From Boltzmann’s kinetic theory to Euler’s equations

Physica D 237 (2008) 2028–2036 www.elsevier.com/locate/physd

From Boltzmann’s kinetic theory to Euler’s equations Laure Saint-Raymond ∗ D´epartement de Math´ematiques et Applications, Ecole Normale Sup´erieure, 45 rue d’Ulm, 75005 Paris, France Available online 14 December 2007

Abstract The incompressible Euler equations are obtained as a weak asymptotics of the Boltzmann equation in the fast relaxation limit (the Knudsen number Kn goes to zero), when both the Mach number Ma (defined as the ratio between the bulk velocity and the speed of sound) and the inverse Reynolds number Kn/Ma (which measures the viscosity of the fluid) go to zero. The entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled Boltzmann equation to its expected limit (provided that it is sufficiently smooth). It thus leads to some strong convergence result. One of the main points to be understood is how to deal with the corrections to the weak limit, i.e. the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves. c 2007 Elsevier B.V. All rights reserved.

PACS: 47.10.A; 47.15.ki Keywords: Kinetic theory; Hydrodynamic limits; Incompressible Euler equations; Entropy method

The topic of this paper is to discuss the connections between the various models describing the motion of fluids. Actually there is a number of ways to describe that motion, depending on the space and time scales we consider (see Fig. 1). At the atomic level, the fluid is a large system of particles interacting according to repulsive forces. We therefore have a complex system of coupled ordinary differential equations, for which essentially no qualitative behavior can be predicted. However, in general, we are not interested in the exact positions and velocities of all particles, so that a statistical approach is suitable. This is precisely the point of view of kinetic theory. Note however that it can be applied only for rarefied gases (in the sense that the size of particles has to be small compared to the mean free path). Now, if we consider typical length scales which are large compared with the mean free path, the collision process is dominating, and local thermodynamic equilibrium is reached almost instantaneously everywhere. The state of the fluid can therefore be described by macroscopic variables such as the pressure, density, and bulk velocity, which are governed by some hydrodynamic equations. Considering still larger time and space scales, it should be possible to deal

again statistically with the nonlinearity, which should lead to turbulence models. A natural question is therefore to understand the connections between the different levels of modeling, and to get a unified theory of fluids, which is part of the sixth problem proposed by Hilbert on the occasion of the International Congress of Mathematicians held at Paris in 1900 [14]. In the present paper, we will actually focus on the transition from kinetic theory to hydrodynamics, and more precisely from the Boltzmann equation to the incompressible Euler equations. Let us just mention that the derivation of the Boltzmann equation from Newtonian mechanics has been justified by Lanford for short times [17], whereas to our knowledge there is no such rigorous study for the transition from determinist hydrodynamics to turbulence (see [16] for a formal derivation, and [8] for an attempted mathematical approach). Let us also mention the contributions in statistical physics which study directly the transition between stochastic systems of particles and hydrodynamics (see [20] for instance). Actually such direct connections are compulsory to derive real constitutive equations (other than the law of perfect gases). 1. The formal derivation

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E-mail address: [email protected]. c 2007 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2007.11.023

Our first objective is to explain the formal expansions leading to the incompressible Euler limit. Note that it has

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momentum and energy, or in other words the first principle of thermodynamics. Using the same symmetries, we also obtain Z def D( f ) = − Q( f, f ) ln f (v)dv ≥ 0 (5) and thus the local decay of entropy (note that the mathematical entropy is the opposite of the physical entropy!): Z Z Ma∂t f ln f dv + ∇x · f ln f (v)vdv ≤ 0. (6)

Fig. 1. Description of fluids.

been only recently observed ([1] for time-independent problems and [2,6] for time-dependent problems) that, under convenient scaling, incompressible equations could be directly derived from the Boltzmann equation. 1.1. The Boltzmann equation Our starting point is the non-dimensional Boltzmann equation, which expresses some balance between the free transport of particles (left-hand side) and the collision process (right-hand side):

We therefore obtain a Lyapunov functional for the Boltzmann equation, which expresses the irreversibility predicted by the second principle of thermodynamics. Collisions are responsible for that relaxation process. Each elementary process loses some information on the precise microscopic configuration that is realized, so that the global effect of collisions is to increase the uncertainty. The asymptotic distribution, which minimizes the entropy, and cancels the entropy dissipation Z Q( f, f ) ln f (v)dv = 0 ⇔ ∀v ∈ R3 , Q( f, f )(v) = 0, R is the Gaussian having the same mass R = f dv, momentum R R RU = f vdv and energy 12 R(U 2 + 3Θ) = 12 f |v|2 dv. In other words, the thermodynamic equilibrium obeys Maxwell’s statistics. 1.2. The incompressible inviscid regime

1 Ma∂t f + v · ∇x f = Q( f, f ) (1) Kn where Ma denotes the Mach number (measuring the compressibility of the fluid) and Kn is the Knudsen number, defined as the ratio between the mean free path and the observation length scale. The operator Q is localized in t and x, and describes elastic binary collisions. Its precise formulation is rather complicated Z Z
Q( f, f )(v) = f (v 0 ) f (v10 ) − f (v) f (v1 ) bdv1 dω, (2) R3

S2

v = v + (v − v1 ) · ωω, 0

v10 = v1 − (v − v1 ) · ωω

(3)

for some nonnegative function b ≡ b(v − v1 , ω), called the cross-section, giving the statistical repartition of pre-collisional velocities (v 0 , v10 ) leading to (v, v1 ). This exact formulation will not be useful in what follows of our presentation. What is needed is the physics encoded in this mathematical operator. As collisions are assumed to be elastic, Q has some symmetry properties leading to the following identities Z Z Q( f, f )(v)dv = Q( f, f )vi dv Z = Q( f, f )|v|2 dv = 0. (4) In particular, integrating the kinetic Eq. (1) against 1, v and 12 |v|2 , we recover the local conservations of mass,

In the fast relaxation limit, i.e. when the Knudsen number Kn – defined as the ratio between the mean free path and the typical length scale – tends to zero, we thus expect the gas to be at local thermodynamic equilibrium. The distribution f is therefore completely determined by the macroscopic quantities R = R(t, x), U = U (t, x) and Θ = Θ(t, x). Let us first recall that, at leading order with respect to the Knudsen number Kn, the hydrodynamic equations, obtained from the local conservation laws replacing f by the corresponding local Maxwellian   R(t, x) |v − U (t, x)|2 , M R,U,Θ ∼ exp − 2Θ(t, x) (2π Θ(t x, x))3/2 are, up to terms of order O(Kn), Ma∂t R + ∇x · (RU ) = 0, Ma∂t (RU ) + ∇x · (RU ⊗ U + RΘ I d) = 0,     1 1 3 5 2 2 RU + RΘ + ∇x · RU U + RΘU = 0, Ma∂t 2 2 2 2 (7) known as the compressible Euler system for perfect gases. Of course such an asymptotics does not remain relevant if the Mach number Ma also goes to zero in the regime to be considered. The Mach number Ma – defined as the ratio

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between the bulk velocity of the fluid and the speed of sound – measures indeed the compressibility of the fluid: if Ma → 0 the first equation in (7) is nothing else than the incompressibility constraint ∇ · (RU ) = 0. The equations of motion are then obtained by a systematic multiscale expansion. Their precise formulation depends further on another important feature of the fluid, namely on its viscosity, which is measured by the Reynolds number Re. Note that, for perfect gases (i.e. for all gases which can be described by Boltzmann’s equation), Von Karmann’s relation states Ma Re = Kn so that all features of the fluid are completely determined by the two non-dimensional parameters Ma and Kn (see [1] for more details). 1.3. Taking limits as ε → 0 In all what follows we are interested in the regimes leading to the incompressible Euler equations. We will thus choose Ma = ε → 0, √ meaning in particular that U/ Θ = O(ε), and Kn = εq

with q > 1

in order that the Reynolds number Ma/Kn = ε 1−q tends to infinity. We will further restrict our attention to the homogeneous case, in the sense that the density R and temperature Θ will be assumed to be fluctuations of order ε around their equilibrium values, say without loss of generality around 1. Denoting by ρ, u and θ the fluctuations of mass, momentum and temperature, and plugging the expansions R = 1 + ερ,

U = εu,

Θ = 1 + εθ,

in the previous hydrodynamic equation (7), we get at leading order with respect to ε ∇ · u = 0, ∇(ρ + θ ) = 0,

(8)

which are the macroscopic constraints (incompressibility and Boussinesq relations), then at second order the equations of motion ∂t u + (u · ∇x )u + ∇ p = 0, ∂t θ + ∇x · (θ u) = 0.

(9)

where p is the pressure, defined as the Lagrange multiplier associated with the incompressibility constraint ∇ · u = 0. Dealing with the more general case when R and Θ have variations of order 1 is not really more difficult from a formal point of view. We actually get the following asymptotics ∇(RΘ) = 0, ∇ · u = 0,

and ∂t R + ∇x · (Ru) = 0, 1 ∂t u + (u · ∇x )u + ∇ p = 0. R The point is that the asymptotic analysis would require to control quantities of different sizes (namely R = O(1), Θ = O(1) and U = O(ε)), and thus to introduce new mathematical tools. Obtaining full proofs valid in all physical configurations is a major problem due in particular to our limited knowledge concerning the solutions of the 3D Euler equations. The main difficulty encountered when trying to justify the previous asymptotic process is to determine the limits of nonlinear terms. Indeed the weak compactness inherited from the physical a priori bounds provides some weak convergence, or in other words some convergence in average. In particular, it is not sufficient to study nonlinear terms as shown for instance by the following example   t sin * 0, ε    2   2t t 1 1 = 1 − cos * . sin ε 2 ε 2 From a physical point of view such a phenomenon can be interpreted in terms of interferences. The question is to decide whether or not high frequency waves bring some contribution to low frequency modes. In order to get a rigorous derivation of the incompressible Euler equations, we will therefore use a stronger notion of convergence. 2. The modulated entropy method The main idea behind energy and entropy methods is to compare – in some appropriate metrics – the distribution under consideration (for instance the solution to the scaled Boltzmann equation) and its formal asymptotics (here the Gaussian M1+ερ,εu,1+εθ with ρ, u, θ satisfying the homogeneous incompressible Euler equation (8) and (9). Note that such a method requires to describe precisely the asymptotic distribution since the remainder has to converge strongly to zero. The first step is to determine a suitable functional measuring the stability of the original system. The convenient quantity here is the scaled relative entropy  ZZ  1 1 fε H ( f |M) = f log − f + M dvdx, ε ε ε M ε2 ε2 where M denotes the centered reduced Gaussian M = M1,0,1 . By Boltzmann’s H theorem (6), the relative entropy is indeed a Lyapunov functional for the Boltzmann equation (1). Furthermore it controls the size of the fluctuation √ !2 ZZ √ 1 fε − M H ( f ε |M) ≥ 2 dvdx. (10) ε ε2

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The idea of using the relative entropy for this type of asymptotic study goes back to Yau [25] in the framework of the Ginzburg–Landau equation, then to Bardos, Golse and Levermore for the Boltzmann equation [3]. The second step is to obtain a precise approximate solution f app using some formal analysis. For hydrodynamic limits, the main term of the approximation is given by some asymptotic expansions due to Hilbert [15] in the inviscid case, and to Chapman and Enskog [4] in the viscous case. Correcting terms are either terms of higher order in the expansion, or terms having rapid variations with respect to time or space variables and thus having no contribution to the mean flow. These fast variations can be oscillations which can be recovered by filtering methods (introduced independently by Schochet [24] and Grenier [11]), or localized phenomena such as boundary and initial layers which require a multiscale treatment (see [5] or [12] for instance). These correctors depend of course on the scaling to be considered. For instance, in the regime leading to the incompressible Euler equations, and in a spatial domain Ω without boundary, they come both from the acoustic waves (fast oscillating) and from the relaxation layer (rapidly decaying). The last step is to establish some stability inequality for the modulated entropy  ZZ  fε 1 1 f log + f H ( f | f ) = − f ε ε app dvdx. ε app f app ε2 ε2 which is the natural quantity to compare the distribution f ε and its formal asymptotics f app in view of the first step above. The expected convergence result arises then as a direct consequence of that stability inequality provided that the family of initial data converges in the appropriate sense. Note that this last step contains all the mathematical contribution in the proof of convergence. It uses technical computations and estimates, which depend strongly on the properties of the solutions to the scaled Boltzmann equation, and thus require a deep understanding both of the transport and collision processes.

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size of the initial data but are not known to be unique, nor to satisfy the local conservations of momentum and energy. Note however that they coincide with the unique classical solution whenever the latter does exist. Let us then state our main convergence result first in the setting of renormalized solutions, which is the most general framework, then in the setting of classical solutions, for which the asymptotics can be described more precisely. For the sake of simplicity, we restrict our attention to spatial domains Ω without boundary namely the whole space R3 or the threedimensional torus T3 . In the regime we consider here, i.e. in the regime leading to the incompressible Euler equations, we then formally expect the approximate solutions to be decomposed as the sum of - a purely kinetic part (determined by the relaxation process in the initial layer); - a fast oscillating hydrodynamic part (governed by the acoustic equations); - a non-oscillating hydrodynamic part (obtained by formal expansion) satisfying the incompressible Euler equations, supplemented by some suitable equation for the temperature. 3.1. In the framework of renormalized solutions We start by precising a little bit the notion of renormalized solution: Definition 1. A renormalized solution of the Boltzmann equation (1) is a function f ∈ C(R+ , L 1loc (Ω × R3 )) which satisfies in the sense of distributions     f 1 0 f M (Ma∂t + v · ∇x ) Γ Γ = Q( f, f ) M Kn M √ for any Γ ∈ C 1 (R+ ) such that |Γ 0 (z)| ≤ C/ 1 + z. Let us then recall that the only requirement for renormalized solutions to exist globally in time is for instance that the initial relative entropy is finite (see [18]):

3. Main results At the present time the mathematical theory of the Boltzmann equation is not really complete, insofar as there is no global existence and uniqueness result for general initial data with finite mass, energy and entropy. The main difficulty comes from the fact that the nonlinearity is quadratic whereas the functional space determined by the physical a priori estimates is roughly speaking the Orlicz space L log L. In particular, for such functions, the collision term does not even make sense. We have therefore at our disposal either strong solutions with higher regularity which require smoothness and smallness assumptions on the initial data, or very weak solutions, called renormalized solutions, which are not known to satisfy the kinetic equation in the sense of distributions but verify a family of formally equivalent equations obtained by some truncation process. These renormalized solutions, built by DiPerna and Lions [7], exist globally in time without restriction on the

Proposition. Assume that the collision cross-section b satisfies Grad’s cutoff assumption [10] (which holds for instance if particles collide like hard spheres). Given any initial data f in satisfying H ( f in |M) < +∞, there exists a renormalized solution f to the Boltzmann equation (1) with initial data f in , satisfying further the entropy inequality Z tZ 1 H ( f |M)(t) + D( f )(s, x)dsdx ≤ H ( f in |M). MaKn 0 Ω Now if we consider some family of suitably scaled renormalized solutions to the Boltzmann equation, we are able to prove that it satisfies the expected asymptotics in the incompressible Euler limit provided that initial data are well prepared, i.e. in the case when the purely kinetic part, the fast oscillating hydrodynamic part and the non-oscillating part of both the density and temperature vanish asymptotically:

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Theorem 1. Let ( f εin ) be a family of nonnegative functions of L 1loc (Ω × R3 ) satisfying the scaling condition 1 H ( f εin |M) ≤ Cin , ε2 and such that 1 H ( f εin |M1,εu in ,1 ) → 0 ε2

(11)

as ε → 0,

(12)

for some given divergence-free smooth vector field u in . Let f ε be a family of renormalized solutions to the scaled Boltzmann equation 1 Q( f ε , f ε ), εq f ε (0, x, v) = f εin (x, v),

ε∂t f ε + v · ∇x f ε =

(13)

where q > 1, meaning in particular that the ratio between the Knudsen number and the Mach number goes to 0. Then the family of fluctuations (gε ) defined by f ε = M(1 + εgε ) converges (entropically) to g = u·v where u is the solution to the incompressible Euler equations ∂t u + u · ∇x u + ∇x p = 0, u(0, x) = u in (x)

∇x · u = 0

on Ω ,

on R+ × Ω ,

(14)

as long as the latter is Lipschitz continuous. Note that (12) is a very strong assumption on the family of initial data, meaning that “well prepared” has to be understood as follows: gεin = u in · v + o(1) in the sense of entropic convergence. We thus require that the initial distribution has a velocity profile close to the local thermodynamic equilibrium |v|2 − 3 2 in order that there is no relaxation layer. We further ask the asymptotic initial thermodynamic fields to satisfy the incompressibility and Boussinesq constraints ρ in + u in · v + θ in

∇ · u in = 0,

∇(ρ in + θ in ) = 0,

which ensures that there is no acoustic wave. We also require that the initial temperature fluctuation (and thus mass fluctuation) is negligible ρ in = θ in = 0. We therefore expect the temperature fluctuation to remain negligible. We finally need some spatial regularity on the limiting bulk velocity, more precisely we require some Lipschitz continuity. We are thus able to consider very general initial data (satisfying only the physical estimate (11)), but in the vicinity of a small set of asymptotic distributions. A natural question is then to know whether or not it is possible to get rid of these restrictions on the asymptotic

distribution. In the sketch of proof we will give in Section 4, we will see that the first two assumptions come from the poor understanding of the Boltzmann equation, in particular from the fact that renormalized solutions to the Boltzmann equation are not known to satisfy the local conservation of energy (the heat flux is not even defined), whereas the last assumption concerning the regularity of the limiting distribution is inherent to the modulated entropy method. Considering solutions to the Boltzmann equation satisfying rigorously the basic physical properties, we expect to control the energy flux and extend the convergence result to take into account acoustic waves. In order to also deal with the relaxation layer, we further need to understand the dissipation mechanism, which will be done by slight modifications of the method. On the contrary, relaxing the regularity assumption requires new ideas. The stability in energy and entropy methods is indeed controlled by the Lipschitz norm of the limiting field. In 3D, the incompressible Euler equations are not even known to have weak solutions, so that we do not expect to extend our convergence result for distributions with lower regularity. In return, in 2D, the mathematical theory of the incompressible Euler equations is much better understood and singular solutions such as vortex patches are known to exist globally in time. It should be then relevant to study the hydrodynamic limit of the Boltzmann equation in this setting. By analogy with the compressible Euler equations, we would expect the spatial discontinuities to dissipate entropy, or in other words to create layers where the distribution is far from local thermodynamic equilibrium. The difficulty should be to split the space–time domain according to these layers. 3.2. In the framework of classical solutions The second result we will state here answers the previous question in the case of smooth limiting fields. Considering a stronger notion of solution for the Boltzmann equation (1), for instance using the classical solutions built by Guo [13], we can prove the convergence to the incompressible Euler equations for general initial data. We indeed recall that nonlinear energy methods allow us to build global smooth solutions to the Boltzmann equation for smooth small data (see [13]): Proposition. Consider the collision cross-section b of hard spheres. Given any initial data f in satisfying

  in

(1 + |v|)1/2 D s f √− M ≤δ (15) x

2

M L (Ω ×R3 ) for s ≥ 4 and δ sufficiently small, there exists a unique classical solution f to the Boltzmann equation (1) with initial data f in (such that the previous norm remains bounded for all time). In particular it satisfies the local conservation laws as well as the local entropy inequality. Note that, for our asymptotic study, the smallness and regularity assumption (15) is not really a restriction since it does not provide any uniform bound on the sequence of fluctuations

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(gεin ). Actually we even do not need so much regularity. We will only require that the solutions of the Boltzmann equation to be considered satisfy the non-uniform nonlinear estimate   Z fε − M 2 1 C dv ≤ 2 a.e. on R+ × Ω M 2 M ε ε (to be compared to (10)). The previous proposition just ensures that such solutions exist. Theorem 2. Let ( f εin ) be a family of nonnegative functions of L 1loc (Ω × R3 ) satisfying the scaling condition (11) 1 H ( f εin |M) ≤ Cin . ε2 Let f ε be some family of solutions to the scaled Boltzmann equation (13) with q > 1, satisfying further   Z fε − M 2 dv ≤ C a.e.on R+ × Ω . M (16) M Then, up to the extraction of a subsequence, the family of fluctuations (gε ) defined
by f ε = M(1+εgε ) converges weakly to u · v + 12 θ |v|2 − 5 , where (u, θ ) is the solution to the incompressible Euler equations ∂t u + u · ∇x u + ∇x p = 0, ∂t θ + u · ∇ x θ = 0

∇x · u = 0

on R+ × Ω ,

on R+ × Ω

(17) 1 u(0, x) = Pu in (x), θ (0, x) = (3θ in − 2ρ in ) on Ω , 5 as long as the latter is Lipschitz continuous. Furthermore the difference gε − g behaves asymptotically in L 1loc (dtdx, L 1 (Mdv)) as     t t gosc , x, v = (ρosc , u osc , θosc ) ,x ε ε   1 · 1, v, (|v|2 − 3) 2 where (ρosc , u osc , θosc ) is the fast oscillating part of the solution of the acoustic system (21) stated in Section 5. Note that the purely kinetic part does not appear in that convergence statement since its contribution to the L 1 norm is negligible. The entropic convergence we will establish is actually stronger. 4. Proof of Theorem 1 Theorem 1 has been established by the author [22], and results from different contributions we will present briefly. The incompressible Euler limit of the Boltzmann equation has been first investigated by Golse in [9]. He proved the entropic convergence of scaled renormalized solutions for wellprepared data assuming further (H1) the local conservation of momentum; (H2) some uniform nonlinear a priori estimate on the fluctuation gε giving both a control for large v, and some equiintegrability with respect to x.

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Assumption (H1) was removed by Lions and Masmoudi in [19]; their argument uses the local momentum conservation with matrix-valued defect measure satisfied by renormalized solutions of the Boltzmann equation. That this defect measure vanishes in the incompressible Euler limit follows from the strong convergence result to be proved. For the sake of simplicity we will not give the details of this argument below. Assumption (H2) was removed by the author first in the framework of the BGK equation [21], then in the case of the original Boltzmann equation [22] using refined dissipation estimates. The argument is based on loop estimates instead of a priori estimates, and the conclusion follows from Gronwall’s inequality. 4.1. The modulated entropy inequality In order to establish the stability inequality leading to the entropic convergence stated in Theorem 1, the starting point is the derivation with respect to time of the modulated entropy. For the sake of simplicity, we will omit here the defect measure occurring both in the global entropy inequality and in the local conservation of momentum. A simple computation based on the entropy inequality and on the local conservations of mass and momentum leads then by integration by parts to Z tZ 1 1 D( f ε )(s, x)dxds H ( f |M )(t) + ε 1,εu,1 ε2 εq+3 0 1 ≤ 2 H ( f εin |M1,εu in ,1 ) ε Z Z Z 1 t A(u) · (εu − v) f ε (s, x, v)dvdxds + ε 0 Z Z Z 1 t Du : (v − εu)⊗2 f ε (s, x, v)dvdxds (18) − 2 ε 0 (note that we consider spatial domains without boundary), where the acceleration operator A(u) is given by A(u) = ∂t u + u · ∇x u. Owing to the assumption on the initial data, the first term on the right-hand side will converge to 0 as ε → 0. The convergence of the second term will be given (up to the√extraction √ of a subsequence) by the weak compactness on 1 ( f − M) (see (10)) coming from the uniform entropy ε ε bound. The difficulty is thus to control the last term, referred to as the flux term. Actually we are not able to obtain directly some convergence. In such an inviscid regime, the entropy dissipation does not control the transport term v · ∇x gε , and thus does not provide any additional regularity on the bulk velocity. This lack of strong compactness is also the reason why weak solutions to the 3D incompressible Euler equations are not known to exist. 4.2. Control of the flux term The method consists then in introducing a suitable decomposition of the momentum flux, and estimating each term

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in that decomposition either by the modulated entropy, or by the entropy dissipation, to get Z t ZZ Z 1 − 2 ∇x u : (v − εu)⊗2 f ε (s, x, v)dvdxds 2ε 0 Z C t kDuk L 2 ∩L ∞ (Ω ) H ( f ε |M1,εu,1 )(s)ds + o(1). (19) ≤ 2 ε 0 The main idea behind this result is that the local thermodynamic equilibrium M fε is expected to give a good approximation of the distribution f ε in the fast relaxation limit, at least if the moments remain bounded. Now, for Maxwellian distributions, the flux term can be computed explicitly in terms of the moments Z (v − εu)⊗2 M fε dv   1 + εθε 2 ⊗2 = (1 + ερε ) ε (u ε − u) + Id 3 and estimated by the modulated entropy 1 1 (1 + ερε )|u ε − u|2 ≤ 2 H ( f ε |M1,εu,1 ) 2 ε (the trace part of the matrix has no contribution since the test velocity field is divergence-free.) The first difficulty to apply this strategy is to obtain a control on the relaxation to local Maxwellians. Indeed, in the case of the Boltzmann equation, the entropy production is not known to measure the distance between f ε and M fε . We cannot give here the details of the argument which is rather technical. Let us just mention that the suitable decomposition looks like some linearized Chapman–Enskog’s expansion:   |v|2 − 3 , gε = Π⊥ gε + ρε + u ε · v + θε 2 Z Z ρε = Mgε dv, u ε = Mgε vdv, Z |v|2 − 3 θε = Mgε dv, 2 where Π⊥ denotes the orthogonal projection parallel to the kernel of the linearized collision operator. (Note that, as gε is not in L 2 , we need to introduce some renormalized fluctuation). The first term is then controlled by the entropy dissipation while the second one can easily be estimated in terms of the modulated entropy. A second difficulty to be addressed is related to cases where moments are far from their asymptotic values (i.e. when they become very large pointwise or when the macroscopic density or temperature vanish). In that case, the flux term is estimated directly by the modulated entropy, using both the Young and Bienaym´e–Chebyshev inequalities. 4.3. Convergence Combining (18) and (19), we then conclude by Gronwall’s lemma: 1 H ( f ε |M1,εu,1 )(t) ε2

Z t  1 in ≤ 2 H ( f ε |M1,εu in ,1 ) exp kDuk L 2 ∩L ∞ ds ε 0 Z tZ Z 1 + A(u) · (εu − v) f ε (s, x, v)dvdx ε 0  Z t kDuk L 2 ∩L ∞ dσ ds × exp s  Z t Z t kDuk L 2 ∩L ∞ dσ ds. o(1) exp + 0

s

If u is Lipschitz continuous, the first term on the right-hand side converges to 0 by the assumption (12) on the initial data. The weak convergence on 1ε ( f ε − M) inherited from the uniform entropy bound (11) ensures that there exists some u¯ such that, up to the extraction of a subsequence, Z 1 (εu − v) f ε dv * (u − u). ¯ ε Taking limits in the local conservation of mass, we then get the incompressibility constraint ∇x · u¯ = 0. As u is the solution to the incompressible Euler equations, we have A(u) = −∇x p. Integrating by parts, we conclude that the second term also converges to 0. We thus get the entropic convergence for all t ≥ 0 1 H ( f ε |M1,εu,1 )(t) → 0. ε2 (For the details we refer to [22]). 5. Proof of Theorem 2 Theorem 2 requires some improvements in the relative entropy method developed in [23]. The main idea is that, in domains where the distribution is expected to present rapid variations, the formal hydrodynamic approximation is not relevant, and that correctors have to be added in order to obtain the convenient asymptotics. The point is indeed to obtain a refined description of the asymptotics taking into account both the relaxation in the initial layer and the acoustic waves. 5.1. Description of acoustic waves Since acoustic waves only contribute to the hydrodynamic part of the distribution, relaxing the constraints on the initial thermodynamic fields does not require strong modifications of the method. Outside from the initial layer, the strategy consists then in modulating the entropy by any fluctuation of Maxwellian, meaning that we assume neither the incompressibility constraint nor the Boussinesq constraint on the test functions. We define the approximate solution f app by   3 3 1 log f app = − log(2π ) + ε ρ − θ − e−εθ |v − εu|2 . 2 2 2 We then expect the modulated entropy inequality to differ from the usual one by some penalization arising in the

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acceleration operator. More precisely, (18) has to be replaced by Z t ZZ 1 1 H ( f ε | f app )(t) + q+3 D( f ε )dsdx ε2 ε 0 Z tZ 1 1 in )+ 2 ∂t exp(ερ)dxds ≤ 2 H ( f εin | f app ε ε 0    Z ZZ 1 |v − εu|2 1 t −εθ f ε 1, e (v − εu), −3 + ε 0 2 eεθ · Aε (ρ, u, θ )dvdxds Z tZZ   1 1 + 2 f ε Dx u : Φε + e 2 εθ Dx θ · Ψε dxdvds (20) ε 0 denoting by Aε (ρ, u, θ ) the (five components) generalized acceleration operator, and by Φε and Ψε the kinetic momentum and energy fluxes — which are scaled translated variants of   1 2 ⊗2 Φ = v − |v| I d , 3  1  2 Ψ = v |v| − 5 . 2 Note that such an inequality is established only for solutions to the Boltzmann equation satisfying the local conservations of mass, momentum and energy. The difficult point is to build some suitable approximate solutions f app , or in other words some family (ρapp , u app , θapp ) of smooth thermodynamic fields satisfying approximately Aε (ρapp , u app , θapp ) = 0, i.e. the acoustic system  1 ∂t ρ + u · ∇ x ρ + ∇ x · u   ε    ∂t u + u · ∇x u + θ∇x ρ − 3 θ + 1 ∇x (ρ + θ )   = 0. (21) 2 ε     2 ∂t θ + u · ∇x θ + ∇x · u 3ε 

Such a construction is done by a filtering method (see [24] or [11] for instance). Let us first rewrite the previous system (21) on (ρapp , u app , def

θapp ) = Vapp in a more abstract way: ∂t V +

1 L V + Q(V, V ) = 0 ε

where Q describes the nonlinear part of the system, and L is the linear penalization defined by   2 L : (ρ, u, θ ) 7→ ∇x · u, ∇x (ρ + θ ), ∇x · u . 3 The first step is to conjugate the system by the semi-group generated by the linear penalization L       tL tL ∂t exp V + exp Q(V, V ) = 0, ε ε or equivalently         tL tL ˜ tL ˜ ˜ ∂t V + exp Q exp − V , exp − V = 0. ε ε ε

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The first-order approximation, i.e. the envelope equation, is then obtained by taking limits in that filtered system: ˜ V˜0 , V˜0 ) = 0 ∂t V˜0 + Q( where Q˜ is defined as some projection of Q on the resonant modes of the linear penalization L. Nevertheless, because of the high frequency oscillations, we do not expect the error in the first-order approximation to converge strongly to 0. We therefore have to add some correctors (i.e. the second- and third-order approximations) in order to establish the convenient convergence statement :

 

Vapp − exp − t L (V˜0 + ε V˜1 + ε 2 V˜2 ) → 0.

2

ε L The conclusion of the proof follows from the same arguments as in the previous case, i.e. from Gronwall’s lemma, except that the control of the energy flux (which is a third moment in v) requires some additional estimate, for instance (16). (For the details we refer to [23]). 5.2. Description of the Knudsen layer In the initial layer, the purely kinetic part of the fluctuation is expected to be of order O(1) and to converge to 0 exponentially in time. In order to take into account the relaxation process in the relative entropy method, one thus has to construct a refined approximation f app , and then to introduce it in the modulated entropy inequality (20). This requires in particular to also modulate the entropy dissipation. The modulated entropy inequality becomes indeed Z t ZZ 1 1 (22) D( f ε | f app )dsdx H ( f | f )(t) + ε app ε2 εq+3 0 1 in ≤ 2 H ( f εin | f app ) ε  Z t ZZ 1 1 − γε ∂t f app − q+1 Q( f app , f app ) ε 0 ε  1 + v · ∇x f app dvdxds ε Z t ZZ ZZ   1 0 0 f app f app1 − f app f app1 + q+1 4ε 0
0 × γε γε1 − γε0 γε1 dvdv1 dωdxds denoting by γε the modulated fluctuation defined by f ε = f app (1 + εγε ) and by D( f ε | f app ) the modulated entropy dissipation. Note that the integrand defining the modulated entropy dissipation is always nonnegative, which is crucial to get some stability. It remains then to build a suitable approximate solution f app . Let us recall that, in the initial layer, the dominating process is expected to be the relaxation, so that the transport can be neglected in first approximation. We thus solve the homogeneous equation ∂t f app =

1 εq+1

Q( f app , f app )

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using a fixed-point argument in some functional space with exponential time decay. Up to some spatial regularization of the initial data and truncation of large velocities, we are then able to prove that the second term on the right-hand side of (22) converge to 0, provided that t = o(ε). The conclusion is again based on some Gronwall’s type 0 argument. The point is to prove that the L 2x (L p ( f app dv)) norm of γε is controlled by the square root of the modulated entropy, and to obtain a uniform bound on Z t 1 0 0 p k f app f app1 − f app f app1 k L ∞ (L v,v χε (t) = q+1 ) ds. x 1 ,ω ε 0 We then obtain, for any τε  ε, 1 1 in H ( f ε | f app )(τε ) ≤ 2 H ( f εin | f app ) exp(χε (τε )) + o(1). 2 ε ε This concludes the proof inside the initial layer. It remains then to put together both estimates (inside and outside the initial layer) using the fact that the local thermodynamic equilibrium is a good approximation in entropic sense, provided that τε  εq+1 . (For the details we refer again to [23]). References [1] K. Aoki, Y. Sone, Steady gas flows past bodies at small Knudsen numbers — Boltzmann and hydrodynamic systems, Trans. Theory Stat. Phys. 16 (1987) 189–199. [2] C. Bardos, F. Golse, C.D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys. 63 (1991) 323–344. [3] C. Bardos, F. Golse, C.D. Levermore, Fluid dynamic limits of the Boltzmann equation II: Convergence proofs, Commun. Pure Appl. Math. 46 (1993) 667–753. [4] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York, 1960. [5] F. Coron, F. Golse, C. Sulem, A classification of well-posed kinetic layer problems, Comm. Pure Appl. Math. 41 (1988) 409–435. [6] A. De Masi, R. Esposito, J.L. Lebowitz, Incompressible Navier–Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42 (1989) 1189–1214.

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