Vol. 39, N 3, 2005, pp. 459–475 DOI: 10.1051/m2an:2005019
LOW MACH NUMBER LIMIT FOR VISCOUS COMPRESSIBLE FLOWS
Rapha¨ el Danchin
1Abstract. In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case ofbarotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions.
We focus on the case ofill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations when the Mach number goes to 0. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small . The reader is referred to [R. Danchin,Ann. Sci. ´Ec. Norm. Sup. (2002)] for the detailed proof in the whole space case, and to [R. Danchin,Am. J. Math. 124(2002) 1153–1219] for the case of periodic boundary conditions.
Mathematics Subject Classification. 35B25, 35B40, 76N10.
Plenary lecture, Low Mach Number Flows Conference, June 21-25, 2004, Porquerolles, France.
1. Introduction 1.1. Governing equations
It is common sense that slightly compressible flows and incompressible flows do not differ much from one another. As a matter of fact, the so-called incompressible Navier-Stokes equations
(N SI)
∂tu+u· ∇u−ν∆u+∇Π =g, divu= 0,
are often considered as relevant for describing slightly compressible liquids as water for instance. In this paper, we discuss to what extent this heuristic may be made rigorous from the mathematical viewpoint.
For the sake of simplicity, we restrict ourselves to the case of a barotropic fluid. The flow at point (t, x) (throughout, it is understood that t ∈R+ denotes the time variable whereasx∈Ω ⊂RN denotes the space variable) is thus characterized by its densityρ=ρ(t, x)∈R+ and its velocityu=u(t, x)∈RN. The governing equations write:
∂tρ+ div(ρu) = 0,
∂t(ρu) + div (ρu⊗u)−µ∆u−(λ+µ)∇divu+∇P =ρf.
Keywords and phrases. Low Mach number limit, compressible Navier-Stokes.
1Laboratoire de Math´ematiques et Applications, Universit´e Paris 12, 61 avenue du G´en´eral de Gaulle, 94010 Cr´eteil Cedex 10, France. [email protected]
c EDP Sciences, SMAI 2005
Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2005019
Above f = f(t, x) stands for a given external force. The viscosity coefficients λ and µ are assumed to be constant1 and to satisfy µ > 0 and ν def= λ+ 2µ > 0, a condition which insures ellipticity of the operator
−µ∆−(λ+µ)∇div and is satisfied for physical fluids. Thebarotropic hypothesis amounts to assuming that the pressureP is a (suitably smooth) function of the density: P =P(ρ).
In the present paper, we assume that the space variablexbelongs to the whole spaceRN or to the periodic boxTNa with period 2πai in theith variable. We are concerned with the multi-dimensional case only: N≥2.
1.2. The small Mach number regime
In the small Mach number regime, the relevant time scale is of order 1/ (where stands for the Mach number). Now, making the change of variable t =tand the change of unknown (ρ, u)(t, x) = (ρ, u)(t, x) the compressible Navier-Stokes equations rewrite:
(N SC)
∂tρ+ div(ρu) = 0,
∂t(ρu) + div(ρu⊗u)−µ∆u−(λ+µ)∇divu+∇P
2 =ρf,
with P =P(ρ). For notational convenience we dropped the exponent over the time variable. More details on the derivation of (N SC) may be found in the introduction of [20], or in [14].
Given a converging sequence of solutions (ρ, u), what can be said about the system satisfied by the limit (ρ, v)? At the formal level, a necessary condition for the convergence of the momentum equation is that ∇P→0 whengoes to zero. Henceρmust be a (positive) constant ¯ρ. Then passing to the limit in the mass conservation equation, we get divu→0. Coming back to the momentum equation, we conclude thatv must satisfy the incompressible Navier-Stokes equation
∂tv+v· ∇v−µ∆v+∇Π =f, divv= 0,
with initial datumPu0 (whereP stands for the L2orthogonal projector over solenoidal vector-fields).
From now on, we make the stability assumptionP( ¯ρ)>0. Besides, performing a harmless change of variable, one can thus assume with no loss of generality that
¯
ρ= 1 and P(1) = 1.
1.2.1. The case of well-prepared data
Passing to the limit in the momentum equation is easier if ∇P =O(2), or, in other words, if we assume thatρ−1 =O(2), and if besides divu=O(). Indeed, this entails that∂tρand∂tuare uniformly bounded so that the dangerous time oscillations cannot occur. Starting from this simple consideration, different authors have studied (N SC) with data of the typeρ0= 1 +2ρ0,1andu0=u0+u0,1with divu0= 0, and (ρ0,1, u0,1) uniformly bounded in a suitable functional space2.
In the usual PDE’S terminology, such data are referred aswell-prepared data. Indeed, they are well-prepared in the sense that they belong (up to lower order terms) to the kernel of the singular operator appearing in (N SC). Hence, they are unlikely to produce highly oscillating terms.
It is then possible to calculate an asymptotic expansion for (ρ, u) in terms of powers of. This approach (historically the first one in the framework of compressible fluids) has been followed by different authors:
Klainerman and Majda in [18], Kreiss, Lorenz and Naughton in [19] and Hoff in [15] amongst others.
1This assumption may be somewhat relaxed.
2We here takef≡0 to simplify.
1.2.2. The case of ill-prepared data
In the present paper, we make the weaker assumption that the initial density satisfiesρ0= 1 +b0 and that (b0, u0, f) tends to some limit in a suitable sense that we refrain from making precise since we are going to assume from now on (for simplicity only) thatb0,u0andf areindependentof .
Denotingρ= 1 +b, it is found that (b, u) satisfies
(N SC)
∂tb+divu
=−div(bu),
∂tu+u· ∇u− Au
1 +b + (1+k(b))∇b =f, (b, u)|t=0= (b0, u0),
withA:=µ∆ + (λ+µ)∇div andka smooth function satisfyingk(0) = 0.
The problem of passing to the limit in (N SC) has been studied by different authors in the last decade.
Roughly, two kinds of approach have been used.
The first one is based on the use of the following (formal) “energy equality”:
(√
ρu)(t)2
L2+
π(t, x) dx+ 2 t
0 µ∇u2L2+ (λ+µ)divu2L2
dτ = √
ρ0u02
L2+
π0(x) dx+ 2 t
0
ρf·u(τ, x) dxdτ (1) whereπstands for the “free energy per unit volume” and may be computed fromP. For example, ifP=P(ρ) withP(ρ) =aργ, we have
π=a(ρ)γ−1−γ(ρ−1) 2γ(γ−1) ·
For fixed, by combining (1) and very fine compactness arguments, the existence ofglobal weak solutionswith finite energy (i.e. finite left-hand side in (1)) has been stated by Lions in the viscous case for pressure laws of the type P(ρ) =aργ withγlarge enough (see [21] for more precise statements).
Next, taking advantage of the uniform estimates provided by (1), it is possible to pass to the limit when goes to 0. This latter result has been stated by Lions, Desjardins, Grenier and Masmoudi in various contexts.
The proof strongly depends on the boundary conditions. The reader may refer to [22] for the case of periodic boundary conditions, to [9] for the whole space and to [10] for the case of bounded domain with homogeneous Dirichlet conditions. Let us mention that Lions and Masmoudi have also proved somelocal weak convergence results regardless of the boundary conditions (see [23]).
From the practical viewpoint however, these nice and very general results of convergence for global weak solutions do not give much information in dimensionN ≥3. Indeed, owing to the use of compactness arguments, and to the fact that the uniqueness of the global weak solution of the limit system – the incompressible Navier- Stokes equation – has not been stated, convergence holds true onlyup to a subsequence. Of course, in dimension N = 2 the global weak solution of the limit system is actually unique so that the whole sequence actually converges!
Let us also stress that this approach cannot be applied to elucidate the inviscid case.
Another possible approach for passing to the limit in (N SC) amounts to assuming that the data are smooth enough so that the initial value problem (N SC) is well-posed (i.e. to each data corresponds a unique solution on a small time interval, and small changes on the data induce small changes on the solution). As regards the study of the small Mach number limit in the framework of ill-prepared data, one can motion the works by Ukai, (see [29]), Schochet [28], M´etivier, Schochet (see [26, 27]) in the inviscid case, and the papers by Gallagher [12]
and Hagstr¨om, Lorenz (see [14]) in the viscous case.
In the present paper, we state the following results in the caseRn or Tna (seeThms. 4.2 and 5.2 for the full statements):
• For a suitable class of data, system (N SC) has a unique solution on some non trivial time interval [0, T].
• The timeTmay be chosen so that lim inf→0T≥T whereT denotes the lifespan of the solutionv to the incompressible Navier-Stokes equations3:
(N SI)
∂tv+P(v· ∇v)−µ∆v=Pf, v|t=0=Pu0.
In particular, ifT = +∞then alsoT= +∞for small.
• When goes to 0, the incompressible partPu ofu tends strongly to v in an appropriate functional space whereas the gradient partQuofu, andbtend to 0 in a weaker sense (which strongly depends on the boundary conditions, the periodic case being worse than the whole space case).
The whole proof of these results (which is quite long and technical) may be found in [6,7]. In the present survey, we skip the details in order to stay as far as possible at the elementary level.
2. Well-posedness results for compressible flows 2.1. A good functional framework
Finding a good functional framework is one of the key points in the study of slightly compressible fluids.
The difficulty lies in the fact that we have to exhibit a functional framework which suits both (N SC) and the limit system (N SI), despite the fact that (N SC) is of hyperbolic/parabolic type whereas (N SI) is definitely parabolic.
Let us first focus on the incompressible Navier-Stokes equations which have been more studied than (N SC).
It is well known that a good functional framework for (N SI) must be related to itsscaling invariance. More precisely, observing that the transformation
v(t, x)−→λv(λ2t, λx) (2)
does not affect (N SI) (provided the data have been modified accordingly), one can guess that an appropriate functional space for (N SI) must have a norm which is invariant (up to an irrelevant constant) for allλ >0 by the above transformation. This induces to take initial data in a functional space invariant byv0(x)−→λv0(λx) and to show well-posedness in a functional space E which iscritical insofar as its norm is invariant by (2). It turns out that once well-posedness has been stated in a critical spaceE, it holds also true in any “reasonable”
subspaceE such thatE →E.
The first example of solving (N SI) in a critical space has been exhibited by Fujita and Kato in 1964 (see [11]). There, the initial velocity belongs to the homogeneous Sobolev space ˙HN2−1(RN) and well-posedness is stated in a subspace ofC([0, T]; ˙HN2−1(RN)). Since then, well-posedness has been stated in a plethora of critical spaces: Lebesgue spaceLN(RN) (see [16]), Besov spaces ˙B
Np−1
p,r (see [2]) etc.
Fortified by these results on the incompressible model, one can guess that the initial velocityu0 for (N SC) must belong to a space whose norm is invariant byu0(x)−→λu0(λx). We expect scaling considerations to help us to find the relevant space for the density. Now, we notice that if the pressure term is neglected in (N SC), then the transformation
(b(t, x), u(t, x))−→(b(λ2t, λx), λu(λ2t, λx)) does not affect (N SC).
3Remind that the operatorPstands for theL2 projector on divergence-free vector-fields.
Hence, choosing data (b0, u0) in acriticalfunctional spaceE×F whose norm is invariant by (b0(x), u0(x))−→(b0(λx), λu0(λx))
seems appropriate.
Note that it roughly means thatu0and∇b0must have the same regularity. If we restrict ourselves to Sobolev spaces, this leads to the assumption thatu0and∇b0 are in ˙HN2−1(RN).
2.2. Existence of solutions with critical regularity
Whether local well-posedness may be proved foru0and∇b0in ˙HN2−1(RN) is unclear. The main reason why is that assuming∇b0∈H˙N2−1(RN) does not provide any control on theL∞ norm ofb so that the appearance of vacuum in the flow cannot be excluded.
At least for small time, this trouble may be avoided if one supposes thatb0belongs to a slightly smaller space which is continuously embedded in L∞. If one wishes to keep the scaling invariance, one can assume that b0 belongs to the homogeneous Besov ˙B2,1N2 which satisfies ˙B2,1N2 →L∞ and whose norm is invariant by dilation.
We here recall that for |s| ≤ N/2, homogeneous Besov spaces B˙2,1s may be defined as the completion of smooth compactly supported functions for the norm
zB˙s2,1 def=
q∈Z
2qs∆qzL2,
where (∆q)q∈Zstands for a homogeneous dyadic partition in Fourier variables, that isF(∆qz)(ξ) =ϕ(2−qξ)Fz(ξ) for some smoothϕsupported in an annulus and such that
q∈Zϕ(2−qξ) = 1 wheneverξ= 0.
Remark that
zH˙s ≈
q∈Z
22qs∆qz2L2
12
.
Hence ˙B2,1s →H˙s. The spaces ˙B2,1s and ˙Hsare very close to one another though.
In [3, 5], we prove that (N SC) with= 1 are well-posed in critical Besov spaces4. In particular, we get the following result:
Theorem 2.1. Assume thatb0 belongs toB˙2,1N2 ∩B˙2,1N2−1 (with small norm), thatu0 belongs to B˙2,1N2−1 and that f ∈L1(0, T; ˙B2,1N2−1). Then (N SC1)has a unique local solution(b, u)such that
b∈ C [0, T]; ˙B2,1N2 ∩B˙2,1N2−1
∩L2 0, T; ˙B2,1N2
, u∈ C [0, T]; ˙B2,1N2−1
∩L1 0, T; ˙B2,1N2+1
. Besides, ifu0˙
B2,1N2−1 is small compared tomin(µ, λ+ 2µ)then T= +∞.
Remark 2.2. By performing a change of variable, system (N SC) may be shown to be equivalent to (N SC1).
Hence a similar statement holds true for (N SC).
Remark 2.3. As expected, one can prove that more regular data yield more regular solutions.
4See [8] for the proof of uniqueness in dimensionN= 2.
3. The linearized equations
Investigating the linearized equations at (b, u) = (0,0) is a key step in the study of the incompressible limit for (N SC).
Since system (N SC) writes
∂tb+divu
=−div(bu),
∂tu+u· ∇u− Au
1 +b+ (1+k(b))∇b =f
withka smooth function such thatk(0) = 0 andA=µ∆ + (λ+µ)∇div, the linearized equations about (0,0) read
(L)
∂tb˙+div ˙u = ˙F ,
∂tu˙− Au˙ +∇b˙ = ˙G.
For a more accurate analysis of this system, we shall decompose the second equation on solenoidal and gradient- like vector-fields. This may be done by introducing the L2 projector on solenoidal vector-fields P and its conjugate projector5Qdef= Id− P.
Denotingνdef= λ+ 2µ, and applyingP andQto the last line of the above linear system, we get
∂tb˙+divQu˙ = ˙F ,
∂tQu˙−ν∆Qu˙+∇b˙
=QG,˙
∂tPu˙ −µ∆Pu˙ =PG.˙
On the one handPu˙ satisfies a mere heat equation with constant diffusion coefficient. On the other hand, there is a coupling between the first two equations through the singular first order terms−1div ˙Quand−1∇b.˙
Introducing ˙ddef= |D|−1div ˙u(where|D|α stands for the Fourier multiplier |ξ|α) so thatQu˙ =∇|D|−1d, the˙ coupling between ˙bandQu˙ may be described by the following baby system:
(BM)
∂tb˙+|D|d˙ = ˙F ,
∂td˙−ν∆ ˙d−|D|b˙ = ˙G.
By performing a Fourier transform with respect to the space variables, the above model may be solved explicitly.
Indeed, in Fourier variables, (BM) reads d
dt Fb˙
Fd˙
(t, ξ) =
0 −−1|ξ|
−1|ξ| −ν|ξ|2
A(ξ)
Fb(t, ξ)˙ Fd(t, ξ)˙
+
FF(t, ξ)˙ FG(t, ξ)˙
.
By computing the eigenvalues of the matrixA(ξ), we discover that the system behaves quite differently at low and high frequencies.
5 In RN or TNa, Q coincides with the 0-th order pseudo-differential operator −∇div (−∆)−1 whose vector-valued symbol isξ|ξ|−2ξ·
The low frequency regime: forν|ξ|<2, the matrixA(ξ) has two complex conjugate eigenvalues with negative real part:
λ±(ξ) =−ν|ξ|2 2
1±i
4 2ν2|ξ|2−1
. Hence in the asymptoticν|ξ| →0, we have
λ±(ξ)∼ −ν|ξ|2 2 ∓i|ξ|
·
In the low frequency regime, we thus expect the pseudo-differential operatorA(D) to behave like−ν2∆∓i|D| . We conclude that at low frequencies the semi-group generated by A(D) should inherit from the properties of the heat equation with diffusion ν2 and from the ones of the (half) wave equation with velocity−1.
The high frequency regime: forν|ξ|>2, the matrixA(ξ) has two negative eigenvalues, namely
λ±(ξ) =−ν|ξ|2 2
1±
1− 4
2ν2|ξ|2
·
In the asymptotic ν|ξ| →+∞, we have
λ+(ξ)∼ −ν|ξ|2 and λ−(ξ)∼ − 1 2ν·
Hence, system (BM) has one parabolic mode with diffusion ν (a further computation shows that this mode tends to be collinear to ˙d), and one damped mode with coefficient 12ν (which tends to be collinear to ˙b).
According to this rough analysis, the tentative conclusion is that the velocity ˙uhas a parabolic behaviour.
The behaviour of ˙b is more complex: parabolic for low frequencies and damped for high frequencies. One also has to keep in mind that both ˙u and ˙b benefit from highly oscillating properties at low frequency, a regime which tends to overrun high frequencies whengoes to 0...
By combining the above heuristics with a few calculation, one eventually gets the following estimate for all s∈Randt∈R+ and some universal constantC:
b(t)˙ Bs,∞
ν +u(t)˙ B˙s−1
2,1 + t
0 νb˙ Bs,1
ν + min(µ, ν)u˙ B˙s+1
2,1
dτ≤ C
b˙0Bνs,∞+u˙0B˙s−12,1 + t
0 νF˙Bs,∞ν + min(µ, ν)G˙ B˙s−12,1
dτ
. (3) Above,zBσ,1α andzBασ,∞ stand forhybridBesov norms defined as follows:
zBσ,1α def=
q∈Z
2qσmin(α−1,2q)∆qzL2, zBασ,∞
def=
q∈Z
2qσmax(α,2−q)∆qzL2.
Note that ·Bs,1ν is equivalent to ·B˙2,1s+1 at low frequencies and to (ν)−1·B˙s2,1 at high frequencies whereas
·Bνs,∞ is equivalent to·B˙2,1s−1at low frequencies and toν·B˙s2,1 at high frequencies. In both cases, the cut-off frequency between the two regimes is at|ξ| ∼(ν)−1.
Remark 3.1. Hybrid Sobolev spacesHνs,1 andHνs,∞may be defined in a similar way. It is then easy to state a similar estimate as (3) in the Sobolev framework (see [6]).
Remark 3.2. Inequality (3) provides us with uniform estimates inC([0, T]; ˙Bs−1)∩L2(0, T; ˙Bs) for (˙b,u). Note˙ however that we did not take advantage of the oscillating properties of (BM) at low frequencies. This is due to the use ofL2type estimates which kills thei|D|/part of the operatorA(D) at low frequency. In the next two sections, we shall see how to use those oscillating properties to pass to the limit in (N SC).
4. The incompressible limit in the case of the whole space 4.1. The heuristic
The basic idea is that the highly oscillating acoustic waves go at infinity with the high speed −1 and polynomial decay6. This fact has been used before by S. Ukai in the inviscid case (see [29]) and in the viscous case by B. Desjardins and E. Grenier in the framework of global weak solutions (see [9]).
More concretely, proving that the compressible part (b,Qu) of the solution tends to zero relies on dispersive estimates ofStrichartz typefor the equation of acoustics:
(W)
∂tb˙+|D|d˙ = ˙F ,
∂td˙−|D|b˙ = ˙G.
Let us mention in passing that Strichartz estimates have been discovered by Strichartz (see the original pa- per [28]) and have been extensively used lastly for dispersive equations such as the Schr¨odinger or the wave equation (seee.g. [13, 17]).
Stating dispersive estimates for (W) requires the use of Besov norms of type Lp: zB˙p,1σ def=
q∈Z
2qσ∆qzLp if 1≤p <∞, and zB˙σ∞,1 def= sup
q∈Z2qσ∆qzL∞. Proposition 4.1. Let (˙b,d)˙ be a solution to(W). Then we have the inequality
(˙b,d)˙
LrT( ˙Bs+N( 1p,1 p−12 )+1r)≤C1r (˙b0,d˙0)B˙2,1s +( ˙F ,G)˙ L1 T( ˙B2,1s )
for some constant C=Cr,p,N whenever p≥2, 2
r ≤min
1,(N−1) 1 2 −1
p
and (r, p, N)= (2,∞,3).
Example. For data (b0, u0) in ˙B2,1N2−12 and in the case where dimension is N ≥4, one can choosep=∞and r= 2. As we have the embedding ˙B0∞,1 →L∞, one gets
(b, d)L2
T(L∞)≤C12
(b0, d0)˙
B2,1N2−12 +(F, G)
L1T( ˙B2,1N2−12)
. (4)
Of course the above proposition also provides some decay inin the more physical cases whereN = 2,3.
6Note that this heuristic would be false in dimensionN= 1.
4.2. Results of convergence
Let us first state a result of convergence in high dimension:
Theorem 4.2. LetN ≥3,α∈(0,1/2)andT∈(0,+∞]. Assume thatb0∈B˙2,1N2−1∩B˙2,1N2+α,u0∈B˙2,1N2−1∩B˙2,1N2+α−1 andf ∈L1(R+; ˙B2,1N2−1∩B˙2,1N2−1+α), and that system
∂tv+v· ∇v−µ∆v+∇Π =f, divv= 0, v|t=0=Pu0
has a unique solution7v∈ C([0, T]; ˙B2,1N2−1∩B˙2,1N2−1+α)∩L1(0, T; ˙B2,1N2+1∩B˙2,1N2+1+α)on [0, T]×RN.
There exists an0>0 such that for all ∈(0, 0)system(N SC)has a unique solution(b, u)with u∈ C [0, T]; ˙B2,1N2−1∩B˙2,1N2+α−1
∩L1 0, T; ˙B2,1N2+1∩B˙2,1N2+α+1
, b∈C [0, T]; ˙B2,1N2 ∩B˙2,1N2+α
and b∈L2 0, T; ˙B2,1N2 ∩B˙2,1N2+α
∩ C [0, T]; ˙B2,1N2−1∩B˙2,1N2+α−1
uniformly with respect to.
Besides, the incompressible part of the velocity, Pu, tends tov in C [0, T]; ˙B2,1N2−1∩B˙2,1N2+α−1
,∩L1 0, T; ˙B2,1N2+1∩B˙2,1N2+α+1
whereas|D|α−1+1p(b,Qu)tends to0 inLp(0, T;L∞)with the rate1p wheneverp≥2 ifN ≥4,p >2ifN = 3.
In the case N = 2 and under the assumptions made in the theorem above (even if α = 0 in fact), the incompressible solutionvis always global. We end up with the following statement:
Theorem 4.3. Let α∈(0,1/6],b0∈B˙2,10 ∩B˙2,11+α,u0∈B˙02,1∩B˙α2,1 andf ∈L1(R+; ˙B2,10 ∩B˙2,1α ).
The incompressible system (N SI) with initial datum Pu0 and external force Pf has a global solution v ∈ C(R+; ˙B2,10 ∩B˙2,1α )∩L1(R+; ˙B2,12 ∩B˙2+α2,1 ). Besides, there exists an 0 > 0 such that for all ∈ (0, 0), sys- tem (N SC) has a unique global solution which belongs to the space described in Theorem 4.2 with T = +∞. In particular,Pu tends tov inL∞(R+; ˙B2,10 ∩B˙2,1α )∩L1(R+; ˙B22,1∩B˙2+α2,1 )whereas|D|α−34(b,Qu)tends to0 in L4(R+;L∞) with the rate14.
4.3. Sketch of the proof
Let us describe briefly the main steps of the proof. To avoid technicalities, we assume that N ≥ 4 and α= 1/2 so that inequality (4) may be used. The reader is referred to [7] for the general caseN ≥2 andα >0.
Step 1(local well-posedness for fixed). Under the assumptions of Theorem 4.2, it has been proved in [5] that there exists a positiveT such that (N SC) has a unique solution (b, u) on [0, T]×RN which belongs to the spaceET defined as follows:
ET def= L1 0, T;BνN2,1∩BνN2+12,1
∩ C [0, T];BνN2,∞∩BνN2+12,∞
× L1 0, T; ˙B2,1N2+1∩B˙2,1N2+32
∩ C [0, T]; ˙B2,1N2−1∩B˙2,1N2−12 N
. Note however that the results of [5] do not supply a positive lower bound forTwhengoes to 0.
7It is well known that this assumption is always fulfilled for some positiveT. Besides, one can takeT = +∞if the norm ofPu0
in ˙B2,1N2−1 is small compared toµ.
Step 2(uniform estimates for the solution). We use the fact that (b, u) satisfies
∂tb+u· ∇b+divu =F,
∂tu+u· ∇u− Au+∇b =G,
withF=−bdivu andG=f+1+bbAu−k(b)∇b wherek satisfiesk(0) = 0.
Up to the convection termsu· ∇b andu· ∇u, this system is of type (L). It is actually possible to extend inequality (3) to handle the convection terms (see [7]). Taking advantage of tame estimates, one eventually gets the following inequality fors∈[N/2, N/2 + 1/2]:
b(t)Bνs,∞+u(t)B˙2,1s−1+ t
0 bBs,1ν +uB˙s+12,1
dτ≤ CeCV(t)
b0Bs,∞ν +u0B˙s−12,1 + t
0 fBs,∞ν +bL∞uB˙s+12,1
dτ
(5) with
V(t)def= t
0
u2L∞+23∇uL43∞+b2L∞
dτ .
Step 3. (convergence of the compressible part of the solution). Since (Qu, b) satisfies
∂tb+divQu =F,
∂tQu+∇b =G, with
F=−div(bu) and G=−ν∆Qu+Q
f+ Au
1 +b −k(b)∇b−u· ∇u
, Strichartz estimate (4) combined with various product estimates in Besov spaces lead to
(b,Qu)L2
t(L∞)≤C12
b0˙
B2,1N2−12 +u0˙
B2,1N2−12 +Qf
L1t( ˙B2,1N2−12)+(b, u)E t
1 +(b, u)E t
. (6)
Note that we expect the previous step to provide a uniform control on (b, u)E
T. Hence (Qu, b) should tend to 0 inL2(0, T;L∞) with the rate 12.
Step 4. (convergence of the incompressible part of the solution). In order to study the convergence of Pu, one can use the fact thatwdef=Pu−v satisfies
∂tw+P
w·∇v+u·∇w
−µ∆w=−P Qu·∇v+u·∇Qu+1+bbAu , w|t=0= 0.
Note that we expect the right-hand side to be small whengoes to 0. Indeed, the first two terms containQu which, according to the previous step is of size12 in L2(0, T;L∞) and the last term should also be small since it begins with an.
More concretely, standard results for the heat equation (seee.g. Prop. 7.4 in [7]) yield:
wFt ≤Cexp
C t
0 (v˙
B2,1N2 +1+u˙
B2,1N2 +1) dτ
t
0 Qu·∇v+u·∇Qu+ b 1 +bAu
B˙2,1N2−1∩B˙2,1N2−12 dτ
with
Ftdef= L1 0, t; ˙B2,1N2+1∩B˙2,1N2+32
∩ C [0, t]; ˙B2,1N2−1∩B˙2,1N2−12 N
.
At a price of a bit of paradifferential calculus, it is possible to show that the right-hand side is indeed small.
For instance, one has
Qu·∇v
L1T B˙2,1N2−1
≤C2N1
Qu
L2T B˙2,1N2
+−12QuL2 T(L∞)
.
Hence one can expectwto tend to zero, or in other words, Pu to tend tov.
Step 5. (bootstrap and continuation argument). In this last step, one combines the different estimates obtained hitherto and a standard bootstrap argument to prove that if v ∈ FT is a solution to (NSI) on [0, T]×RN then (b, u) may be continued beyond T for small enough (in other word, one can take T ≥ T). Since the various estimates proved in the previous steps do notdepend on T, the bootstrap argument still works if T = +∞, yielding a global solution for (N SC) with small.
5. The incompressible limit in the periodic case 5.1. The filtering method
Prescribing periodic boundary conditions precludes from using dispersive properties in order to pass to the limit in (N SC). Indeed, there is no chance that the acoustic waves go at infinity. Hence, there could be resonances which may hinder the convergence to the incompressible model.
We are going to see however that these resonances are very few and that the filtering method (introduced by Schochet in the framework ofinviscid compressible fluids, see [28]) enables us to prove convergence toward the homogeneous incompressible Navier-Stokes equations.
In order to introduce the filtering method, let us rewrite (N SC) as d
dt b
u
+L
b u
=
−div(bu)
f −u· ∇u−κb∇b−K(b )b∇b−I(b)Au
where the skewsymmetric operatorLis defined by L
b u
def=
divu
∇b
,
P(1 +z)/(1 +z) = 1 +κz+zK(z) with K(0) = 0 and I(z)def=z/(1 +z).
In the next proposition, we summarize the important properties ofL:
Proposition 5.1. The operator Lgenerates a unitary groupeτL. Besides, KerL={(b, u)|b= cste and divu= 0}, and thus, eτL
0 Pu
= 0
Pu
.
For Z : TN → RN+1, we denote by (eτLZ)1 the first component of eτLZ and by (eτLZ)2 the last N components of eτLZ.
