0 has been changed into

α γ v0,λ : x 7−→ λ v0(λ x).

Hence, either (E) may be solved for any data in X (which is not the most common case) or the condition (1.1) itself has to be invariant by rescaling, that is,

∥v0,λ∥X ≈ ∥v0∥X , ∀ λ > 0.

If so, we shall say that the space X is critical for (E). This simple argument has been used a number of times for solving evolutionary equations such as nonlinear Schr¨odinger,wave or heat equations. Let us explain how to implement it on the incompressible Navier-Stokes equations { ∂tu + div(u ⊗ u) − µ∆u + ∇Π = 0, (NSI) divu = 0. At the formal level, solving (NSI) amounts to solving (E) with ∫ t t∆ (t−τ)∆ v0 := e u0 and B(v, v)(t) = − e Pdiv(v ⊗ v) dτ, 0

t∆ where (e )t>0 stands for the heat semi-group and P the Leray projector over divergence-free vector-fields. Now, it is clear that the above equation is invariant for all λ > 0 by the transform

2 v 7→ vλ with vλ(t, x) = λv(λ t, λx).

Hence, any space X which is norm invariant for every λ > 0 by the above transform is critical for the incompressible Navier-Stokes equation1) . In the context of the incompressible Navier-Stokes equations, this idea has been first implemented by Fujita and Kato [20] (see also the work by Chemin [5]). Here { } + d −1 1 + d − 1 X = v ∈ C(R ; H˙ 2 ) such that t 4 v ∈ C(R ; H˙ 2 2 )

d −1 and the initial data is in the homogeneous Sobolev space H˙ 2 . There are a number of critical functional spaces in which (NSI) may be globally solved for small data, for instance, •C(R+; Ld) (see the works by Giga [23], Kato [24] and Furioli-Lemari´e-Terraneo [21]); d −1 d +1 •C R+ ˙ p ∩ 1 R+ ˙ p ( ; Bp,1 ) L ( ; Bp,1 ) and more general Besov spaces (see the works by Cannone-Meyer- Planchon [4] and by Kozono and Yamazaki [29]). Presenting similar results in the context of compressible viscous fluids is the main goal of these notes. To simplify the presentation, we focus on the barotropic Navier-Stokes equations: { ∂tρ + div(ρu) = 0, ′ ∂t(ρu) + div (ρu ⊗ u) − µ∆u − µ ∇div u + ∇P = 0,

where • ρ = ρ(t, x) ∈ R+ (with t ∈ R+ and x ∈ Rd) is the density, • u = u(t, x) ∈ Rd is the velocity field,

1) Which, of course, does not mean that (NSI) may be solved in any such space! Danchin R Sci China Math February 2012 Vol. 55 No. 2 247

• the viscosity coefficients µ and µ′ satisfy µ > 0 and ν := µ + µ′ > 0. In order to close the system, we assume the pressure P to be a given (suitably smooth) function of ρ. This is the so-called barotropic assumption. In the viscous case that we shall consider, this assumption is somewhat irrelevant from a physical viewpoint. However, the above system already contains many features of the full model as far as mathematical results are concerned. We supplement the system with the following boundary conditions: • u decays to zero at infinity, • ρ tends to some positive constantρ ¯ at infinity. We shall takeρ ¯ = 1 for simplicity. Denoting ρ = 1+a and assuming that the density is positive everywhere, the barotropic system rewrites { ∂ta + u · ∇a = −(1 + a)divu, (1.2) ∂tu − Au = −u · ∇u − J(a)Au − ∇G(a),

where A := µ∆ − µ′∇div,J(a) := a/(1 + a) and G is a smooth function with G(0) = 0. Strictly speaking, the above system does not own any scaling invariance property in the above meaning. However, we notice that up to a change of G into λ2G, System (1.2) is invariant by the rescaling

a(t, x) → a(λ2t, λx), u(t, x) → λu(λ2t, λx). (1.3)

To some extent, the term G is lower order. Therefore, we expect the critical functional spaces for the velocity to be the same ones as in the incompressible case, whereas one more derivative has to be taken for the density. This heuristics will be made more explicit later on in the paper. Our paper unfolds as follows. The first section is dedicated to a short presentation of the Fourier analysis toolbox. There, the Littlewood-Paley decomposition plays a central role. In passing, we shall prove a priori estimates in Besov spaces for the heat and transport equations. In the second section, we focus on the basic local well-posedness theory in a “critical” functional framework for (1.2). As a warm-up, we shall consider the case where the initial density is “almost” a constant, which may be achieved by perturbative methods. Next, we shall consider the case of large data with truly nonconstant density (bounded away from zero), which is more demanding. In all this section, we will not use much the structure of the equations. Roughly, our method would apply to any reasonable coupling between a transport and a heat equation. In the third section, we present a global existence result for small perturbations of a constant stable equilibrium. This will require a deeper analysis of the structure of (1.2) (at the linear level in particular). In the last section, we shall establish the convergence to the incompressible Navier-Stokes equations for general ill-prepared data in the low Mach number regime. There, the dispersive properties of the linearized system in the whole space will play a fundamental role. In contrast with the results of the previous sections, those properties are specific to the whole space case and proving convergence in the periodic framework would require other techniques (see [13]).

2 The Fourier analysis toolbox

Here we introduce the Littlewood-Paley decomposition, define Besov spaces, state product estimates and a priori estimates for the heat equation and the transport equation. The detailed proofs may be found in [2].

2.1 A primer on Littlewood-Paley theory

The Littlewood-Paley decomposition is a localization procedure for tempered distributions in the fre- quency space. The interest of this localization is that the derivatives act almost as homotheties on distributions the Fourier transform of which is supported in a ball or an annulus. 248 Danchin R Sci China Math February 2012 Vol. 55 No. 2

Owing to Parseval’s formula, this fact is obvious in the L2 framework. The Bernstein inequalities state that it is also true in the Lp framework. More precisely, • the direct Bernstein inequality states that for any R > 0, a constant C exists so that, for any k ∈ N, any couple (p, q) in [1, ∞]2 with q > p > 1 and any function u of Lp with Supp ub ⊂ B(0, λR) for some λ > 0, we have k k+1 k+d( 1 − 1 ) ∥D u∥Lq 6 C λ p q ∥u∥Lp ; • the reverse Bernstein inequality asserts that for any 0 < r < R, there exists a constant C so that for any k ∈ N, p ∈ [1, ∞] and any function u of Lp with Supp ub ⊂ {ξ ∈ Rd / rλ 6 |ξ| 6 Rλ} for some λ > 0, we have k k+1 k λ ∥u∥Lp 6 C ∥D u∥Lp . As solutions to nonlinear PDE’s need not be spectrally localized in annuli (even if we restrict to initial data with this property), it is suitable to have a device which allows for splitting any function into a sum of functions with this property. This is exactly what Littlewood-Paley’s decomposition does. ⊂ 4 ≡ 3 To construct it, fix some smooth bump function χ with Supp χ B(0, 3 ) and χ 1 on B(0, 4 ), then set φ(ξ) = χ(ξ/2) − χ(ξ) so that ∑ φ(2−jξ) = 1, if ξ ≠ 0. j∈Z

j For j ∈ Z, the dyadic block ∆˙ j (which is bound to localize about the frequency 2 ) is defined by

−j ∆˙ j := φ(2 D).

The homogeneous Littlewood-Paley decomposition for u reads ∑ u = ∆˙ ju. (2.1) j

Note that for a general tempered distribution, Equality (2.1) holds true modulo polynomials, a fact which is quite unpleasant when dealing with nonlinear problems. As in our analysis, we shall consider only solutions which tend to 0 (or to some given positive constant) at infinity, it is natural to restrict our attention to distributions which tend to 0 at infinity. Following S′ the approach of [2], we shall thus consider the set h of tempered distributions u such that

lim ∥S˙ju∥L∞ = 0, j→−∞

−j where S˙j stands for the low frequency cut-off defined by S˙j := χ(2 D). Note that Equality (2.1) holds S′ true whenever u is in h. One can now define what a homogeneous Besov space is. Definition 2.1. For s ∈ R and 1 6 p, r 6 ∞, we set

( ) 1 ∑ r rjs ˙ r ∥u∥ ˙ s := 2 ∥∆ju∥ p Bp,r L j

if r < ∞ and js ˙ ∥u∥ s := sup 2 ∥∆ u∥ p . B˙ ∞ j L p, j ˙ s ′ We then define the space B as the subset of distributions u ∈ S such that ∥u∥ ˙ s is finite. p,r h Bp,r ˙ s p Roughly, having u in Bp,r means that u has s fractional derivatives in L . In particular, it is obvious that ˙ s ˙ s ˙ s B2,2 coincides with the homogeneous Sobolev space H and it is true that B∞,∞ is the homogeneous H¨olderspace C˙ 0,s if s ∈ (0, 1). Danchin R Sci China Math February 2012 Vol. 55 No. 2 249

Here are some important properties of Besov spaces: • ˙ s 6 Bp,r is a Banach space whenever s < d/p or s d/p and r = 1, • for any p ∈ [1, ∞], we have the following chain of continuous embedding:

˙ 0 → p → ˙ 0 Bp,1 , L , Bp,∞,

s−d( 1 − 1 ) • if s ∈ R, 1 6 p 6 p 6 ∞ and 1 6 r 6 r 6 ∞, then B˙ s ,→ B˙ p1 p2 , 1 2 1 2 p1,r1 p2,r2 d • ∞ ˙ p if p < , then Bp,1 is an algebra continuously embedded in the set of continuous functions decaying to 0 at infinity,

• the following real interpolation property is satisfied for all 1 6 p, r1, r2, r 6 ∞, s1 ≠ s2 and θ ∈ (0, 1): − [B˙ s1 , B˙ s2 ] = B˙ θs2+(1 θ)s1 , p,r1 p,r2 (θ,r) p,r • for any smooth homogeneous of degree m function F on Rd \{0}, the Fourier multiplier F (D) maps ˙ s ˙ s−m ˙ s ˙ s−1 Bp,r in Bp,r . In particular, the gradient operator maps Bp,r in Bp,r .

2.2 Product estimates

Formally, any product of two distributions u and v may be decomposed into

uv = Tuv + R(u, v) + Tvu (2.2) with ∑ ∑ ∑ Tuv := S˙j−1u∆˙ jv and R(u, v) := ∆˙ ju ∆˙ j′ v. j j |j′−j|61 The above operator T is called “paraproduct” whereas R is called “remainder”. The decomposition (2.2) has been first introduced by Bony [3]. The bilinear operators R and T possess a score of continuity properties in standard functional spaces (see in particular [32]). In these notes, we shall make an extensive use of the following ones: Proposition 2.2. For any (s, p, r) ∈ R × [1, ∞]2 and t < 0, we have2)

∥Tuv∥ ˙ s . ∥u∥L∞ ∥v∥ ˙ s and ∥Tuv∥ ˙ s+t . ∥u∥ ˙ t ∥v∥ ˙ s . Bp,r Bp,r Bp,r B∞,∞ Bp,r

2 For any (s1, p1, r1) and (s2, p2, r2) in R × [1, ∞] , we have

• if s1 + s2 > 0, 1/p := 1/p1 + 1/p2 6 1 and 1/r := 1/r1 + 1/r2 6 1, then ∥ ∥ . ∥ ∥ ∥ ∥ R(u, v) ˙ s1+s2 u ˙ s1 v ˙ s2 ; Bp,r Bp1,r1 Bp2,r2

• if s1 + s2 = 0, 1/p := 1/p1 + 1/p2 6 1 and 1/r1 + 1/r2 > 1, then

∥R(u, v)∥ ˙ 0 . ∥u∥ ˙ s1 ∥v∥ ˙ s2 . Bp,∞ Bp1,r1 Bp2,r2 Putting together decomposition (2.2) and the above results of continuity, one may deduce a number of continuity results for the product of two functions. For instance, one may get the following tame estimate: ∞ ∩ ˙ s ∈ ∞ 2 Corollary 2.3. Let a and b be in L Bp,r for some s > 0 and (p, r) [1, ] . Then there exists a constant C depending only on d, p and s such that ( ) ∞ ∞ ∥ab∥ ˙ s 6 C ∥a∥L ∥b∥ ˙ s + ∥b∥L ∥a∥ ˙ s . Bp,r Bp,r Bp,r Finally, the following result (the proof of which relies on Meyer’s first linearization method) will be needed for handling the terms J(a) and G(a) in (NSC).

2) The sign . means that the l.h.s. is bounded by the r.h.s. up to a harmless multiplicative constant. 250 Danchin R Sci China Math February 2012 Vol. 55 No. 2

Proposition 2.4. Let I be a bounded interval of R and F : I → R be a smooth function with F (0) = 0. Then for all compact subset J of I, all (p, r) ∈ [1, ∞]2 and all positive s with s < d/p if r > 1 and s 6 d/p ∈ ˙ s ∈ ˙ s if r = 1, there exists a constant C such that for all a Bp,r with values in J, we have F (a) Bp,r and

∥F (a)∥ ˙ s 6 C∥a∥ ˙ s . Bp,r Bp,r 2.3 A priori estimates for transport equations

The mass equation in (NSC) enters in the class of transport equations. Hence it is suitable to have good a priori estimates for such equations. More precisely, consider the transport equation3) { 1 ∂ta + v · ∇a = f ∈ L (X), T (T) a|t=0 = a0 ∈ X. Roughly, if v is a Lipschitz time-dependent vector-field and if X is a “reasonable” Banach space, then we expect (T) to have a unique solution a ∈ C([0,T ); X) satisfying ( ∫ ) t CV (t) −CV (τ) ∥a(t)∥X 6 e ∥a0∥X + e ∥f(τ)∥X dτ (2.3) 0 ∫ t ∥∇ ∥ ∞ 0,ε ∈ with (say) V (t) := 0 v(τ) L dτ. This is quite obvious if X is the H¨olderspace C (with ε (0, 1)) as (in the case f ≡ 0 to simplify) the solution to (T) is given by

−1 a(t, x) = a0(ψt (x)),

where ψt stands for the flow of v at time t. Therefore, | − | | −1 − −1 | a(t, x) a(t, y) = a0(ψt (x)) a0(ψt (y)) 6 ∥ ∥ | −1 − −1 |ε a0 C˙ 0,ε ψt (x) ψt (y) 6 ∥ ∥ ∥∇ −1∥ε | − |ε a0 C˙ 0,ε ψt L∞ x y . −1 ∥∇ ∥ ∞ 6 As ψt L exp(V (t)), we get the result in this particular case. Littlewood-Paley’s decomposition will enable us to prove a similar result in a much more general framework. ˙ s Theorem 2.5. The above result holds true for X = Bp,r with ∫ t V (t) = ∥∇v(τ)∥ d dτ, ˙ p1 0 Bp ,1 ( ) 1 d d d whenever 1 6 p 6 p1 6 ∞, 1 6 r 6 ∞ and − min , ′ < s < 1 + · If r = 1 (resp. r = ∞), then the ( ) p1 p p1 d d case s = 1 + d/p1 (resp. s = − min , ′ ) also works. p1 p

Proof. Applying ∆˙ j to (T ) gives

∂t∆˙ ja + v · ∇∆˙ ja = ∆˙ jf + R˙ j (2.4)

p−2 with R˙ j := [v · ∇, ∆˙ j]a. In the case p ∈ (1, ∞), multiplying both sides by |∆˙ ja| ∆˙ ja and integrating over Rd yields ∫ ∫ ( ) 1 d ˙ p 1 ˙ p ˙ ˙ p−2 ˙ ∥∆ a∥ p + v · ∇|∆ a| dx = ∆ f + R |∆ a| ∆ a dx. p dt j L p j j j j j Therefore ∫ ( ) t ∥divv∥L∞ ∥∆˙ ja(t)∥Lp 6 ∥∆˙ ja0∥Lp + ∥∆˙ jf∥Lp + ∥R˙ j∥Lp + ∥∆˙ ja∥Lp dτ. (2.5) 0 p

3) ∈ ∞ p p p p R+ Whenever X is a Banach space and p [1, ], we shall denote LT (X) := L ((0,T ); X) and L (X) := L ( ; X). Danchin R Sci China Math February 2012 Vol. 55 No. 2 251

Having p tend to 1 or ∞ implies that (2.5) also holds if p = 1 or p = ∞.

Now, under the above conditions over s, p, the remainder term R˙ j satisfies

∥ ˙ ∥ 6 −js∥∇ ∥ ∥ ∥ Rj(t) Lp Ccj(t)2 v(t) d a(t) B˙ s (2.6) B˙ p1 p,r p1,1 with ∥(cj(t))∥ℓr = 1. This may be proved by taking advantage of Bony’s decomposition. Indeed we have (with the summation convention over repeated indices):

k k k k R˙ = [T k , ∆˙ ]∂ a + T v − ∆˙ T v + R(v , ∂ ∆˙ a) − ∆˙ R(v , ∂ a). j v j k ∂k∆˙ j a j ∂ka k j j k

Let us just explain how to bound the first term which is the only one where having a commutator improves the estimates (bounding the other terms stems mostly from Proposition 2.2). Owing to the properties of spectral localization, we have ∑ ˙ ˙ k ˙ ˙ [Tvk , ∆j]∂ka = [Sj′−1v , ∆j]∂k∆j′ a. |j−j′|64

Let h := F −1φ. Remark that ∫ ( ) k jd j k k [S˙j′−1v , ∆˙ j]∂k∆˙ j′ a(x) = 2 h(2 (x − y)) S˙j′−1v (x) − S˙j′−1v (y) ∂k∆˙ j′ a(y) dy. Rd Hence, according to the mean value formula,

k [S˙j′−1v ,∆˙ j]∂k∆˙ j′ a(x) ∫ ∫ 1 jd j k = 2 h(2 (x − y))((x − y) · ∇S˙j′−1v (y + τ(x − y)))∂k∆˙ j′ a(y) dτ dy. Rd 0 So finally, ∑ ∑ −j ∥ ˙ ∥ . ∥∇ ∥ ∥ ˙ ′ ∥ . ∥∇ ∥ ∥ ˙ ′ ∥ [Tvk , ∆j]∂ka Lp 2 v L∞ ∂k∆j a Lp v L∞ ∆j a Lp . |j′−j|64 |j′−j|64

d Hence, given that B˙ p1 ,→ L∞, this term may be bounded according to (2.6). p1,1 Let us resume to (2.4). Using (2.5) and (2.6), multiplying by 2js, then summing up over j yields ∫ ∫ t t ∥ ∥ 6 ∥ ∥ 6 ∥ ∥ ∥ ∥ ′∥ ∥ a L∞(B˙ s ) a Le∞(B˙ s ) a0 B˙ s + f B˙ s dτ + C V a B˙ s dτ t p,r t p,r p,r 0 p,r 0 p,r

∥ ∥ js∥ ˙ ∥ ∞ with a e∞ ˙ s := 2 ∆ja L (Lp) r . Lt (Bp,r ) t ℓ Then applying Gronwall’s lemma yields the desired inequality for a.

2.4 A maximal regularity estimate for the heat equation

Consider the heat equation

∂tu − ∆u = f, u|t=0 = u0. We want to find a Banach space X for which

2 ∥∂tu, D u∥L1(X) 6 C∥f∥L1(X) (2.7)

1 if u0 ≡ 0. This gain of two derivatives compared to the source term when performing a L -in-time integra- tion will be the key to our well-posedness results in a critical functional framework for the compressible Navier-Stokes equations. 252 Danchin R Sci China Math February 2012 Vol. 55 No. 2

Now, it is well known that if r ∈ (1, ∞) and X = Lq or W˙ s,q for some s ∈ R and q ∈ (1, ∞), then

2 ∥∂tu, D u∥Lr (X) 6 C∥f∥Lr (X).

The inequality fails for the endpoint case r = 1 for those spaces X, though. However, it has been noticed by Chemin [6] that Inequality (2.7) is true for Besov spaces with third index 1. This is stated in the following theorem. ∈ ∞ ∈ R ˙ s Theorem 2.6. For any p [1, ] and s , Inequality (2.7) is true with X = Bp,1. More generally, ˙ s 1 R+ ˙ s if the initial data u0 is in Bp,1 and if f is in L ( ; Bp,1), then the above heat equation has a unique C R+ ˙ s ∩ 1 R+ ˙ s+2 solution u in ( ; Bp,1) L ( ; Bp,1 ) and for all t > 0,

∥u∥e∞ ˙ s + ∥u∥ 1 ˙ s+2 6 C(∥u0∥ ˙ s + ∥f∥ 1 ˙ s ). (2.8) Lt (Bp,1) Lt (Bp,1 ) Bp,1 Lt (Bp,1) Proof. We focus on the proof of the inequality. For any j ∈ Z, we have

∂t∆˙ ju − ∆∆˙ ju = ∆˙ jf.

Hence, according to Duhamel’s formula ∫ t t∆ (t−τ)∆ ∆˙ ju(t) = e ∆˙ ju0 + e ∆˙ jf(τ) dτ. 0 Let us admit for a while that there exist two positive constants c and C such that for any j ∈ Z, p ∈ [1, ∞] and λ ∈ R+, we have λ∆ −cλ22j ∥e ∆˙ jv∥Lp 6 Ce ∥∆˙ jv∥Lp . (2.9)

Then we get ∫ t −c22j t −c22j (t−τ) ∥∆˙ ju(t)∥Lp . e ∥∆˙ ju0∥Lp + e ∥∆˙ jf(τ)∥Lp dτ. (2.10) 0 Multiplying by 2js and summing up over j yields ∫ ∑ ∑ t ∑ js −c22j t js −c22j (t−τ) 2 ∥∆˙ ju(t)∥Lp . e 2 ∥∆˙ ju0∥Lp + e ∥∆˙ jf(τ)∥Lp dτ, j j 0 j whence

∥u∥ ∞ ˙ s 6 ∥u∥e∞ ˙ s . ∥u0∥ ˙ s + ∥f∥ 1 ˙ s . Lt (Bp,1) Lt (Bp,1) Bp,1 Lt (Bp,1) Note that combining (2.10) with convolution inequalities also yields

2j −c22j t 2 ∥∆˙ u∥ 1 p . (1 − e )(∥∆˙ u ∥ p + ∥∆˙ f∥ 1 p ). j Lt (L ) j 0 L j Lt (L ) Now, multiplying by 2js and summing over j yields ∑ ∥ ∥ . − −c22j t js ∥ ˙ ∥ ∥ ˙ ∥ u 1 ˙ s+2 (1 e )2 ( ∆ju0 Lp + ∆jf L1(Lp)), (2.11) Lt (Bp,1 ) t j

which is slightly better than what we wanted to prove4) . For the sake of completeness, let us prove (2.9). If p = 2 this is a mere consequence of Parseval’s formula. In the general case, one may first reduce the proof to the case j = 0 (just perform a suitable change of variable) then consider a function ϕ in D(Rd \{0}) with value 1 on a neighborhood of the support of φ so as to write

λ∆ −1 −λ|ξ|2 d e ∆˙ 0u = F (ϕ(ξ)e ∆˙ 0u(ξ)) = gλ ⋆ u

2j 4) As obviously (1 − e−c2 t) is bounded by 1 and tends to 0 when t goes to 0+. This fact will be used later on when proving local existence for the compressible Navier-Stokes equations. Danchin R Sci China Math February 2012 Vol. 55 No. 2 253

with ∫ −d i(x|ξ) −λ|ξ|2 gλ(x) := (2π) e ϕ(ξ)e dξ.

From repeated integration by parts, one easily gets ∫ 2 −d i(x|ξ) d −λ|ξ|2 gλ(x) = (1 + |x| ) e (Id − ∆ξ) (ϕ(ξ)e )dξ. Rd

So using Leibniz and Fa´a-di-Bruno’sformulae, we conclude that

2 −d −cλ |gλ(x))| 6 C(1 + |x| ) e .

Therefore λ∆ ˙ ˙ −cλ ˙ ∥e ∆0u∥Lp 6 ∥gλ∥L1 ∥∆0u∥Lp 6 Ce ∥∆0u∥Lp , and we are done. In the whole space case (and also for periodic boundary conditions), it turns out that the above estimates for the heat equation extend to the Lam´esystem

′ ∂tu − µ∆u − µ ∇divu = f, u|t=0 = u0, (L)

whenever µ > 0 and ν := µ + µ′ > 0. Indeed, if P and Q denote the projector over divergence-free and potential vector fields, respectively, then we have

∂tPu − µ∆Pu = Pf and ∂tQu − ν∆Qu = Qf. Therefore, given that P and Q are homogeneous multipliers of degree 0 (as FQu(ξ) = |ξ|−2ξ (ξ ·Fu(ξ)) and P = Id − Q), the above result for the heat equation implies ∈ ˙ s ∈ 1 R+ ˙ s C R+ ˙ s ∩ Corollary 2.7. If u0 Bp,1 and f L ( ; Bp,1), then (L) has a unique solution u in ( ; Bp,1) 1 R+ ˙ s L ( ; Bp,1) and

∥u∥e∞ ˙ s + ∥u∥ 1 ˙ s+2 6 C(∥u0∥ ˙ s + ∥f∥ 1 ˙ s ). Lt (Bp,1) Lt (Bp,1 ) Bp,1 Lt (Bp,1)

3 The local existence theory

This section is dedicated to the local well-posedness issue for the barotropic Navier-Stokes equations   ∂ta + u · ∇a = −(1 + a)divu,  (NSC) ∂tu − Au = −u · ∇u − J(a)Au − ∇G(a),

in the whole space, for data in “critical” functional spaces. The important fact is that, if one restricts to local-in-time results for the Cauchy problem, then the density and the velocity equations may be considered separately, that is, it will be enough to use the priori estimates for the transport equation and Lam´esystem that have been proved in the previous section. Recall that up to a change of the pressure law, the barotropic compressible Navier-Stokes equations 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

d d −1 ∈ ˙ p1 ∈ ˙ p2 a0 Bp1,r1 and u0 Bp2,r2 . 254 Danchin R Sci China Math February 2012 Vol. 55 No. 2

However, it is not clear that one may solve (NSC) 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 d ∞ ˙ p1 velocity equation), an L control for a is needed. In the scale of Besov spaces Bp1,r1 , having r1 = 1 is the only choice which ensures (continuous) inclusion in L∞ As for the velocity equation, a gain of two derivatives is required to handle the term J(a)Au (as A is

second order). According to Corollary 2.7, we thus have to take r2 = 1. This is all the more appropriate d +1 because this will ensure that u ∈ L1 (B˙ p2 ), hence that u ∈ L1 (C0,1), a property that is needed to T p2,1 T transport the initial Besov regularity of a. Finally, owing to the coupling between the mass and velocity equations, it is natural (but not mandatory, see [25]) to take p1 = p2 = p. So, in short, we want to solve (NSC) locally for

d d −1 ∈ ˙ p ∈ ˙ p a0 Bp,1 and u0 Bp,1 .

Before going further into the proof of existence, let us emphasize that one cannot expect to reduce System (NSC) to the model problem presented in the introduction. This is due to the hyperbolic nature of the transport equation which entails a loss of one derivative in the Lipschitz-type stability estimates. Hence, existence will rather stem from bounds in high norm for the solution and stability in low norms or, alternately, from Schauder-Tikhonov type fixed point arguments. The exact procedure will be described with more details below.

3.1 A first local existence result

As a warm-up, we concentrate on the case where a0 is small (after [14] and [15]). This case is simpler because it requires only estimates for the transport equation and for the Lam´esystem with constant d d −1 coefficients (namely, Theorem 2.5 and Corollary 2.7). Given that a ∈ B˙ p and u ∈ B˙ p , we expect ( ) 0 p,1 0 p,1 d ∈ C ˙ p to construct a solution (a, u) such that a [0,T ]; Bp,1 and

d −1 d +1 ∈ C ˙ p ∩ 1 ˙ p u ([0,T ]; Bp,1 ) L ([0,T ]; Bp,1 ).

So, as a first step, we shall prove a priori estimates in “large norm” (namely in the above space) for the solutions to (NSC). A priori estimates in large norm for the density ∥ ∥ Let U(t) := u d +1 . According to Theorem 2.5, we have 1 ˙ p Lt (Bp,1 ) ( ∫ ) t CU(t) −CU ∥a∥ d 6 e ∥a0∥ d + e ∥(1 + a)divu∥ d dτ . ∞ ˙ p ˙ p ˙ p Lt (Bp,1) Bp,1 0 Bp,1

From product laws in Besov spaces, we have:

∥ ∥ . ∥ ∥ ∥ ∥ (1 + a)divu d (1 + a d ) u d +1 . ˙ p ˙ p ˙ p Bp,1 Bp,1 Bp,1

Inserting this in the above inequality and applying Gronwall’s lemma, we thus get

CU(T ) CU(T ) ∥a∥ d 6 e ∥a0∥ d + e − 1. ∞ ˙ p ˙ p LT (Bp,1) Bp,1

It is also obvious that ∫ ∫ t ∥div u∥ ∞ dτ t ∥div u∥ ∞ dτ ∥a∥ ∞ ∞ 6 ∥a ∥ ∞ e 0 L + e 0 L − 1. Lt (L ) 0 L Danchin R Sci China Math February 2012 Vol. 55 No. 2 255

Hence if for some suitably small η > 0, we have

CU(T ) 6 log 2 and eCU(T ) − 1 6 η, (3.1) then

∥a∥ d 6 2∥a0∥ d + η and ∥a∥L∞(L∞) 6 2∥a0∥L∞ + η. (3.2) ∞ ˙ p ˙ p T LT (Bp,1) Bp,1 A priori estimates in large norm for the velocity From Corollary 2.7, we get ∫ T ∥ ∥ . ∥ ∥ ∥ · ∇ A ∇ ∥ u d −1 d +1 u0 d −1 + u u + J(a) u + (G(a)) d −1 dt. ∞ ˙ p ∩ 1 ˙ p ˙ p ˙ p LT (Bp,1 ) LT (Bp,1 ) Bp,1 0 Bp,1 Product and composition laws in Besov spaces yield if d > 1 and 1 6 p < 2d,

∥ · ∇ ∥ . ∥ ∥ ∥ ∥ ∥ A ∥ . ∥ ∥ ∥ ∥ u u d −1 u d −1 u d +1 , J(a) u d −1 a d u d +1 , ˙ p ˙ p ˙ p ˙ p ˙ p ˙ p Bp,1 Bp,1 Bp,1 Bp,1 Bp,1 Bp,1 ∥∇ ∥ . ∥ ∥ (G(a)) d −1 a d . ˙ p ˙ p Bp,1 Bp,1 Hence ∫ T ∥ ∥ ∥ ∥ . ∥ ∥ ∥ ∥ ∥ ∥ u d −1 + u d +1 u0 d −1 + u d −1 u d +1 dt ∞ ˙ p 1 ˙ p ˙ p ˙ p ˙ p LT (Bp,1 ) LT (Bp,1 ) Bp,1 0 Bp,1 Bp,1 ∥ ∥ ∥ ∥ ∥ ∥ + a d u d +1 + T a d . ∞ ˙ p 1 ˙ p ∞ ˙ p LT (Bp,1) LT (Bp,1 ) LT (Bp,1)

The last-but-one term may be absorbed by the left-hand side if ∥a∥ d is small. According to (3.2), ∞ ˙ p LT (Bp,1) this may be ensured if ∥a0∥ d and η are small enough. Note also that applying Gronwall lemma shows ˙ p Bp,1 that the term with the integral may be eliminated for T small enough by virtue of (3.1). From this, we conclude that ( ) ∥ ∥ 6 ∥ ∥ ∥ ∥ u d −1 C u0 d −1 + T ( a0 d + η) . ∞ ˙ p ˙ p ˙ p LT (Bp,1 ) Bp,1 Bp,1

In order to ensure (3.1) for small enough T, one may split u into uL +u ¯ with uL solution to

∂tuL − AuL = 0, uL|t=0 = u0.

We have 6 ∥ ∥ ∥ ∥ U(T ) uL d +1 + u d +1 . 1 ˙ p 1 ˙ p LT (Bp,1 ) LT (Bp,1 ) The first term goes to 0 for T tending to 0. As for the second term, it is small for T small asu ¯(0) = 0. In order to get more information on the smallness of T, one may use the fact that u satisfies   ∂tu − Au = −u · ∇u − uL · ∇u − uL · ∇uL − J(a)Au − ∇(G(a)),  u|t=0 = 0.

By combining Corollary 2.7 and the product laws in Besov spaces, we thus get ∫ t ∥ ∥ ∥ ∥ . ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ u d +1 + u d −1 u d −1 u d +1 dτ + uL d u d 1 ˙ p ∞ ˙ p ˙ p ˙ p 2 ˙ p 2 ˙ p Lt (Bp,1 ) Lt (Bp,1 ) 0 Bp,1 Bp,1 Lt (Bp,1) Lt (Bp,1) ∥ ∥ ∥ ∥ + uL d +1 uL d −1 1 ˙ p ∞ ˙ p Lt (Bp,1 ) Lt (Bp,1 ) 256 Danchin R Sci China Math February 2012 Vol. 55 No. 2

∥ ∥ ∥ ∥ ∥ ∥ + a d u d −1 + t a d . (3.3) ∞ ˙ p 1 ˙ p ∞ ˙ p Lt (Bp,1) Lt (Bp,1 ) Lt (Bp,1) Note that arguing by interpolation yields for any β > 0,

∥ ∥ ∥ ∥ 6 ∥ ∥ ∥ ∥ −1∥ ∥ ∥ ∥ uL d u d β uL d −1 u d +1 + Cβ uL d +1 u d −1 . 2 ˙ p 2 ˙ p ∞ ˙ p 1 ˙ p 1 ˙ p ∞ ˙ p Lt (Bp,1) Lt (Bp,1) Lt (Bp,1 ) Lt (Bp,1 ) Lt (Bp,1 ) Lt (Bp,1 ) Therefore, taking β small enough, using Gronwall’s lemma, (3.1) and (3.2), we conclude by a standard bootstrap argument that the l.h.s. of (3.3) may be made smaller than any given ε, for all t ∈ [0,T ] if, for some α (depending on ε) we have ∥ ∥ 6 max(T, uL d +1 ) α. 1 ˙ p LT (Bp,1 ) Hence, according to (2.11), it suffices to choose T ∈]0, α] so that ∑ −c22j T js (1 − e )2 ∥∆˙ ju0∥Lp . α. j

This gives a (non so) explicit lower bound for the time interval on which the norm of the solution (a, u) may be bounded in terms of the initial data. Stability estimates in small norm Consider two solutions (a1, u1) and (a2, u2) of (NSC) with the above regularity. The difference (δa, δu) := 2 1 2 1 (a −a , u −u ) satisfies   ∑3  2  ∂tδa + u · ∇δa = δFi, i=1  ∑5 (3.4)   ∂tδu − Aδu = δGi, i=1 with

1 2 1 δF1 := −δu · ∇a , δF2 := −δa divu , δF3 := −(1 + a )divδu, 1 2 2 1 2 1 δG1 := (J(a ) − J(a ))Au , δG2 := −J(a ) Aδu, δG3 := −∇(G(a ) − G(a )), 2 1 δG4 := −u · ∇δu, δG5 := −δu · ∇u .

Owing to the hyperbolic nature of the mass equation, one loses one derivative in the stability estimates: 1 indeed, δF1 has (at most) the same regularity as ∇a . This induces also a loss of one derivative for δu (look at δG1 for instance). Hence, at most, we expect to prove stability estimates in

d −1 d −2 d C ˙ p × C ˙ p ∩ 1 ˙ p d FT := ([0,T ]; Bp,1 ) ( ([0,T ]; Bp,1 ) LT (Bp,1)) . The most obvious unpleasant effect of this loss of one derivative is that when applying composition

and product laws in Besov spaces for bounding the norms of δFi and δGj, one has to impose stronger conditions on p and on d. Roughly, Proposition 2.2 (which is optimal) guarantees that product makes sense whenever the sum of regularity indices is nonnegative. So in our case, if we assume that d > 2 and 1 1 6 p < d, then, after a few computation, we conclude that if T and ∥a ∥ d are small enough, then ∞ ˙ p LT (Bp,1) we have the following stability estimate: ∥ ∥ ∥ ∥ ∥ ∥ . ∥ ∥ ∥ ∥ δa d −1 + δu d −2 + δu d δa(0) d −1 + δu(0) d −2 , ∞ ˙ p ∞ ˙ p 1 ˙ p ˙ p ˙ p LT (Bp,1 ) LT (Bp,1 ) LT (Bp,1) Bp,1 Bp,1 which implies uniqueness. The limit case d = 2 or p = d is more involved as, for instance,

∈ ˙ 0 A 2 ∈ ˙ 0 ⇒ A 2 ∈ ˙ −1 (δa Bd,1 and u Bd,1) = δa u Bd,∞ only. (3.5) Danchin R Sci China Math February 2012 Vol. 55 No. 2 257

Hence estimates have to be performed in a Besov space with third index ∞. On the one hand, this is not a trouble for δa as applying Theorem 2.5 and product laws implies that

∥ 2∥ u L1 (B˙ 2 ) 1 T d,1 ∥δa∥ ∞ ˙ 0 6 (∥δa(0)∥ ˙ 0 + (1 + ∥a ∥ ∞ ˙ 1 )∥δu∥ 1 ˙ 1 )e . LT (Bd,∞) Bd,∞ LT (Bd,1) LT (Bd,1) On the other hand, owing to (3.5), one has to generalize Corollary 2.7 to Besov spaces with third index ∞. This is not quite possible. In fact, one may only get a control on the following norm:

∥ ∥ j∥ ˙ ∥ δu Le1 (B˙ 1 ) := sup 2 ∆jδu L1 (Ld). T d,∞ j T

This norm is definitely weaker than ∥δu∥ 1 ˙ 1 . However, one may prove the following logarithmic LT (Bd,1) interpolation inequality: ( ) ∥δu∥e1 ˙ 0 + ∥δu∥e1 ˙ 2 LT (Bd,∞) LT (Bd,∞) ∥δu∥ 1 ˙ 1 . ∥δu∥e1 ˙ 1 log e + . (3.6) L (B ) L (B ∞) T d,1 T d, ∥δu∥e1 ˙ 1 LT (Bd,∞) This is just a matter of splitting the l.h.s. into ∑ ∑ ∑ ∥ ∥ 6 j∥ ˙ ∥ j∥ ˙ ∥ j∥ ˙ ∥ δu 1 ˙ 1 2 ∆jδu L1 (Ld) + 2 ∆jδu L1 (Ld) + 2 ∆jδu L1 (Ld). LT (Bd,1) T T T j6−M −MN Owing to the definition of the Besov norms, one may write that ∑ j∥ ∥ . −M ∥ ∥ 2 δu L1 (Ld) 2 δu e1 ˙ 0 , T LT (Bd,∞) 6− j ∑M j∥ ∥ . − ∥ ∥ 2 δu L1 (Ld) (M + N 1) δu e1 ˙ 1 , T LT (Bd,∞) − MN Then taking the “best” integers M and N yields (3.6). Let us resume to the proof of stability estimates in the endpoint case d = 2 or p = d. Inserting (3.6) in the estimate for δa yields for some function α in L1(0,T ) depending only on the norms of the two solutions over [0,T ], ( ∫ ) t

X(t) 6 C ∥δa0∥ ˙ 0 + ∥δu0∥ ˙ −1 + α(τ)X(τ) log(e + 1/X(τ)) dτ , B ∞ B ∞ d, d, 0 ∥ ∥ ∥ ∥ with X(t) := δa ∞ ˙ 0 + δu ∞ ˙ −1 ∩e1 ˙ 1 . Hence Osgood’s lemma ensures that for small enough Lt (Bd,∞) Lt (Bd,∞) Lt (Bd,∞) t, we have ∫ exp(− t α dτ) ∥ ∥ ∥ ∥ . ∥ ∥ ∥ ∥ − 0 · δa ∞ ˙ 0 + δu ∞ ˙ −1 ∩e1 ˙ 1 ( δa(0) ˙ 0 + δu(0) ˙ 1 ) Lt (Bd,∞) Lt (Bd,∞) Lt (Bd,∞) Bd,∞ Bd,∞ From the above (somewhat formal) computations, we thus expect to get the following local existence result. d d −1 ∈ ˙ p ∈ ˙ p 6 Theorem 3.1. Assume that a0 Bp,1 and that u0 Bp,1 with 1 p < 2d. There exists a positive real number η such that if

∥a0∥ d 6 η, (3.7) ˙ p Bp,1 then (NSC) has a local-in-time solution (a, u) with

d d −1 d +1 ∈ C ˙ p ∈ C ˙ p ∩ 1 ˙ p a ([0,T ]; Bp,1) and u ([0,T ]; Bp,1 ) L ([0,T ]; Bp,1 ). Uniqueness holds true if p 6 d. 258 Danchin R Sci China Math February 2012 Vol. 55 No. 2

Let us just give the scheme of the proof. • If Lipschitz stability estimates are available (that is, in the case 1 6 p < d and d > 2 or if the data have more regularity), then one may

− first, construct inductively a sequence of approximate solutions (an, un) (with an and un solutions to a linear transport equation and Lam´esystem, respectively), for instance, if (an, un) has been constructed, then one may define (an+1, un+1) as the solution to { ∂tan+1 + un · ∇an+1 = −(1 + an)divun, (NSC) ∂tun+1 − Aun+1 = −un · ∇un − J(an)Aun − ∇G(an),

with smoothed out data; − second, exhibit uniform bounds in high norm (as explained above);

− third, show that (an, un)n∈N is a Cauchy sequence in small norm (similar as stability estimates). • In the general case, one has to resort to Schauder-Tikhonov type arguments. The needed compactness is given by uniform bounds in suitable norms for the first order time derivative which stem from the large norm estimates. It is also possible to truncate and smooth out the data so as to solve the system by the first approach then prove uniform bounds and compactness by looking at the time derivatives. The two approaches are, in fact, equivalent and, unfortunately, quite cumbersome if written out completely (see for example [14, 16]). Remark 3.2. The above theorem states uniqueness only in the case where the initial velocity has nonnegative regularity whereas existence may be proved for some data with negative regularity.

3.2 A local existence result for truly nonconstant initial density

We now want to solve (NSC) locally for large a0, that is, to replace the smallness condition (3.7) by the nonvacuum assumption 1 + a0 > 0. The following theorem states that one may do so: d d −1 ∈ ˙ p ∈ ˙ p 6 Theorem 3.3. Assume that a0 Bp,1 and that u0 Bp,1 with 1 p < 2d. If in addition

1 + a0 > 0

then (NSC) has a local-in-time solution (a, u) with

d d −1 d +1 ∈ C ˙ p ∈ C ˙ p ∩ 1 ˙ p a ([0,T ]; Bp,1) and u ([0,T ]; Bp,1 ) L ([0,T ]; Bp,1 ).

Uniqueness holds true if p 6 d. In the case p = 2, this statement has been first proved in [18]. It has been extended to any p ∈ (1, 2d) in [8]. In this section, we give an insight of the two methods as we believe that they are both of interest and likely to be useful in other contexts. Note also that once a priori estimates (in large and small norm) are available, the general scheme for proving the statement will be the same as in the previous section. So we concentrate on a priori estimates. ˙ s 3.2.1 An approach based on the decay properties of functions in Besov spaces Bp,1 This approach is borrowed from [18]. There only the case p = 2 has been considered, for simplicity. The proof however may be adapted to any p ∈ [1, 2d). As a may be large, using estimates for the constant coefficient Lam´esystem is no longer appropriate. One has to consider ′ ∂tu − div(¯µ∇u) − ∇(¯µ divu) = f, (3.8) ′ ′ where nowµ ¯ andµ ¯ depend on (t, x) and satisfy the nondegeneracy conditionµ ¯ > µ∗ andµ ¯ +µ ¯ > µ∗ for ′ ′ some positive constant µ∗. In addition, we assume that (¯µ, µ¯ ) tends to some constant (µ, µ ) at infinity. Danchin R Sci China Math February 2012 Vol. 55 No. 2 259

In what follows, we assume that µ′ = 0 and p = 2 for simplicity. As a preliminary step, let us establish d − ∞ ˙ 2 that if c :=µ ¯ µ is more regular than LT (B2,1), then one may prove estimates for (3.8) without any smallness condition. d +1 ∈ ∞ ˙ 2 ∈ Z Assume for instance that c LT (B2,1 ). For all j , one may write

∂t∆˙ ju − div(¯µ∆˙ ju) = ∆˙ jf + R˙ j with R˙ j := div([∆˙ j, c]∇u). Hence ∫ ∫ 1 d 2 2 ∥∆˙ u∥ 2 + µ¯|∇∆˙ u| dx = ∆˙ u(∆˙ f + R˙ ) dx. 2 dt j L j j j j Asµ ¯ > µ∗, we deduce that ∫ ∫ t t ˙ 2j ˙ ˙ ˙ ˙ ∥∆ju(t)∥L2 +µ∗2 ∥∆ju∥L2 dτ 6 ∥∆ju0∥L2 + (∥∆jf∥L2 +∥Rj∥L2 ) dτ. (3.9) 0 0 Taking advantage of the extra regularity of c, one may write that5) js∥ ˙ ∥ . ∥ ∥ ∥ ∥ 2 Rj L2 cj c d +1 u ˙ s+1 ˙ 2 B2,1 B2,1 ∥ ∥ − d 6 d · js with (cj) ℓ2 = 1 if 2 < s 2 Therefore, multiplying the two sides of (3.9) by 2 and summing up over j, we get ∫ t ∥ ∥ ∥ ∥ 6 ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ u e∞ ˙ s + µ∗ u 1 ˙ s+2 u0 ˙ s + f 1 ˙ s + C c d +1 u ˙ s+1 dτ. Lt (B2,1) Lt (B2,1 ) B2,1 Lt (B2,1) ˙ 2 B2,1 0 B2,1 Using interpolation, we may write ′ µ∗ C 2 ∥ ∥ d ∥ ∥ s+1 6 ∥ ∥ s+2 ∥ ∥ ∥ ∥ C c +1 u B˙ u B˙ + c d +1 u B˙ s . B˙ 2 2,1 2,1 ∗ ˙ 2 2,1 2,1 2 µ B2,1 Plugging this in the above inequality and using Gronwall’s lemma, we end up with ∫ C t ∥c∥2 dτ µ∗ 0 d +1 ˙ 2 B2,1 ∥u∥e∞ ˙ s + µ∗∥u∥ 1 ˙ s+2 6 (∥u0∥ ˙ s + ∥f∥ 1 ˙ s )e . Lt (B2,1) Lt (B2,1 ) B2,1 Lt (B2,1) Now let us consider the rough case. In order to reduce the study to the smooth case, we notice that for any m ∈ Z, we have

∂tu − div(S˙mµ¯∇u) = f + div((c − S˙mc)∇u). (3.10) d +1 ˙ ∈ ∞ ˙ 2 It is clear that Smc LT (B2,1 ). So if we assume in addition m to be chosen so that ˙ ∥c − Smc∥ d (3.11) ∞ ˙ 2 LT (B2,1) is suitably small, then the last term in (3.10) may be absorbed by the l.h.s. and we eventually get ∫ C t 2 ∥S˙mc∥ dτ µ∗ 0 d +1 ˙ 2 B2,1 ∥u∥e∞ ˙ s +µ∗∥u∥ 1 ˙ s+2 6 (∥u0∥ ˙ s +∥f∥ 1 ˙ s )e . Lt (B2,1) Lt (B2,1 ) B2,1 Lt (B2,1) Now, in order to proving a priori estimates in large norm for some solution (a, u) to (NSC), it is mostly a matter of checking that if a solves the mass equation, then one may find m ∈ Z such that (3.11) is d ∈ ˙ 2 ∀ ∃ fulfilled. This is true initially because having a0 B2,1 implies that ε > 0, m > 0 such that ∑ d ˙ j 2 ˙ ∥a0 − Sma0∥ d 6 2 ∥∆ja0∥L2 6 ε. B˙ 2 2,1 j>m−1 From the time-continuity properties of the mass equation, we gather that if ˙ ∥a0 − Sma0∥ d 6 ε, ˙ 2 B2,1 then (3.11) is fulfilled with 2ε for small enough T. The proof of stability estimates in small norm follows from similar considerations. 5) The proof is in the same spirit as that of Inequality (2.6). 260 Danchin R Sci China Math February 2012 Vol. 55 No. 2

3.2.2 An approach based on the properties of the heat equation This method which relies on the smoothing properties of the heat equation and, more generally, of the Lam´esystem

∂tu − Au = f has been introduced by Chen, Miao and Zhang in [8]. It is based on the fact that if u satisfies the Lam´esystem, then (2.11) holds true. In order to take 2j advantage of the smallness of (1 − e−c2 t) for t going to 0, it is thus natural to introduce weighted Besov norms as follows: ′ Definition 3.4. Let (ej)j∈Z be a nondecreasing nonnegative sequence such that ej ∼ ej′ if j ∼ j . Then we set ∑ js j−j′ ∥ ∥ ∥ ˙ ∥ ′ f ˙ s := 2 ωj ∆jf Lp r with ωj := 2 ej . Bp,r (ω) ℓ j′>j

The weight sequence ω = (ωj)j∈Z has the following important properties: • it is bounded and “almost” nondecreasing; ′ • ωj ∼ ωj′ if j ∼ j ; j−j′ ′ • ωj 6 2 ωj′ if j > j . Thanks to the above properties, all the functional calculus (continuity results for paraproduct and remainder, composition estimates, . . . ) extends to weighted Besov norms. Ditto for the priori estimates for the transport equation. The main motivation for introducing weighted norms lies in the fact that one may find some weight sequence ω adapted to our problem satisfying

∥f∥ ˙ s . ∥f∥ ˙ s Bp,r (ω) Bp,r

and so that the l.h.s. is very small compared to the r.h.s. Let us be a bit more specific: take ω = ωT with ∑ j−j′ −c22j T ωT,j := 2 eT,j′ and eT,j = 1 − e . j′>j With this notation, Inequality (2.11) rewrites

∥u∥ 1 ˙ s+2 6 ∥u0∥ ˙ s + ∥f∥ 1 ˙ s (3.12) Lt (Bp,1 ) Bp,1(ωt) Lt (Bp,1(ωt))

and it is clear that the terms on the r.h.s. tend to zero for t going to zero. As in the first method, to handle large perturbations of constant density, the above inequality has to be generalized to the variable coefficient Lam´esystem

′ ∂tu − div(¯µ∇u) − ∇(¯µ divu) = f,

whereµ ¯ andµ ¯′ have the same properties as in (3.8). ′ Assume that µ = 0 and p = 2 for simplicity. As before, applying ∆˙ j to the equation and using an energy method yields ∫ ∫ t t( ) ˙ 2j ˙ ˙ ˙ ˙ ∥∆ju(t)∥L2 + cµ∗2 ∥∆ju∥L2 dτ 6 ∥∆ju0∥L2 + ∥∆jf∥L2 + ∥Rj∥L2 dτ, 0 0

where R˙ j satisfies a good estimate in weighted Besov spaces, namely, ∑ js∥ ˙ ∥ . ∥ − ∥ ∥ ∥ ωj2 Rj L1 (L2) µ µ¯ d u L1 (B˙ s+2) T Le∞(B˙ 2 (ω)) T 2,1 j T 2,1 Danchin R Sci China Math February 2012 Vol. 55 No. 2 261

− d 6 d − if 1 2 < s 2 1. In the caseµ ¯ = µ/(1 + a) with a bounded away from 0, taking ω = ωT (the weight adapted to the Lam´esystem) and s = d/2 − 1, we get

∥ ∥ . ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ u d+1 u0 d−1 + f d+1 + a d u d+1 . 1 ˙ 2 ˙ 2 1 ˙ 2 e∞ ˙ 2 1 ˙ 2 LT (B2,1 ) B2,1 (ωT ) Lt (B2,1 (ωT )) LT (B2,1(ωT )) LT (B2,1 )

d d ∈ e∞ ˙ 2 ∈ ˙ 2 Now, given that a LT (B2,1) (because a0 B2,1 and a solves the transport equation), we deduce that limT →0 ∥a∥ d = 0. Hence one may absorb the last term by the l.h.s. if T is small enough, and e∞ ˙ 2 LT (B2,1(ωT )) d +1 d 1 ˙ 2 ˙ 2 thus get an estimate for u in LT (B2,1 ) even though a is not small in B2,1.

4 The global existence theory

This section is devoted to proving a global existence result for the barotropic Navier-Stokes equation { ∂ta + u · ∇a = −(1 + a)divu, (NSC) ∂tu − Au = −u · ∇u − J(a)Au − ∇G(a).

If we resume our comparison with the incompressible Navier-Stokes equations, then we expect to get

global existence in the case where the initial data a0 and u0 are small for some critical norm. In view of the local-in-time results that we have presented so far, one may thus give the following: Tentative statement There exists some constant η such that whenever

∥ ∥ ∥ ∥ 6 a0 d + u0 d −1 η ˙ p ˙ p Bp,1 Bp,1

with 1 6 p < 2d and d > 2, System (NSC) has a global solution. Recall however that the scaling invariance exhibited in (1.3) was imperfect inasmuch as it did not take the pressure term into consideration. This fact was quite obvious when proving local a priori estimates for (NSC) in the previous section. On the one hand, estimates for transport equation naturally provide

bounds for ∥a∥ d , whereas ∥a∥ d is needed when bounding u by means of Corollary 2.7. So ∞ ˙ p 1 ˙ p LT (Bp,1) LT (Bp,1) far, we used that ∥∇ ∥ . ∥ ∥ (G(a)) d −1 T a d . 1 ˙ p ∞ ˙ p LT (Bp,1 ) LT (Bp,1) Of course this is a good estimate if T is small. At the same time, the r.h.s. grows linearly in time, hence we cannot expect to get any global-in-time control on u by this device. In other words, while the pressure term may be “neglected” in the linear analysis leading to local-in-time existence results, it has to be included in the linear analysis for the global existence theory. This is the aim of the next subsection.

4.1 The linearized system

The linearized system about (a, u) = (0, 0) reads { ∂ta + divu = 0, with α := P ′(1). (4.1) ∂tu − Au + α∇a = 0,

Let ν := µ + µ′. Applying operators P and Q to the second equation, the above system translates into   Q  ∂ta + div u = 0, Q − Q ∇  ∂t u ν∆ u + α a = 0, (4.2)  ∂tPu − µ∆Pu = 0. 262 Danchin R Sci China Math February 2012 Vol. 55 No. 2

In the homogeneous Besov spaces setting, it is equivalent to bound Qu or v := |D|−1div u, and Pu or w := |D|−1curl u. So we are led to considering   | |  ∂ta + D v = 0, − − | |  ∂tv ν∆v α D a = 0, (4.3)  ∂tw − µ∆w = 0.

Note that the last equation (that is the linearized equation for the vorticity part of the velocity field) is a mere heat equation with constant diffusion. So we have to concentrate on the linearized system for the density and the potential part of the velocity, namely, { ∂ta + |D|v = 0, (4.4) ∂tv − ν∆v − α|D|a = 0.

Taking the Fourier transform with respect to the space variable yields ( ) ( ) ( ) d ba ba 0 −|ξ| = A(ξ) with A(ξ) := . dt vb vb α|ξ| −ν|ξ|2

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 that α > 0 (i.e., P ′(1) > 0), that is, we focus on the case where the pressure law is increasing in some neighborhood of the reference density. Note also that a convenient change of variable reduces the study to the case α = 1, an assumption that we shall make from now on. The low frequency regime ν|ξ| < 2 There are two distinct complex conjugated eigenvalues: √ ν|ξ|2 4 λ±(ξ) = − (1 ± iS(ξ)) with S(ξ) := − 1, 2 ν2|ξ|2

and we find that ( ( ) ) 1 i i ba(t, ξ) = etλ−(ξ) 1 + ba (ξ) − vb (ξ) 2 S(ξ) 0 ν|ξ|S(ξ) 0 ( ( ) ) 1 i i + etλ+(ξ) 1 − ba (ξ) + vb (ξ) , 2 S(ξ) 0 ν|ξ|S(ξ) 0 ( ( ) ) i 1 i vb(t, ξ) = etλ−(ξ) ba (ξ) + 1 − vb (ξ) ν|ξ|S(ξ) 0 2 S(ξ) 0 ( ( ) ) i 1 i + etλ+(ξ) − ba (ξ) + 1 + vb (ξ) . ν|ξ|S(ξ) 0 2 S(ξ) 0

For ξ → 0, we have 1 1 ba(t, ξ) ∼ etλ−(ξ)(ba (ξ) − ivb (ξ)) + etλ+(ξ)(ba (ξ) + ivb (ξ)), 2 0 0 2 0 0 1 1 vb(t, ξ) ∼ etλ−(ξ)(iba (ξ) + vb (ξ)) + etλ+(ξ)(−iba (ξ) + vb (ξ)). 2 0 0 2 0 0 Hence, the low frequencies of a and v have a similar behavior. Danchin R Sci China Math February 2012 Vol. 55 No. 2 263

2 Note that |etλ±(ξ)| = e−νt|ξ| /2 and that

ν|ξ|2 Re λ±(ξ) ∼ − , Im λ±(ξ) ∼ ∓|ξ| for ξ → 0. 2 Hence we expect the system to have both parabolic and wave-like behavior. For the time being, we just take advantage of the parabolic behavior. More precisely, according to Parseval’s formula, we use the fact that

˙ ˙ −cνt22j ˙ ˙ ∥(∆ja, ∆jv)(t)∥L2 6 Ce ∥(∆ja0, ∆jv0)∥L2 (4.5)

whenever 2jν 6 1. The high frequency regime ν|ξ| > 2 There are two distinct real eigenvalues: √ ν|ξ|2 4 λ±(ξ) := − (1 ± R(ξ)) with R(ξ) := 1 − , 2 ν2|ξ|2

and after a lengthy computation, we find that ( ( ) ) 1 1 1 ba(t, ξ) = etλ−(ξ) 1 + ba (ξ) − vb (ξ) 2 R(ξ) 0 ν|ξ|R(ξ) 0 ( ( ) ) 1 1 1 + etλ+(ξ) 1 − ba (ξ) + vb (ξ) , 2 R(ξ) 0 ν|ξ|R(ξ) 0 ( ( ) ) 1 1 1 vb(t, ξ) = etλ−(ξ) ba (ξ) + 1 − vb (ξ) ν|ξ|R(ξ) 0 2 R(ξ) 0 ( ( ) ) 1 1 1 + etλ+(ξ) − ba (ξ) + 1 + vb (ξ) . ν|ξ|R(ξ) 0 2 R(ξ) 0

| | → ∞ → − ∼ 2 ∼ − | |2 ∼ − 1 For ξ , we have R(ξ) 1 and 1 R(ξ) 2/(νξ) . Hence λ+(ξ) ν ξ and λ−(ξ) ν . In other words, a parabolic and a damped mode coexist and the asymptotic behavior of (a, v) for |ξ| → ∞ is given by

− t −1 −νt|ξ|2 −2 −1 ba(t, ξ) ∼ e ν (ba0(ξ) − (ν|ξ|) vb0(ξ)) + e (−(ν|ξ|) ba0(ξ) + (ν|ξ|) vb0(ξ)), − t −1 −2 −νt|ξ|2 −1 vb(t, ξ) ∼ e ν ((ν|ξ|) ba0(ξ) − (ν|ξ|) vb0(ξ)) + e (−(ν|ξ|) ba0(ξ) + vb0(ξ)).

−νt|ξ|2 − t At first, one would expect the damped mode to dominate as e is negligible compared to e ν 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 easily get Lemma 4.1. There exist two positive constants c and C such that for any j ∈ Z satisfying 2jν > 3 and t ∈ R+, we have

− t − ˙ 2ν ˙ j 1 ˙ ∥∆ja(t)∥L2 6 Ce (∥∆ja0∥L2 + (2 ν) ∥∆jv0∥L2 ), − − t − 2j − − t ˙ j 1 2ν ˙ cνt2 j 2 2ν ˙ ∥∆jv(t)∥L2 6 C((2 ν) e ∥∆ja0∥L2 + (e +(ν2 ) e )∥∆jv0∥L2).

In fact, the same inequalities hold true for any p ∈ [1, ∞]. Indeed, following the proof of (2.9) yields

˙ j ∗ ˙ j ∗ | | −1 ˙ j ∗ | | | −2 ˙ ∆ja(t) = h1(t) ∆ja0 + h2(t) (ν D ) ∆jv0 + h3(t) ( ν D ) ∆ja0 j ∗ | | −1 ˙ + h4(t) (ν D ) ∆jv0, ˙ j ∗ | |−1 ˙ j ∗ | | −2 ˙ j ∗ | |−1 ˙ ∆jv(t) = k1(t) ( νD ∆ja0) + k2(t) (ν D ) ∆jv0 + k3(t) ( νD ∆ja0) 264 Danchin R Sci China Math February 2012 Vol. 55 No. 2

j ∗ ˙ + k4(t) ∆jv0 with

j j j j − t ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ 6 2ν h1(t) L1 + h2(t) L1 + k1(t) L1 + k2(t) L1 Ce , ∥ j ∥ ∥ j ∥ ∥ j ∥ ∥ j ∥ 6 −cνt22j h3(t) L1 + h4(t) L1 + k3(t) L1 + k4(t) L1 Ce .

This implies that

−1 ∥∆˙ a∥ ∞ p + ν∥∆˙ a∥ 1 p . ∥∆˙ a ∥ p + ∥(ν|D|) ∆˙ v ∥ p , j Lt (L ) j Lt (L ) j 0 L j 0 L 2j ∥∆˙ u∥ ∞ p + ν2 ∥∆˙ u∥ 1 p . ∥ν|D|∆˙ a ∥ p + ∥∆˙ v ∥ p . j Lt (L ) j Lt (L ) j 0 L j 0 L

Hence, we recover that for high frequencies, it is suitable to work at the same level of regularity for ∇a and v. At the same time, according to (4.5), one has to work at the same level of regularity for low frequencies, a fact which does not follow from our scaling considerations for (NSC). Putting together all the estimates for the dyadic blocks and using Duhamel’s formula, we conclude that whenever (a, u) satisfies { ∂ta + divu = F, (LPH) ∂tu − Au + ∇a = G, we have for the low frequencies:

ℓ ℓ ℓ ∥(a, u)∥ ′ ′ . ∥(a , u )∥ ′ + ∥(F,G)∥ ′ e∞ ˙ s ∩e1 ˙ s +2 0 0 B˙ s e1 ˙ s Lt (B2,r ) Lt (B2,r ) 2,r Lt (B2,r ) and for the high frequencies:

∥ ∥h ∥ ∥h a e∞∩e1 ˙ s+1 + u e∞ ˙ s ∩e1 ˙ s+2 (Lt Lt )(Bp,r ) Lt (Bp,r ) Lt (Bp,r ) . ∥ ∥h ∥ ∥h ∥ ∥h ∥ ∥h a0 ˙ s+1 + u0 ˙ s + F e1 ˙ s+1 + G e1 ˙ s , Bp,r Bp,r Lt (Bp,r ) Lt (Bp,r ) where the index ℓ (resp. h) means that only low (resp. high) frequencies have been taken into account when computing the norm.

4.2 The paralinearized system

Unfortunately, having proved the above a priori estimates for (LPH) is not the end of the story: indeed, if one rewrites the compressible Navier-Stokes equations as follows { ∂ta + divu = −div(au),

∂tu − Au + ∇a = −u · ∇u − J(a)Au − ∇(aK(a)) with K(0) = 0, then one may tempt to apply the preceding estimates for System (LPH) in the case where F and G stand for the nonlinear terms in the above right-hand side. However, one has to keep in mind that there is no gain of regularity whatsoever for the first equation. Hence div (au) will cause a loss of one derivative for a. In order to avoid this loss, we shall include (a part of) this term in the linearized equations. In order to track the bad part of it, one may take advantage of Bony’s decomposition for div(au), namely,

div(au) = div(Tua) + divR(a, u) + divTau.

The continuity results of Proposition 2.2 assert that, under the regularity properties for a and u given in the local existence statement, only the first term is responsible for the loss of one derivative. Danchin R Sci China Math February 2012 Vol. 55 No. 2 265

Therefore, we have to better understand how behaves the following paralinearized system: { ( ) ∂ta + div Tva + divu = F, (PL) ∂tu + Tv · ∇u − Au + ∇a = G, ( ) ∑ ( ) · ∇ with div Tva := i ∂i Tvi a and Tv u := Tvi ∂iu. Of course, it would be more natural to keep the whole convection terms. In fact, the advantage of paralinearizing the system is that low and high frequencies are (almost) uncoupled for ∑ ( ∆˙ jdiv(Tva) = ∆˙ j S˙j′−1v∆˙ j′ a). |j′−j|64

Moreover, we have for all j ∈ Z, { ∂t∆˙ ja + S˙j−1v · ∇∆˙ ja + div∆˙ ju = ∆˙ jF + nice remainder,

∂t∆˙ ju + S˙j−1v · ∇∆˙ ju − A∆˙ ju + ∇∆˙ ja = ∆˙ jG + nice remainder.

At this point, it is natural to perform a Lagrangian change of variables: (τ, y) = (t, ψj(t, x)) with ψj the flow of S˙j−1v so as to cancel out the convection terms. However, this change of variables will spoil the constant coefficients first and second order terms for any positive time. Whether this damage is not too important is the question that we shall tackle in the next paragraph.

4.3 A priori estimates for a convection-diffusion equation

In these notes, in order to avoid technicalities as much as possible, we present how the method proposed in the previous paragraph (paralinearization, localization and Lagrangian change of variables) may be implemented on the simpler case of the convection-diffusion equation. Here we use the method introduced by Hmidi in [28] after an idea of Vishik in [34] (and generalized to non divergence free vector fields v in [17]). So, let us consider the following convection-diffusion equation:

∂tu + v · ∇u − ν∆u = f. (TDν )

The following theorem asserts that one may get a family of a priori estimates in Besov spaces, which are the optimal ones in the limit cases v ≡ 0 or ν = 0.

Theorem 4.2. Let 1 6 p 6 p1 6 ∞ and 1 6 r 6 ∞. Let s ∈ R satisfies ( ) − d d d · min , ′ < s < 1 + p1 p p1

Then for any smooth solution of (TDν ) with ν > 0, we have

∥ ∥ ∥ ∥ 6 CV (T ) ∥ ∥ ∥ ∥ u e∞ ˙ s + ν u e1 ˙ s+2 Ce ( u0 ˙ s + f e1 ˙ s ) LT (Bp,r ) LT (Bp,r ) Bp,r LT (Bp,r ) ∫ T ∥∇ ∥ with V (T ) := 0 v(t) d dt. B˙ p1 p1,1 j Proof. Let us first localize the equation about the frequency 2 . We see that uj := ∆˙ ju satisfies

∂tuj + vj · ∇uj − ν∆uj = fj + R˙ j

with vj := S˙j−1v, fj := ∆˙ jf and

R˙ j = (S˙j−1v − v) · ∇uj + [v · ∇, ∆˙ j]u. 266 Danchin R Sci China Math February 2012 Vol. 55 No. 2

A slight variation over the proof of (2.6) yields

∥ ˙ ∥ . −js∥∇ ∥ ∥ ∥ Rj Lp cj2 v d u B˙ s B˙ p1 p,r p1,1

with ∥(cj)∥ℓr = 1. Next, perform a Lagrangian change of coordinates e e uej := uj ◦ ψj, fj := fj ◦ ψj, Rj := R˙ j ◦ ψj.

We get e e ∂tuej − ν∆uej = fj + Rj + νTj

with Tj := (∆uj) ◦ ψj − ∆uej. From the chain rule and H¨olderinequality, we infer that ( ) 2 ∥Tj∥Lp . 1 + ∥∇ψj∥L∞ ∥Id − ∇ψj∥L∞ ∥D uj ◦ ψj∥Lp + ∥∆ψj∥L∞ ∥∇uj ◦ ψj∥Lp .

The r.h.s. may be bounded according to the following classical flow estimates: ( ∫ ) t ∥∇ψj(t)∥L∞ 6 exp ∥∇vj∥L∞ dτ , 0 ( ∫ ) t ∥Id − ∇ψj(t)∥L∞ 6 exp ∥∇vj∥L∞ dτ − 1, ( ∫ 0 ) ∫ t t 2 2 ∥∇ ψj(t)∥L∞ 6 exp 2 ∥∇vj∥L∞ dτ ∥∇ vj∥L∞ dτ. 0 0 Note that according to Bernstein inequality,

k j(k−1) ∥∇ vj∥L∞ . 2 ∥∇v∥L∞ , ∀ k > 1.

Hence 2j CV (t)−1 ∥Tj(t)∥Lp . 2 (e )∥uj(t)∥Lp . (4.6)

j If uej were spectrally localized in an annulus of size 2 , then the regularity estimates for the heat equation would enable us to gain the factor 22j and we would be done for t small as the term (eCV (t) − 1) goes to 0 when t tends to 0. As the Lagrangian change of variable destroys the spectral localization, the next idea is to localize

again the equation for uej, namely, e e ∂tuej − ν∆uej = fj + Rj + νTj.

We may write e e ∂t∆˙ j′ uej − ν∆∆˙ j′ uej = ∆˙ j′ fj + ∆˙ j′ Rj + ν∆˙ j′ Tj for j′ ∈ Z and use the smoothing properties of the heat equation for bounding each block, then sum over ′ j to bound uej. If we simply use that e e ∥∆˙ j′ fj∥Lp . ∥fj∥Lp , e then, after summation, the contribution given by the terms ∆˙ j′ fj is infinite. To overcome this, one may, in the light of Bernstein inequalities, write that

e −j′ e −j′ ∥∆˙ j′ fj∥Lp . 2 ∥∇∆˙ j′ fj∥Lp = 2 ∥∆˙ j′ ((∇fj ◦ ψj) · ∇ψj)∥Lp , −j′ CV j−j′ . 2 ∥∇fj ◦ ψj∥Lp ∥∇ψj∥L∞ . e 2 ∥fj∥Lp . Danchin R Sci China Math February 2012 Vol. 55 No. 2 267

e One may proceed in the same way for ∆˙ j′ Rj and ν∆˙ j′ Tj. Therefore, using the smoothing properties of the heat equation, we get the following inequality for all (j, j′) ∈ Z2:

2j′ j−j′ CV (t) ∥∆˙ ′ ue ∥ ∞ p + ν2 ∥∆˙ ′ ue ∥ 1 p . ∥∆˙ ′ ∆˙ u ∥ p + 2 e ∥f ∥ 1 p j j Lt (L ) j j Lt (L ) j j 0 L j Lt (L ) 2(j−j′) 2j′ CV (t) − ∥ ∥ + 2 ν2 (e 1) uj L1(Lp) ∫ t t j−j′ −js ′ CV ∥ ∥ + 2 cj2 V e u B˙ s dτ. 0 p,r ′ This inequality is suitable if j > j − N0 (where N0 fixed integer). To handle the low frequencies, one ˙ e may merely bound Sj−N0 uj according to the following lemma (in the spirit of Vishik’s in [34]):

Lemma 4.3. For any p ∈ [1, ∞],N0 ∈ N and j ∈ Z, we have ( ) −j −N0 ∥ ˙ ˙ ◦ ∥ p . ∥ −1 ∥ ∞ ∥ ˙ ∥ p ∥∇ −1 ∥ ∞∥ ∥ ∞ ∥∇ ∥ ∞ Sj−N0 (∆jv ϕ) L Jϕ L ∆jv L 2 Jϕ L Jϕ L + 2 ϕ L .

Here we thus get

CV (t) −N0 CV (t) ∥S˙ − ue ∥ q p . e (2 + e − 1)∥u ∥ q p , ∀ 1 6 q 6 ∞. j N0 j Lt (L ) j Lt (L )

In order to bound uj we split it into (N0 is any fixed integer) ∑ −1 −1 ˙ e ◦ ˙ ′ e ◦ uj = Sj−N0 uj ψj + ∆j uj ψj . ′ j >j−N0 Then putting together the previous computations yields

js 2j js 2 ∥u ∥ ∞ p +ν2 2 ∥u ∥ 1 p j Lt (L ) j Lt (L )

CV (t) js 3N0 js . e (2 ∥∆˙ u ∥ p + 2 2 ∥f ∥ 1 p j 0 L j Lt (L ) − N0 2N0 CV (t) − js∥ ∥ ∞ 2j js∥ ∥ + (2 + 2 (e 1))(2 uj L (Lp) + ν2 2 uj L1(Lp))) ∫ t t t ′ 3N0 CV ∥ ∥ + 2 cjV e u B˙ s dτ. 0 p,r

−N0 In order to conclude, it is only a matter of choosing N0 large enough (say such that 16C2 ∈ [1, 2)), then t is so small as the second line to be absorbed by the l.h.s. After performing a ℓr summation and using Gronwall’s lemma, we end up with

∥ ∥ ∥ ∥ 6 ∥ ∥ ∥ ∥ u e∞ ˙ s + ν u e1 ˙ s+2 C0( u0 ˙ s + f e1 ˙ s ), Lt (Bp,r ) Lt (Bp,r ) Bp,r Lt (Bp,r ) ∫ ∈ T1 ′ ≈ whenever t [0,T1] with T1 such that 0 V dt ε with ε small enough. Then one may split [0,T ] into

[0,T ] = [0,T1] ∪ · · · ∪ [Tk−1,T ] ∫ with Tj V ′ dt ≈ ε and repeat the argument on every subinterval. As kε ≈ V, this completes the Tj−1 proof. We can now resume to the initial paralinearized barotropic system (PL). From a similar method, we prove (see [7]) the following: ′ ∈ R2 ∈ ∞ ∈ ∞ Proposition∫ 4.4. Let (s, s ) , r [1, ] and (a, u) be a solution of (PL). Assume that p [2, ]. t ′ ′ ∥∇ ∥ ∞ Let V (t) := 0 v(τ) L dτ. There exists a constant C depending only on (µ, µ ), d and (s, s ) such that the following estimate holds for all t > 0:

ℓ h h ∥ ∥ ′ ∥ ∥ ′ ′ ∥ ∥ ′ (a, u) e∞ ˙ s′ ∩e1 ˙ s +2 + a e∞ ˙ s +1 ∩e1 ˙ s +1 + u e∞ ˙ s′ ∩e1 ˙ s +2 Lt (B2,r ) Lt (B2,r ) Lt (B2,r ) Lt (B2,r ) Lt (B2,r ) Lt (B2,r ) 268 Danchin R Sci China Math February 2012 Vol. 55 No. 2

∥ ∥h ∥ ∥h + a e∞ ˙ s+1 ∩ 1 ˙ s+1 + u e∞ ˙ s ∩ 1 ˙ s+2 Lt (Bp,1 ) Lt (Bp,1 ) Lt (Bp,1) Lt (Bp,1 ) CV (t) ℓ ℓ h h 6 Ce (∥(a , u )∥ ′ + ∥(F,G)∥ ′ + ∥a ∥ ′ + ∥u ∥ ′ 0 0 B˙ s e1 ˙ s 0 ˙ s +1 0 B˙ s 2,r Lt (B2,r ) B2,r 2,r h h h h +∥F ∥ ′ + ∥G∥ ′ +∥a ∥ s+1 + ∥u ∥ e1 ˙ s +1 e1 ˙ s 0 B˙ 0 B˙ s Lt (B2,r ) Lt (B2,r ) p,1 p,1 ∥ ∥h ∥ ∥h + F 1 ˙ s+1 + G 1 ˙ s ). Lt (Bp,1 ) Lt (Bp,1) Remark 4.5. At first sight, this statement may seem slightly better than the corresponding one for the transport-diffusion equation (look at the exponential term). This is due to the fact that only the paralinearized convection terms have been included in (PL). A similar improvement would hold for the paralinearized convection-diffusion equation.

4.4 Statement of the global existence results

Granted with the above a priori estimates for the paralinearized system, one may prove the following statement. d d −1 ∈ ˙ p ∈ ˙ p ∈ ℓ ∈ ˙ s h ∈ Theorem 4.6. Assume that a0 Bp,1 and u0 Bp,1 for some p [2, 2d) and that a0 B2,r, a0 ˙ s+1 ∈ ˙ s ∈ ∞ ∈ R B2,r , u0 B2,r for some r [1, ] and s such that − min(1, d/p) < s < d/2 − 1 if r > 1, − min(1, d/p) < s 6 d/2 − 1 if r = 1.

There exist two constants c and M depending only on d, p2, s and on the physical parameters of the system such that if ∥ ℓ ∥ ∥ h∥ ∥ h∥ ∥ ∥ ∥ h∥ 6 a0 B˙ s + a0 ˙ s+1 + a0 d + u0 B˙ s + u0 d −1 c, 2,r B2,r ˙ p 2,r ˙ p Bp,1 Bp,1 then (NSC) has a global-in-time (unique if p 6 d) solution (a, u) with6)

d d ℓ ∈ Ce ˙ s ∩ e1 ˙ s+2 h ∈ Ce ˙ s+1 ∩ ˙ p ∩ e1 ˙ s+1 ∩ ˙ p a b(B2,r) L (B2,r ), a b(B2,r Bp,1) L (B2,r Bp,1), d −1 d +1 ∈ Ce ˙ s ∩ e1 ˙ s+2 h ∈ Ce ˙ p ∩ 1 ˙ p u b(B2,r) L (B2,r ), u b(Bp,1 ) L (Bp,1 ). The above theorem deserves a few comments. • The case p = 2, s = d/2 − 1 and r = 1 goes back to [11]. • The L2 type condition is lower order and one may take s negative. • The above statement has been proved in a recent joint work with Charve (see [7]). Almost the same result has been proved independently by Chen, Miao and Zhang in [9]. The general strategy is the same. However, as the authors chose to include the whole convection term in the linear analysis, there, p has to be taken smaller than 4. In a recent paper, Haspot [27] has proposed another method based on the use of Hoff’s viscous effective flux. d d • Recall that the incompressible Navier-Stokes equations are globally well-posed for small u0 in L (R ). The above statement shows that (NSC) is globally well posed if ∥ ∥ ∥ ∥ a0 1∩ ˙ 1 + u0 2∩ ˙ 0 H Bd,1 L Bd,1

is small. This is almost the same assumption on u0 as in the incompressible case (where u0 has just to ˙ 0 be in Bd,∞, see [4]). • As one may take s < 0 and p so that d/p − 1 < 0, the smallness condition is satisfied for small −1 densities and large highly oscillating velocities: indeed, if u0 : x 7→ ϕ(x) sin(ε x · ω) n with ω and n in Sd−1 and ϕ ∈ S(Rd), then (see, for example, [2])

− 1− d ∥ ∥ 6 s ∥ ∥ 6 p u0 B˙ s Cε and u0 d −1 Cε . 2,r ˙ p Bp,1

6) e e∞ Cb means L and continuous in time. Danchin R Sci China Math February 2012 Vol. 55 No. 2 269

Hence such data with small enough ε generate global solutions. However, as here p > d is needed, the above statement does not ensure uniqueness. In [7], we proved a slightly a more accurate statement in which global existence and uniqueness holds true for oscillating data.

Theorem 4.7. Let a0 and u0 satisfy the same regularity conditions as in previous statement. Assume in addition that there exists pe ∈ [p, 2d) such that for some suitably small constant c, we have

∥ ℓ ∥ ∥ h∥ ∥ h∥ ∥ ∥ ∥ h∥ 6 a0 B˙ s + a0 ˙ s+1 + a0 d + u0 B˙ s + u0 d −1 c. (4.7) 2,r B2,r ˙ pe 2,r ˙ pe Bpe ,1 Bp,e 1 Then (NSC) has a global-in-time solution in the same space as before and uniqueness holds true if p 6 d. Compared to the previous statement, here the smallness condition is disconnected from the regularity −1 assumption. As a consequence, if ε is small enough, a0 satisfies (4.7) and, say, u0 : x 7→ ϕ(x) sin(ε x·ω) n, then the corresponding solution is global and unique.

4.5 Sketch of the proof

We just indicate how to get global-in-time a priori estimates. As we plan to use our estimates for the paralinearized system, we rewrite the barotropic Navier-Stokes equations as follows: { ∂ta + div(Tua) + divu = F, (4.8) ∂tu + Tu · ∇u − Au + ∇a = G,

with ′ ′ F := −divT u and G := ∇(aK(a)) − J(a)Au − T∇ · u. ∫ a u t ∥∇ ∥ ∞ Denoting U(t) := 0 u L dτ and

ℓ h h X(t) := ∥a∥ ∞ + ∥a∥ d + ∥a∥ d Le (B˙ s )∩Le1(B˙ s+2) t 2,r t 2,r e∞ ˙ s+1∩ ˙ p e1 ˙ s+1∩ ˙ p Lt (B2,r Bp,1) Lt (B2,r Bp,1) h ∥ ∥e∞ e s+2 ∥ ∥ + u L (B˙ s )∩L1(B˙ ) + u d −1 d +1 , t 2,r t 2,r e∞ ˙ p ∩ 1 ˙ p Lt (Bp,1 ) Lt (Bp,1 ) the paraproduct estimates give

CU(t) ℓ h h h 6 ∥ ∥ ∥ ∥ ∥ ∥e ∥ ∥ ∥ ∥ X(t) Ce (X(0) + F e1 ˙ s + F e1 ˙ s+1 + G L1(B˙ s ) + F d + G d −1 ). Lt (B2,r ) Lt (B2,r ) t 2,r 1 ˙ p 1 ˙ p Lt (Bp,1) Lt (Bp,1 ) ∥ ∥h ∥ ∥h As an example, let us bound F d and G d −1 . Thanks to Propositions 2.2 and 2.4, one 1 ˙ p 1 ˙ p Lt (Bp,1) Lt (Bp,1 ) may write

′ ′ h ∞ ∥divT u∥ d . ∥T u∥ d . ∥a∥ ∞ ∥u∥ d , a a +1 Lt (L ) +1 1 ˙ p L1(B˙ p ) L1(B˙ p ) Lt (Bp,1) t p,1 t p,1 h 2 ∥∇ ∥ . ∥ ∥ d . ∥ ∥ (aK(a)) d−1 aK(a) a d , 1 ˙ p L1(B˙ p ) 2 ˙ p Lt (Bp,1 ) t p,1 Lt (Bp,1) h ∥ A ∥ .∥ ∥ d ∥A ∥ d .∥ ∥ d ∥ ∥ d J(a) u d−1 J(a) u −1 a u +1 , 1 ˙ p L∞(B˙ p ) L1(B˙ p ) L∞(B˙ p ) L1(B˙ p ) Lt (Bp,1 ) t p,1 t p,1 t p,1 t p,1 ′ h ∥ · ∥ . ∥∇ ∥ 1 ∞ ∥ ∥ d . ∥ ∥ d ∥ ∥ d T∇u u d−1 u L (L ) u −1 u +1 u −1 . 1 ˙ p t L∞(B˙ p ) L1(B˙ p ) L∞(B˙ p ) Lt (Bp,1 ) t p,1 t p,1 t p,1 ∥ ∥ℓ ∥ ∥ We get similar estimates for F e1 ˙ s , G Le1(B˙ s ) and so on. Lt (B2,r ) t 2,r By splitting all the norms appearing in the quadratic r.h.s. above into low and high frequencies, we discover that they are all bounded by X(t). Putting together all these informations, we eventually get ( X 6 CeCX X(0) + X2) (4.9) 270 Danchin R Sci China Math February 2012 Vol. 55 No. 2 as long as the solution is defined and has the required regularity. Now it is clear that if eCX(t) 6 2 and 4CX(t) 6 1, (4.10) then Inequality (4.9) ensures that X(t) 6 4CX(0). (4.11) Plugging this in (4.10) and using a classical bootstrap argument, it is easy to conclude that if X(0) is small enough, then (4.10) is satisfied as long as the solution is defined. Hence (4.11) holds true too.

5 On the incompressible limit

We now want to study the convergence of the barotropic Navier-Stokes equations when the Mach number ε tends to 0. Given that the Mach number is the ratio of the typical velocity over the sound speed, in the small Mach number regime, we expect the relevant time scale to be 1/ε. Therefore, it is natural to set

(ρ, u)(t, x) = (ρε, εuε)(εt, x).

With these new variables, the original system (NSC) recasts in   ε ε ε ∂tρ + div(ρ u ) = 0, ∇P ε  ∂ (ρεuε) + div(ρεuε ⊗ uε) − µ∆uε −(λ+µ)∇div uε + = 0. t ε2 In the case of well-prepared data

ε O 2 ε ε O ρ0 = 1 + (ε ) and u0 with div u0 = (ε), the time derivatives may be bounded independently of ε for ε going to 0. Hence no acoustic waves have to be taken into account and one may prove that the solution tends to the incompressible Navier-Stokes equations when ε goes to 0 by a standard approach. Besides, asymptotic expansions may be derived if one has more information on the asymptotic expansions of the data. Here, we shall rather consider ill-prepared data, namely,

ε ε ε ρ0 = 1 + εb0 and u0.

Initially, for such data, the time derivative of the solution is of order ε−1 and highly oscillating acoustic waves do have to be considered. Whether they may interact or not is the main problem. This is the question that we want to address now in the whole space framework. ε ε To simplify, we take (b0, u0) = (b0, u0) independent of ε. Note that it is not assumed that div u0 = 0. We still assume that P ′(1) = 1. Denoting ρε = 1 + εbε, it is found that (bε, uε) satisfies

 ε  ε div u ε ε  ∂tb + = −div(b u ),  ε Auε ∇bε ∂ uε + uε · ∇uε − + (1+k(εbε)) = 0, (NSCε)  t ε  1 + εb ε ε ε (b , u )|t=0 = (b0, u0), with A := µ∆ + (λ+µ)∇div and k a smooth function satisfying k(0) = 0.

According to the previous parts, System (NSCε) is locally well-posed for all small enough ε > 0. We want to study whether Danchin R Sci China Math February 2012 Vol. 55 No. 2 271

ε ε (1) we have lim infε→0 Tε > T , where Tε stands for the lifespan of (b , u ) and T stands for the lifespan of the solution v to the incompressible Navier-Stokes equation: { ∂tv + P(v · ∇v) − µ∆v = 0, (NSI) v|t=0 = Pu0.

(2) Tε = +∞ for small ε if T = +∞, (3) uε tends to v and bε converges to 0.

5.1 Back to the linearized equations

With the above scaling, the linearized compressible Navier-Stokes equations in terms of (bε, uε) read   divuε  ∂ bε + = −div(bεuε), t ε  Auε ∇bε  ∂ uε + uε · ∇uε − + (1+k(εbε)) = 0 t 1 + εbε ε and the linearized equations about (0, 0) are   divQuε  ∂ bε + = 0,  t ε ∇ ε ε ε b  ∂tQu − ν∆Qu + = 0,  ε  ε ε ∂tPu − µ∆Pu = 0.

As pointed out in the previous section, the last equation is the heat equation whereas denoting vε := |D|−1divQuε, the first two equations are equivalent to   |D|vε  ∂ bε + = 0, t ε  |D|vε (BMε)  ∂ vε − ν∆vε − = 0. t ε This latter system may be solved explicitly by using the Fourier transform ( ) ( )( ) b − b d bε(ξ) 0 −ε 1|ξ| bε(ξ) = . dt vbε(ξ) ε−1|ξ| −ν|ξ|2 vbε(ξ)

As in the previous section, we discover that there are two regimes: in the high frequency regime νε|ξ| > 2, the eigenvalues read ( √ ) ν|ξ|2 4 λ±(ξ) = − 1 ± 1 − , 2 ε2ν2|ξ|2 whereas in the low frequency regime νε|ξ| < 2, one has ( √ ) ν|ξ|2 4 λ±(ξ) = − 1 ± i − 1 . 2 ε2ν2|ξ|2

Therefore 1 λ+(ξ) ∼ −ν|ξ|2 and λ−(ξ) ∼ − ε2ν for ξ → ∞, and |ξ|2 |ξ| λ ± (ξ) ∼ −ν ∓ i 2 ε for ξ → 0. Hence, in high frequency, we expect to have 272 Danchin R Sci China Math February 2012 Vol. 55 No. 2

• one parabolic mode with diffusion ν, • 1 one damped mode with coefficient ε2ν , whereas, in low frequency, (BMε) should behave like

d ν |D| z − ∆z ∓ i z = 0. dt 2 ε The important fact is that the low frequency regime tends to invade the whole Rd when ε → 0 as the threshold between the two regimes is at |ξ| = 2(νε)−1. Hence it has to be studied with more care than in the previous section. In Rd, taking advantage of the large imaginary part of the eigenvalues for low frequencies turns out to be the key to proving convergence for ε tending to 0 as it supplies dispersion. Note that as our global existence theorem was based on L2 type estimates as far as low frequencies were concerned, the imaginary part of the eigenvalues was not used so far.

5.2 About dispersion

In the whole space, the following Strichartz estimates are available for the acoustic wave system. Proposition 5.1. Let (bε, vε) solve { ε −1 ε ∂tb + ε |D|v = F, ε −1 ε ∂tv − ε |D|b = G.

Then we have the inequality

1 ∥ ε ε ∥ . r ∥ ∥ ∥ ∥ (b , v ) s+d( 1 − 1 )+ 1 ε ( (b0, d0) ˙ s + (F,G) 1 ˙ s ), er ˙ p 2 r B2,1 LT (B2,1) LT (Bp,1 )

whenever p > 2, ( ( )) 2 1 1 6 min 1, (d − 1) − and (r, p, d) ≠ (2, ∞, 3). r 2 p The proof is quite straightforward (see [12]) for a convenient change of variable reduces the statement to the case ε = 1 and this enters in the classical theory of Strichartz estimates as described in [2, Chapter 8], for instance. The fundamental fact that we shall use for proving convergence is that the above statement implies that, compared to Sobolev embedding, dispersion gives a gain of 1/r derivative and the small factor 1 ε r . For instance, if the dimension is d > 4, then one may take p = ∞ and r = 2 so that, by virtue of functional embedding, one gets

1 ∥ ε ε ∥ . 2 ∥ ∥ ∥ ∥ (b , v ) L2 (L∞) ε ( (b0, v0) d − 1 + (F,G) d − 1 ). T ˙ 2 2 1 ˙ 2 2 B2,1 LT (B2,1 ) Similar gains may be obtained in dimension d = 2, 3. However, the computations get wilder.

5.3 A statement about the incompressible limit

In [12], it is proved that in the whole space case, the lifespan of (NSCε) is greater than or equal to the lifespan of the incompressible Navier-Stokes equations (NSI), and that convergence holds true. In particular, if the limit solution is known to exist for all time (a property that is always satisfied in dimension 2), then it is also true for the solution to (NSCε) for small enough ε. Here is the exact statement. Theorem 5.2. Let α > 0 and T ∈ (0, ∞]. Assume that

d −1 d +α d −1 d +α−1 ∈ ˙ 2 ∩ ˙ 2 ∈ ˙ 2 ∩ ˙ 2 b0 B2,1 B2,1 and u0 B2,1 B2,1 . Danchin R Sci China Math February 2012 Vol. 55 No. 2 273

Let d −1 d −1+α d +1 d +1+α ∈ C ˙ 2 ∩ ˙ 2 ∩ 1 ˙ 2 ∩ ˙ 2 v ([0,T ]; B2,1 B2,1 ) L ([0,T ]; B2,1 B2,1 )

be the solution on [0,T ] of the incompressible Navier-Stokes equations with initial data Pu0. ε ε There exists ε0 > 0 such that for all ε ∈ (0, ε0), system (NSCε) has a unique solution (b , u ) on [0,T ] with, uniformly with respect to ε,

d −1 d +α−1 d +1 d +α+1 ε ∈ C ˙ 2 ∩ ˙ 2 ∩ 1 ˙ 2 ∩ ˙ 2 u ([0,T ]; B2,1 B2,1 ) L (0,T ; B2,1 B2,1 ), (5.1) d −1 d +α d d +α ε ∈ C ˙ 2 ∩ ˙ 2 ε ∈ 2 ˙ 2 ∩ ˙ 2 εb ([0,T ]; B2,1 B2,1 ), b L (0,T ; B2,1 B2,1 ).

Besides,

d −1 d +α−1 d +1 d +α+1 P ε −→ C ˙ 2 ∩ ˙ 2 ∩ 1 ˙ 2 ∩ ˙ 2 u v in ([0,T ]; B2,1 B2,1 ) L (0,T ; B2,1 B2,1 ), α−1+ 1 ε ε p ∞ |D| p (b , Qu ) −→ 0 in L (0,T ; L )

1 with the rate ε p whenever p > 2 if d > 4, p > 2 if d = 3 and p > 4 if d = 2. Remark 5.3. (1) In the case of small data, a similar global-in-time result may be established in the critical functional framework. (2) A similar statement holds true in the periodic setting. The proof is quite different, though (see [13]). Let us just give the structure of the proof and the main ideas. The proof relies on three ingredients:

• estimates independent of ε in large norms for the paralinearized system (NSCε) (which are mostly given by Proposition 4.4 with p = 2, after rescaling); • the dispersion estimates stated in Proposition 5.1; • some maximal regularity estimates for parabolic systems in the spirit of (2.8). The proof goes as follows: Step 1. Uniform estimates for (bε, uε). We rewrite (NSC ) as ε { −1 ∂ta + div(Tua) + ε divu = F, −1 ∂tu + Tu · ∇u − Au + ε ∇a = G.

Uniform estimates may be deduced from our previous study pertaining to the case ε = 1 (make a change of variable). They lead to estimates independent of ε for the norms appearing in (5.1). Step 2. Convergence. We show that Puε → v strongly and that (Quε, bε) → 0 in a weaker norm. In order to prove the convergence to 0 of (Quε, bε), we use the dispersion properties for the acoustic wave equations in Rd. One may prove that a suitable (low) norm of (Quε, bε) decays like a power of ε. By putting together the uniform estimates of the previous step, parabolic estimates and tricky paraproduct estimates, one may d −1+α P ε − 1 ˙ 2 deduce that the nonlinear terms of the Lam´eequation satisfied by u v go to zero in LT (B2,1 ). As a consequence, we conclude that Puε → v strongly in high norm, like some power of ε. Step 3. Bootstrap. By combining the facts that Puε ∼ v and (Quε, bε) ∼ 0 with a bootstrap argument, one may conclude that

lim inf Tε > T ε→0+ and that the uniform estimates and the results of convergence described in the previous steps hold true on [0,T ]. 274 Danchin R Sci China Math February 2012 Vol. 55 No. 2

6 A few open problems

We conclude these notes with a short list of open problems related to the study of the compressible Navier-Stokes system. (1) Uniqueness for a larger set of indices p : owing to the loss of one derivative in the stability estimates, uniqueness holds true for a much smaller family of critical Besov spaces. In particular, it cannot be proved for velocities in Besov spaces with negative index of regularity (see the recent work by Haspot in [26], though). Is there any alternative method for proving uniqueness so as to avoid this loss? (2) For smooth enough data satisfying some compatibility conditions, one may prove existence and uniqueness results (see for example the works by Salvi and Straˇskrabain [33] and by Cho et al. [10]) even though there may be some vacuum. Is it possible to get similar results for critical regularity? (3) Weak-strong uniqueness results: in the case of critical data with no vacuum, one can construct both global weak solutions with finite energy (after the work by Lions in [30]) and local “strong” solutions with critical regularity. Do those two types of solutions coincide? (See the recent work by Germain in [22] for preliminary results). (4) Extend results with critical regularity to domains with, say, Dirichlet boundary conditions for the velocity at the boundary (see a first attempt in [19]). (5) Study well-posedness for anisotropic norms or anisotropic viscosity coefficients (with possibly no viscosity in the vertical direction as in Paicu’s work [31] for the incompressible Navier-Stokes equations). (6) Generalize the local and global existence statements for the full Navier-Stokes equations or for related models with more physics (for example, other type of stress tensors). (7) Find a weaker smallness condition ensuring global existence (for example, in the spirit of what is known in the incompressible case). (8) Is there any way to take advantage of the Lp approach used in the global statement to improve the result pertaining to the incompressible limit? (9) Establish an incompressible convergence result for large data with critical regularity. (10) Study the low Mach number limit in the critical framework for the full Navier-Stokes equations (adapt the work by Alazard in [1] to the critical framework).

Acknowledgements These notes have been written for the Chinese-French summer research institute project on stress tensor effects on fluid mechanics, while visiting the Morningside Center of Mathematics, Beijing, in January 2010. The author is indebted to local and French organizers for their kind invitation.

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Mathematics

CONTENTS Vol. 55 No. 2 February 2012

Articles

Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves ...... 207 ALAZARD Thomas

A mathematical model for unsteady mixed flows in closed water pipes ...... 221

BOURDARIAS Christian, ERSOY Mehmet & GERBI Stéphane

A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations ...... 245

DANCHIN Raphaël

Shallow water equations for power law and Bingham fluids ...... 277 FERNÁNDEZ-NIETO Enrique D., NOBLE Pascal & VILA Jean-Paul

Multicomponent flow modeling ...... 285 GIOVANGIGLI Vincent

Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces ...... 309 HASPOT Boris

Stability of planar diffusion wave for nonlinear evolution equation ...... 337

HE Cheng, HUANG FeiMin & YONG Yan

Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics ...... 353 LE BRIS Claude & LELIÈVRE Tony

On the rigidity of solitary waves for the focusing mass-critical NLS indimensions d ≥ 2 ...... 385 LI Dong & ZHANG XiaoYi

Large shear rate behavior for the Hébraud-Lequeux model ...... 435

OLIVIER Julien

Regularity of the Koch-Tataru solutions to Navier-Stokes system ...... 453

ZHANG Ping & ZHANG Ting

math.scichina.com www.springer.com/scp Indexed by: SCI-CD MR Z Math MathSciNet