Antoine Echelard

Jacques Lévy Véhel

Application of fractal tools for the classification of microscopical images of milk fat


solutions of the Euler equations

Keywords: 1991 Mathematics Subject Classification. 35Q30, 82B40, (76D05, 76B99) Navier-Stokes equations, Euler equations, Boltzmann equation, Fluid dynamic limit, Inviscid limit, Slip coefficient, Maxwell's accommodation boundary condition, Accomodation parameter, Relative entropy method, Dissipative

THE INCOMPRESSIBLE EULER LIMIT

OF THE BOLTZMANN EQUATION WITH ACCOMMODATION BOUNDARY CONDITION CLAUDE BARDOS, FRANC ¸OIS GOLSE, AND LIONEL PAILLARD Abstract. The convergence of solutions of the Navier-Stokes equations set in a domain with boundary to solutions of the Euler equations in the large Reynolds number limit is a challenging open problem both in 2 and 3 space dimensions. In particular it is distinct from the question of existence in the large of a smooth solution of the initial-boundary value problem for the Euler equations. The present paper proposes three results in that direction. First, if the solutions of the Navier-Stokes equations satisfy a slip boundary condition with vanishing slip coefficient in the large Reynolds number limit, we show by an energy method that they converge to the classical solution of the Euler equations on its time interval of existence. Next we show that the incompressible Navier-Stokes limit of the Boltzmann equation with Maxwell's accommodation condition at the boundary is governed by the Navier-Stokes equations with slip boundary condition, and we express the slip coefficient at the fluid level in terms of the accommodation parameter at the kinetic level. This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat. Phys. 63 (1991), 323-344]. Finally, we establish the incompressible Euler limit of the Boltzmann equation set in a domain with boundary with Maxwell's accommodation condition assuming that the accommodation parameter is small enough in terms of the Knudsen number. Our proof uses the relative entropy method following closely [L. Saint-Raymond, Arch. Ration. Mech. Anal. 166 (2003), in the case of the 3-torus, except for the boundary terms, which require special treatment.

To C. David Levermore

Introduction

In a program initiated more than 20 years ago with Dave Levermore [START_REF] Bardos | Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles[END_REF][START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF][START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF], we outlined a strategy for deriving incompressible fluid dynamic equations from the theory of renormalized solutions of the Boltzmann equation invented by R. DiPerna and P.-L. Lions [START_REF] Diperna | On the Cauchy problem for the Boltzmann equation: global existence and weak stability results[END_REF].

At the time of this writing, complete derivations of the Stokes [START_REF] Lions | From Boltzmann Equation to the Navier-Stokes and Euler Equations I[END_REF][START_REF] Lions | From Boltzmann Equation to the Navier-Stokes and Euler Equations II[END_REF], Stokes-Fourier [START_REF] Golse | The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation[END_REF] and Navier-Stokes-Fourier [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF][START_REF] Golse | The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials[END_REF][START_REF] Levermore | From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system[END_REF] have been obtained following that program, in the greatest possible generality allowed by the current existence theories for both the fluid dynamic and the Boltzmann equations: see [START_REF] Villani | Limites hydrodynamiques de l'équation de Boltzmann[END_REF] for a survey of these issues.

The case of the incompressible Euler equations in space dimension 3 stands out, in the first place because there does not exist a satisfactory theory of global weak solutions of these equations analogous to Leray's theory of weak solutions of the Navier-Stokes equations [START_REF] Leray | Essai sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] in space dimension 3. Even if there was a global existence theory of weak solutions of the incompressible Euler equations in the energy space L ∞ t (L 2 x ) in dimension 3, such solutions would not satisfy the weak-strong uniqueness property observed by Leray in the case of the Navier-Stokes equations. (Indeed there exist nontrivial compactly supported solutions of the incompressible Euler equations in energy space: see [START_REF] Sheffer | An inviscid flow with compact support in space-time[END_REF][START_REF] Shnirelman | On the nonuniqueness of weak solution of the Euler equation[END_REF][START_REF] De Lellis | The Euler equations as a differential inclusion[END_REF].) In [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF], P.-L. Lions proposed a notion of dissipative solution of the incompressible Euler equations -in the same spirit of his definition of the notion of viscosity solutions of Hamilton-Jacobi equations, but using the conservation of energy instead of the maximum principle as in the Hamilton-Jacobi case. The weak-strong uniqueness property is verified by dissipative solutions of the incompressible Euler equations (essentially by definition): if there exists a classical (C 1 ) solution of the incompressible Euler equations, all dissipative solutions with the same initial data must coincide with this classical solution on its maximal time interval of existence. Unfortunately, dissipative solutions are not known to satisfy the incompressible Euler equations in the sense of distributions.

Using the relative entropy method pioneered in [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF] and adapted to the case of the Boltzmann equation in [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF][START_REF] Lions | From Boltzmann Equation to the Navier-Stokes and Euler Equations II[END_REF], L. Saint-Raymond [START_REF] Saint-Raymond | Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles[END_REF][START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF] succeeded in deriving dissipative solutions of the Euler equations in arbitrary space dimension (or classical solutions whenever they exist) from weak solutions of the BGK model [START_REF] Saint-Raymond | Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles[END_REF] or from renormalized solutions of the Boltzmann equation [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF].

However, all the derivations of fluid dynamic equations from the Boltzmann equation referred to above are carried out in either the Euclidian space R N of the flat torus T N so as to avoid difficulties that may result from boundary conditions. The theory of renormalized solutions of the Boltzmann equation in the presence of accommodation boundary conditions was obtained only very recently, by S. Mischler [START_REF] Mischler | Kinetic equations with Maxwell boundary ocnditions[END_REF]; subsequently, N. Masmoudi and L. Saint-Raymond established the Stokes-Fourier limit of such solutions [START_REF] Masmoudi | Saint-Raymond From the Boltzmann equation to the Stokes-Fourier system in a bounded domain[END_REF].

In the present paper, we derive dissipative solutions (or classical solutions whenever they exist) of the incompressible Euler equations from renormalized solutions of the Boltzmann equation in some spatial domain satisfying Maxwell's accommodation boundary condition. In particular, we identify a sufficient scaling condition on the accommodation parameter under which the hydrodynamic limit of the family of solutions of the Boltzmann equation is governed by the incompressible Euler equation with its classical boundary condition -i.e. assuming that the velocity field is tangent at the boundary.

The outline of the paper is as follows. Section 2 gives a sufficient condition on the slip coefficient at the boundary under which the incompressible Euler equations are obtained as the inviscid limit of the incompressible Navier-Stokes equations with slip-boundary condition. The main result in this section is Theorem 2.2, based on an energy method. Section 3 provides a formal derivation of the incompressible Navier-Stokes equations with slip boundary condition from the Boltzmann equation with Maxwell's accommodation condition at the boundary of the spatial domain: see Theorem 3.1 for a precise statement of this result. Based on the intuition provided by sections 2 and 3, we identify a scaling limit of the Boltzmann equation with Maxwell accommodation boundary condition leading to the incompressible Euler equations: see Theorem 4.2, whose proof occupies most of section 4.

It is a our great pleasure to offer this modest contribution to our friend Dave Levermore, in recognition of his outstanding influence on the analysis of nonlinear partial differential equations in the past 30 years, especially on the problem of hydrodynamic limits of the Boltzmann equation, directly inspired from Hilbert's 6th problem on the axiomatization of physics.

Inviscid Limit of the Navier-Stokes Equations with Slip Boundary Conditions

A a warm-up, we begin with a simple observation bearing on the inviscid limit of the incompressible Navier-Stokes equations set in some smooth domain with slip boundary condition. In particular, we identify a sufficient scaling condition on the slip coefficient in order to obtain the incompressible Euler equations in the inviscid limit.

Let Ω designate an open set in R N with C 1 boundary ∂Ω, assuming that N = 2 or 3; henceforth the outward unit normal vector at the point x of ∂Ω is denoted by n x . Consider the initial-boundary value problem with unknown u ν = u ν (t, x), set for x ∈ Ω and t ≥ 0:

(1)

               div x u ν = 0 , ∂ t u ν + div x (u ν ⊗ u ν ) + ∇ x p ν = ν∆ x u ν , u ν • n ∂Ω = 0 , ν(Σ(u ν ) • n) τ + λu ν ∂Ω = 0 , u ν t=0 = u in , where ν > 0 is the kinematic viscosity, λ > 0 the slip coefficient, Σ(u) := ∇ x u + (∇ x u) T , while v(x) τ := (I -n(x) ⊗2 ) • v(x) .
Henceforth, we denote

H(Ω) := {v ∈ L 2 (Ω; R N ) | div v = 0 and v • n ∂Ω = 0} .
For each ν > 0 and u in ∈ H(Ω), there exists a weak solution

u ν of (1) in L ∞ (R + ; H(Ω)) ∩ L 2 (R + ; H 1 (Ω)), meaning that, for each test vector field U ∈ C(R + ; H(Ω)) ∩ C ∞ c (R + × Ω), one has (2) ν ∞ 0 Ω 1 2 Σ(u ν ) : Σ(U)dxdt + λ ∞ 0 ∂Ω u ν • UdS(x)dt = ∞ 0 Ω (u ν • ∂ t U + u ν ⊗ u ν : ∇ x U)dxdt + Ω u in (x) • U(0, x)dx ,
and satisfying in addition u ν ∈ C(R + ; w -L 2 (Ω)), together with the Leray-type energy dissipation inequality:

(3) Ω 1 2 |u ν (t, x)| 2 dx + ν t 0 Ω |Σ(u ν )(t, x)| 2 dxdt + λ t 0 ∂Ω |u ν (t, x)| 2 dS(x)dt ≤ Ω 1 2 |u in (x)| 2 dx
for each t ≥ 0. Such a weak solution of (1) will henceforth be referred to as a "Leray solution of (1)". The classical theory of Leray solutions that is well known in the case where the velocity field satisfies the Dirichlet boundary condition on ∂Ω can be adapted to the case of the slip-boundary condition: see [START_REF] Solonnikov | On a boundary value problem for a stationary system of Navier-Stokes equations[END_REF][START_REF] Beirão | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF], and Theorem 2 in [START_REF] Iftimie | Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions[END_REF].

Since u ν ∈ C(R + ; w -L 2 (Ω)), by an elementary density argument, one can choose a sequence of test vector fields U n of the special form

U n (t, x) = χ n (T -t)w(t, x) where w ∈ C(R + ; H(Ω)) ∩ C 1 c (R + × Ω) and χ n (z) = z -∞ χ n (s)ds
where χ ′ n is a regularizing sequence on R, so that the weak formulation (2) of the Navier-Stokes equations becomes

ν T 0 Ω 1 2 Σ(u ν ) : Σ(w)dxdt + λ T 0 ∂Ω u ν • wdS(x)dt = Ω u in (x) • w(0, x)dx - Ω u ν (T, x)w(T, x)dx + T 0 Ω (u ν • ∂ t w + u ν ⊗ u ν : ∇ x w)dxdt ,
for each T > 0. Furthermore, denoting 5) and (4), we find that (6)

E(w) := ∂ t w + w • ∇ x w , one has T 0 Ω u ν • ∂ t wdxdt = T 0 Ω (u ν • E(w) -u ν ⊗ w • ∇ x w)dxdt while (4) T 0 1 2 |w(t, x)| 2 dx - T 0 1 2 |w(0, x)| 2 dx + T 0 Ω w ⊗ w : ∇ x wdxdt = T 0 Ω w • E(w)dxdt . Therefore (5) ν T 0 Ω 1 2 Σ(u ν ) : Σ(w)dxdt + λ T 0 ∂Ω u ν • wdS(x)dt = T 0 Ω (u ν • E(w) + u ν ⊗ (u ν -w) : ∇ x w -w ⊗ u ν : ∇ x w)dxdt + Ω u in (x) • U(0, x)dx - Ω u ν (T, x)w(T, x)dx , since Ω w ⊗ u ν : ∇ x wdx = Ω div x (u ν 1 2 |w| 2 )dx = 0 , because u ν • n ∂Ω = 0. Combining (
ν T 0 Ω 1 2 Σ(u ν ) : Σ(w)dxdt + λ T 0 ∂Ω u ν • wdS(x)dt = T 0 Ω ((u ν -w) • E(w) + (u ν -w) ⊗ (u ν -w) : ∇ x w)dxdt + Ω u in (x) • w(0, x)dx - Ω 1 2 |w(0, x)| 2 dx - Ω u ν (T, x)w(T, x)dx + Ω 1 2 |w(T, x)| 2 dx
Finally, combining (3) and ( 6), we conclude that any Leray solution u ν of (1) satisfies the inequality ( 7)

Ω 1 2 |u ν -w| 2 (t, x)dx + t 0 Ω (u ν -w) ⊗ (u ν -w) : ∇ x wdxds +ν t 0 Ω 1 2 |Σ(u ν )(s, x)| 2 dxds + λ t 0 ∂Ω |u ν (s, x)| 2 dS(x)ds ≤ Ω 1 2 |u in (x) -w(0, x)| 2 dx + t 0 Ω E(w) • (u ν -w)dxdt +ν t 0 Ω 1 2 Σ(u ν ) : Σ(w)dxds + λ t 0 ∂Ω u ν • wdS(x)ds for each w ∈ C(R + ; H(Ω)) ∩ C 1 c (R + × Ω)
. At this point we recall the definition of dissipative solutions of the incompressible Euler equations set in a domain Ω with smooth boundary:

(8)            div x u = 0 , ∂ t u + div x (u ⊗ u) + ∇ x p = 0 , u • n ∂Ω = 0 , u t=0 = u in .
Definition 2.1 (P.-L. Lions [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF], p. 154, C. Bardos, E. Titi [START_REF] Bardos | Euler equations for incompressible ideal fluids[END_REF], p. 16). Given u in ∈ H(Ω), a dissipative solution of ( 8) is an element u ∈ C(R + ; w -H(Ω)) satisfying u t=0 = u in and the inequality

Ω 1 2 |u -w| 2 (t, x)dx ≤ exp t 0 2 σ(w) - L ∞ (Ω) (s)ds Ω 1 2 |u in (x) -w(0, x)| 2 dx + t 0 exp t s 2 σ(w) - L ∞ (Ω) (τ )dτ Ω E(w) • (u -w)(s, x)dxds for each w ∈ C(R + ; H(Ω)) ∩ C 1 (R + × Ω), where (9) σ(w) -(t, x) := sup |ξ|=1 (-Σ(w)(t, x) : ξ ⊗ ξ) .
We recall that, if the Euler equations ( 8) have a classical solution

v ∈ C 1 ([0, T * ) × Ω) satisfying σ(v) -∈ L 1 ([0, T ]; L ∞ (Ω)) and p ∈ L 1 ([0, T ]; H 1 (Ω)) for each T < T * ,
then all dissipative solutions of (8) must coincide with v on [0, T * ) × Ω a.e., since one can use w = v as the test vector field, so that

Ω E(v) • (u -v)(s, x)dx = - Ω ∇ x p • (u -v)(s, x)dx = 0 because (u -v)(s, •) ∈ H(Ω) for each s ∈ [0, T ).
Theorem 2.2. Let u in ∈ H(Ω), and assume that the slip coefficient λ ≡ λ(ν) in [START_REF] Aoki | Slightly rarefied gas flow over a body with small accommodation coefficient[END_REF] scales with the kinematic viscosity ν so that [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF] λ(ν) → 0 as ν → 0 .

Then any family (u ν ) of Leray solutions of ( 1) is relatively compact in the weak-* topology of L ∞ (R + ; H(Ω)) and in C(R + ; w -H(Ω)) for the topology of uniform convergence on bounded time intervals, and each limit point of (u ν ) as ν → 0 is a dissipative solution of [START_REF] Beirão Da Veiga | Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory[END_REF].

Proof. We deduce from [START_REF] Beirão | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF] with w = 0, or equivalently from the Leray energy inequality that

(11) √ νΣ(u ν ) is bounded in L 2 (R + ; L 2 (Ω)) , and
λ(ν)u ν ∂Ω is bounded in L 2 (R + ; L 2 (∂Ω)) .
By Gronwall's inequality ( 12)

Ω 1 2 |u ν -w| 2 (t, x)dx ≤ exp t 0 2 σ(w) - L ∞ (Ω) (s)ds Ω 1 2 |u in (x) -w(0, x)| 2 dx + t 0 exp t s 2 σ(w) - L ∞ (Ω) (τ )dτ Ω E(w) • (u ν -w)(s, x)dxds + t 0 exp t s 2 σ(w) - L ∞ (Ω) (τ )dτ Q ν (s)ds
where, by the Cauchy-Schwarz inequality

Q ν (s) = ν Σ(u ν ) L 2 (Ω) (s) Σ(w) L 2 (Ω) (s) + λ(ν) u ν L 2 (∂Ω) (s) w L 2 (Ω) (s) .
In view of [START_REF] De Lellis | The Euler equations as a differential inclusion[END_REF], one has

Q ν L 1 ([0,T ]) = O( √ ν) + O( λ(ν)) → 0
as ν → 0, and we conclude by passing to the limit in [START_REF] Diperna | On the Cauchy problem for the Boltzmann equation: global existence and weak stability results[END_REF] following the same argument as in [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF].

Several remarks are in order after this result.

In some references, the slip boundary condition is written

(13) ν ∂u ν ∂n τ + λu ∂Ω = 0 , instead of (14) ν(Σ(u ν ) • n) τ + λu ∂Ω = 0 .
Likewise, the boundary condition

curl u ν × n ∂Ω = 0
is also considered by some authors -and referred to as the Navier slip condition -in the context of the inviscid limit of the Navier-Stokes equations: see for instance [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF][START_REF] Bardos | Existence et unicité de la solution de l'équation d'Euler en dimension deux[END_REF][START_REF] Beirão | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF][START_REF] Beirão Da Veiga | A missed persistence property for the Euler equations, and its effect on inviscid limits[END_REF]. If ∂Ω is a straight line, or a plane, or a hyperplane in space dimension N > 3, the normal vector field n is constant, so that

(Σ(u ν ) • n) τ ∂Ω = ∂u ν ∂n τ ∂Ω + ∇ τ (u • n ∂Ω ) = ∂u ν ∂n τ ∂Ω and curl u ν × n ∂Ω = (∇ x u ν -(∇ x u) T ) • n ∂Ω = ∂u ν ∂n τ ∂Ω -∇ t (u • n ∂Ω ) = ∂u ν ∂n τ ∂Ω
(with ∇ τ denoting the tangential component of the ∇ operator), since the velocity field u ν is tangential on ∂Ω. Therefore, in the case of a flat boundary, all these boundary conditions are equivalent.

If ∂Ω is a smooth curve, or a surface or a hypersurface in space dimension N > 3, then

(Σ(u ν ) • n) τ ∂Ω = ∂u ν ∂n τ -∇ t n • u τ ∂Ω while curl u ν × n ∂Ω = ∂u ν ∂n τ + ∇ τ n • u τ ∂Ω ,
so that all these boundary conditions differ by a 0-order operator given by the Weingarten endomorphism of the boundary ∂Ω.

Here, we have chosen the second boundary condition above, as it is the more natural one when looking at the Navier-Stokes equation as a fluid dynamic limit of the kinetic theory of gases.

However, the same argument as in the proof of Theorem 2.2 can be extended to treat the case of a slip coefficient λ which is not nonnegative, provided that

λ(ν) + = max(λ(ν), 0) → 0 and λ -(ν) = max(-λ(ν), 0) = O(ν)
as ν → 0. Indeed, the contribution of λ(ν) -in the estimate (7) can be absorbed in the viscous dissipation term by means of the following classical inequality: for each α > 0, there exists

C α > 0 such that, for each v ∈ H 1 (Ω), ∂Ω |v(x)| 2 dS(x) ≤ α Ω |∇ x v(x)| 2 dx + C α α Ω |v(x)| 2 dx .
With this observation, the term

(Σ(u ν ) • n) τ ∂Ω can be replaced indifferently with either ∂u ν ∂n τ ∂Ω or curl u ν × n ∂Ω
in the slip boundary condition. More precise variants of Theorem 2.2 have been established by various authors, see for instance [START_REF] Xiao | On the vanishing viscosity limit for the 3-D Navier-Stokes equations with a slip boundary condition[END_REF][START_REF] Beirão Da Veiga | Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory[END_REF]. The result given here holds for a very general class of nonnegative slip boundary coefficients λ and is based upon the simplest imaginable energy estimate. The condition λ ≥ 0 in ( 1) is somehow natural when this initial-boundary value problem is considered as some scaling limit of the Boltzmann equation of the kinetic theory of gases.

Another question is whether the condition λ(ν) → 0 as ν → 0 is optimal. Considers instead the Dirichlet boundary condition for u ν , i.e.

u ν ∂Ω = 0 .

Formally, this boundary condition corresponds with any one of the slip boundary conditions above with lim ν→0 λ(ν) > 0 .

In that case, it well known that the Euler equations ( 8) may fail to describe the inviscid limit of the Navier-Stokes equations, even in the simpler 2 dimensional case. Because the Dirichlet boundary condition overdetermines the velocity field in the inviscid limit, the Euler equations ( 8) are expected to govern the inviscid limit of the Navier-Stokes equations only if the effect of viscosity remains confined on a thin layer near the boundary. But it may happen -and does happen under certain circumstances -that the viscous layer detaches from the boundary, as for instance in the case of the so-called von Karman vortex streets in the case of a Navier-Stokes flow past a cylinder, even at moderate Reynolds numbers. While this situation seems beyond the grasp of current mathematical analysis, there exists a least a very interesting criterion due to T. Kato [START_REF] Kato | Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary[END_REF], formulated in terms of the viscous energy dissipation only, identifying situations where the inviscid limit of the incompressible Navier-Stokes equations with Dirichlet boundary condition is described by the Euler equations. This suggests that, unless λ(ν) → 0, the Euler equations ( 8) might also fail to govern the inviscid limit of the Navier-Stokes equations with slip boundary conditions (1).

From the Boltzmann Equation with Accomodation Boundary Condition to the Navier-Stokes Equations with Slip Boundary Conditions

In this section, we revisit the incompressible Navier-Stokes limit for the Boltzmann equation in the case of the initial-boundary value problem. Our main interest is to understand how the slip boundary condition arises from Maxwell's accommodation boundary condition at the kinetic level in the fluid dynamic limit, and especially how the slip coefficient is related to the accommodation parameter. Strictly speaking, this is not needed in the proof of the main result in the present paper. Therefore, the discussion in this section will be only formal, along the line of [START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF].

Consider the Boltzmann equation with the incompressible Navier-Stokes scaling [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] ǫ∂

t F ǫ + v • ∇ x F ǫ = 1 ǫ B(F ǫ , F ǫ ) .
Here the unknown is the distribution function F ≡ F (t, x, v) that is the density at time t of molecules with velocity v ∈ R N at the position x ∈ Ω with respect to the phase space Lebesgue measure dxdv.

3.1.

Formal structure of the Boltzmann equation. The Boltzmann collision integral acts only on the v variable in F ǫ , keeping t, x as parameters. Its expression for φ ∈ C c (R N ) is ( 16)

B(φ, φ)(v) = R N ×S N-1 (φ(v ′ )φ(v ′ * ) -φ(v)φ(v * ))b(v -v * , ω)dv * dω
where v ′ , v ′ * ∈ R N are the velocities of 2 identical particles about to undergo an elastic collision, assuming that their post-collision velocities are v, v * ∈ R N . The set of all possible pre-collision velocities v ′ , v ′ * are parametrized by the unit vector ω as follows:

(17) v ′ ≡ v ′ (v, v * , ω) := v -(v -v * ) • ωω , v ′ * ≡ v ′ * (v, v * , ω) := v * +(v -v * ) • ωω .
The collision kernel b(z, ω) > 0 is a locally integrable function that satisfies the symmetries

(18) b(v -v * , ω) = b(v * -v, ω) = b(v ′ -v ′ * , ω)
a.e. in (v, v * , ω), assuming that v ′ and v ′ * are given in terms of v, v * , ω by the relations [START_REF] Grad | Asymptotic theory of the Boltzmann equation[END_REF]. Depending on the growth of the collision kernel b as |v -v * | → +∞, the collision integral can be extended by continuity to larger classes of functions than C c (R N ). Finally, we denote

B(F, F )(t, x, v) := B(F (t, x, •), F (t, x, •))(v) .
The collision integral satisfies the identities

(19)                R N B(φ, φ)(v)dv = 0 , R N B(φ, φ)(v)vdv = 0 , R N B(φ, φ)(v)|v| 2 dv = 0 , for each φ ∈ C c (R N )
               ǫ∂ t R N F ǫ dv + div x R N vF ǫ dv = 0 , ǫ∂ t R N vF ǫ dv + div x R N v ⊗2 F ǫ dv = 0 , ǫ∂ t R N 1 2 |v| 2 F ǫ dv + div x R N v 1 2 |v| 2 F ǫ dv = 0 , (20)
R N B(φ, φ)(v) ln φ(v)dv ≤ 0 , and (22) R N B(φ, φ)(v) ln φ(v)dv = 0 ⇔ B(φ, φ) = 0 ⇔ φ is a Maxwellian distribution,
meaning that there exists ρ, θ > 0 and u ∈ R N such that

(23) φ(v) = M ρ,u,θ (v) := ρ (2πθ) N/2 e -|v-u| 2 2θ
for all v ∈ R N . As a result, whenever F is a classical solution of the scaled Boltzmann equation [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] with appropriate decay as |v| → ∞, it satisfies the differential entropy inequality [START_REF] Lions | From Boltzmann Equation to the Navier-Stokes and Euler Equations I[END_REF]

ǫ∂ t R N F ǫ ln F ǫ dv + div x R N vF ǫ ln F ǫ dv = - 1 ǫ R N B(F ǫ , F ǫ ) ln F ǫ dv ≤ 0 .
Throughout this paper, we denote ( 25)

M := M 1,0,1 .
Since Maxwellians are equilibrium distributions for the collision integral, it is natural to investigate the linearization thereof about a Maxwellian, say M for simplicity -the case of an arbitrary Maxwellian being similar. We therefore introduce the linearized collision operator in the form

L M φ := -2M -1 δB(F, F ) δF F =M • Mφ , i.e. L M φ(v) := R N ×S N-1 (φ(v)+φ(v * )-φ(v ′ )-φ(v ′ * ))b(v-v * , ω)Mdv * dω .
Under certain assumptions on the collision kernel b, known as Grad's angular cutoff assumption, H. Grad proved in [START_REF] Grad | Asymptotic theory of the Boltzmann equation[END_REF] that L M is an unbounded, self-adjoint Fredholm operator on L 2 (R N ; Mdv) with domain

D(L M ) := {φ ∈ L 2 (R N ; Mdv) | φ(b ⋆ v M) ∈ L 2 (R N ; Mdv)} , where b(z) := S N-1 b(z, ω)dω
and ⋆ v designates the convolution product in the v variable. Moreover, the nullspace of

L M is Ker L M = span{1, v 1 , . . . , v n , |v| 2 } .
In particular, the tensor field A(v) = v ⊗2 -1 N |v| 2 satisfies A⊥ Ker L M , so that, by the Fredholm alternative, there exists a unique tensor field

 ∈ D(L M ) ∩ (Ker L) ⊥ such that L M  = A componentwise.
Henceforth in this paper, we assume that the collision kernel b comes from a hard cutoff potential in the sense of Grad, and more precisely that it satisfies, for some constant

C b > 0 and all (z, ω) ∈ R N × S N -1 , (26) 0 < b(z, ω) ≤ C b (1 + |z|) , and b(z) ≥ 1 C b .
3.2. Boundary value problem and fluid dynamic limit. The incompressible Navier-Stokes limit of the Boltzmann equation bears on solutions of the Boltzmann equation that are of the form ( 27)

F ǫ = M(1 + ǫg ǫ ) ,
where it is understood that the relative number density fluctuation g ǫ is O(1) in some sense to be made precise as ǫ → 0: see [START_REF] Bardos | Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles[END_REF][START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF][START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF] for more details, together with physical justifications for this scaling assumption.

Here, the scaled Boltzmann equation ( 15) is set on the spatial domain Ω, with Maxwell's accommodation at the boundary, that is assumed to be maintained at the constant temperature 1. This boundary condition reads

(28) F ǫ (t, x, v) = (1 -α)R x F ǫ (t, x, v) + αΛ x F ǫ M (t, x)M(v) , x ∈ ∂Ω , v • n x < 0 , where (29) R x F (t, x, v) := F (t, x, v -2v • n x n x )
is the specular reflection operator on the boundary, while

Λ x φ := √ 2π R N φ(v)(v • n x ) + M(v)dv . (30)
In [START_REF] Saint-Raymond | Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles[END_REF], the parameter α satisfies 0 ≤ α ≤ 1, and is called the accommodation coefficient. The case α = 0 corresponds with specular reflection of the gas molecules on ∂Ω without thermal exchange, while the case α = 1 corresponds with diffuse reflection, or total accommodation, in which case gas molecules are instantaneously thermalized at the boundary. Henceforth, we denote, for each φ ∈ L 1 (R N ; Mdv),

φ := R N ψ(v)M(v)dv .
Theorem 3.1. Let (F ǫ ) ǫ>0 be a family of solutions of the scaled Boltzmann equation [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] set on the spatial domain Ω, satisfying the accommodation boundary condition [START_REF] Saint-Raymond | Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles[END_REF] on ∂Ω. Assume that the relative fluctuations

g ǫ = F ǫ -M ǫM → g
a.e. and in weak-L 1 loc (R + ×Ω×R N ; dtdxMdv) (possibly up to extraction of a subsequence), and that

|v| 3 1 |v|>R |g ǫ | + | Â|1 |v|>R |Q(g ǫ , g ǫ )| + | Â||v|1 |v|>R |g ǫ | → 0 in L 1 loc (R + × Ω) as R → +∞ uniformly in ǫ > 0. Then g(t, x, v) = ρ(t, x) + u(t, x) • v + θ(t, x) 1 2 (|v| 2 -N)
, where u is a solution of the incompressible Navier-Stokes equations

div x u = 0 , ∂ t u + div x (u ⊗2 ) + ∇ x p = ν∆ x u ,
and where ν =

1 (N -1)(N +2) Â : A . Assume further that v • n x g ǫ ∂Ω → v • n x g ∂Ω v τ (v • n x ) + g ǫ ∂Ω → v τ (v • n x ) + g ∂Ω in weak-L 1 loc (R + ×∂Ω).
Then, the velocity field u satisfies the boundary condition

u • n x = 0 , x ∈ ∂Ω , ν(Σ(u) • n x ) τ + λu = 0 , x ∈ ∂Ω ,
where the slip coefficient is given by the formula

λ = α 0 N -1 |v τ | 2 (v • n x ) + = α 0 √ 2π .
Proof. Observe that, under the substitution v → w = v -2v • nn, one has, for each unit vector n,

R N φ(v -2v • nn)(v • n) -Mdv = R N φ(w)(w • n) + Mdw ,
so that, for each x ∈ ∂Ω, one has

R N F ǫ (t, x, v)v • n x dv = -α R N F ǫ (t, x, v)(v • n x ) + dv + αΛ x F ǫ M (t, x) R N M(v)(v • n x ) -dv = 0 , since R N M(v)(v • n x ) + dv = R N M(v)(v • n x ) -dv = 1 √ 2π .
Hence, for each x ∈ ∂Ω and each ǫ > 0, one has

vg ǫ (t, x) • n x = 1 ǫ R N F ǫ (t, x, v)v • n x dv = 0 ,
so that, after passing to the limit as ǫ → 0,

u(t, x) • n x = vg (t, x) • n x = 0 , t > 0 , x ∈ ∂Ω .
Next write the local conservation of momentum -the second local conservation law in [START_REF] Levermore | From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system[END_REF] in the form

∂ t vg ǫ + div x 1 ǫ Ag ǫ + ∇ x 1 ǫ 1 N |v| 2 g ǫ = 0 , where A ≡ A(v) := v ⊗2 -1 N |v| 2 . Let now w ≡ w(x) ∈ R N designate a compactly supported C 1 vector field on R N satisfying div w = 0 , and w(x) • n x = 0 , x ∈ ∂Ω .
Taking the inner product of both sides of the local conservation of momentum with w and integrating over Ω leads to (31)

∂ t Ω w • vg ǫ dx + ∂Ω w ⊗ n x : 1 ǫ Ag ǫ dS(x) - Ω ∇w : 1 ǫ Ag ǫ dx = 0 , since Ω w • ∇ x 1 ǫ 1 N |v| 2 g ǫ dx = Ω div x 1 ǫ 1 N |v| 2 g ǫ • w dx = ∂Ω 1 ǫ 1 N |v| 2 g ǫ w • n x dS(x) = 0 .
Next we pass to the limit in each term appearing in [START_REF] Saint-Raymond | Hydrodynamic Limits of the Boltzmann Equation[END_REF]: following the analysis in [START_REF] Bardos | Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles[END_REF][START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF], one finds that

Ω w • vg ǫ dx → Ω w • vg dx = Ω w • udx while Ω ∇w : 1 ǫ Ag ǫ dx → Ω ∇w : (A(u) -νΣ(u))dx = Ω ∇w : (u ⊗2 -νΣ(u))dx .
(Indeed, since w is divergence-free, ∇w :

1 N |u| 2 = 1 N |u| 2 div x u = 0.) It remains to analyze the boundary term ∂Ω w ⊗ n x : 1 ǫ Ag ǫ dS(x) . Since w • n x = 0 on ∂Ω, w ⊗ n x : Ag ǫ = w ⊗ n x : v ⊗2 g ǫ = v τ v • n x g ǫ • w .
At this point, we decompose the boundary term into the contribution of gas molecules about to collide and those having just collided with the boundary

v τ v • n x g ǫ = v τ v • n x 1 v•nx>0 g ǫ + v τ v • n x 1 v•nx<0 g ǫ
and use the accommodation condition to write

v τ v • n x 1 v•nx<0 g ǫ = v τ v • n x 1 v•nx<0 ((1 -α)R x g ǫ + αΛ x (g ǫ ))
.

Observing that

v τ v • n x 1 v•nx<0 R x g ǫ = R x (v τ v • n x 1 v•nx>0 )g ǫ = -v τ v • n x 1 v•nx>0 ((1 -α)g ǫ + αΛ x (g ǫ )) ,
we conclude that

v τ v • n x 1 v•nx<0 g ǫ = -v τ v • n x 1 v•nx>0 ((1 -α)g ǫ + αΛ(g ǫ ))
so that

v τ v • n x g ǫ = α v τ v • n x 1 v•nx>0 (g ǫ -Λ x (g ǫ )) = α v τ v • n x 1 v•nx>0 g ǫ
-where the second equality follows from the fact that the function

v → v τ v • n x 1 v•nx>0 (g ǫ -Λ x (g ǫ ) is odd in v τ .
Therefore the boundary term appearing in [START_REF] Saint-Raymond | Hydrodynamic Limits of the Boltzmann Equation[END_REF] becomes

∂Ω w ⊗ n x : 1 ǫ Ag ǫ dS(x) = α ǫ ∂Ω w • v τ v • n x 1 v•nx>0 g ǫ dS(x) .
Assume that α ≡ α(ǫ) varies with ǫ so that α(ǫ)/ǫ → α 0 as ǫ → 0. Since

g ǫ → g = ρ + u • v + θ 1 2 (|v| 2 -N)
and we already know that

u • n x = 0 on ∂Ω , one has w • v τ v • n x 1 v•nx>0 g ǫ → v ⊗2 τ (v • n x ) + : u t ⊗ w = 1 N -1 |v τ | 2 (v • n x ) + u • w .
Thus, passing to the limit in (31) leads to (32)

∂ t Ω w • udx + α 0 N -1 |v τ | 2 (v • n x ) + ∂Ω u • wdS(x) - Ω ∇w : (u ⊗2 -νΣ(u))dx = 0 .
(Notice that the term |v τ | 2 (v • n x ) + is independent of x and therefore comes out of the boundary integral.)

Whenever u(t, •) ∈ C 2 (Ω), applying Green's formula transforms the last integral above into

Ω ∇w : (u ⊗2 -νΣ(u))dx = - Ω w • (div x (u ⊗2 ) -ν div x (Σ(u)))dx + ∂Ω w • uu • n x dS(x) -ν ∂Ω w • (Σ(u) • n)dS(x) .
Since u • n x = 0 on ∂Ω, the second integral on the right-hand side above vanishes, and since div x u = 0, one has div x (Σ(u)) = ∆ x u, so that

Ω ∇w : (u ⊗2 -νΣ(u))dx = - Ω w • (div x (u ⊗2 ) -ν∆ x u)dx -ν ∂Ω w • (Σ(u) • n)dS(x) . Thus, if u ∈ C 2 ([0, T ] × Ω), the equality (32) becomes ∂ t Ω w • udx + Ω w • (div x (u ⊗2 ) -ν∆ x u)dx + α 0 N -1 |v τ | 2 (v • n x ) + ∂Ω u•wdS(x)+ν ∂Ω w•(Σ(u)•n)dS(x) = 0 .
This identity holds, say, for each w ∈ C ∞ c (Ω; R N ). In particular, it holds for each w ∈ C ∞ c (Ω; R N ), which implies that

∂ t u + div x (u ⊗2 ) -ν∆ x u = -∇ x p
in the sense of distributions (for some p ∈ D ′ (R * + × Ω)). Since the velocity field u ∈ C 2 ([0, T ] × Ω) we conclude that p ∈ C 1 ([0, T ] × Ω). Substituting this in the identity above with w ∈ C ∞ c (Ω; R N ) gives

- Ω w • ∇ x pdx + ∂Ω (νΣ(u) • n x + α 0 N -1 |v τ | 2 (v • n x ) + u) • wdS(x) = 0
and since, by Green's formula,

- Ω w • ∇ x pdx = - Ω div x (pw)dx = ∂Ω pw • n x dS(x) = 0 , we conclude that ν(Σ(u) • n x ) τ + α 0 N -1 |v τ | 2 (v • n x ) + u = 0 on ∂Ω .
In other words, [START_REF] Sheffer | An inviscid flow with compact support in space-time[END_REF] is the weak formulation of

     ∂ t u + div x (u ⊗2 ) -ν∆ x u = -∇ x p , x ∈ Ω , t > 0 , ν(Σ(u) • n x ) τ + λu = 0 , x ∈ ∂Ω , t > 0 , u • n x = 0 , x ∈ ∂Ω , t > 0 . with λ = α 0 N -1 |v τ | 2 (v • n x ) + = α 0 √ 2π .
The argument above is a proof of the (formal) Navier-Stokes limit Theorem 3.1 by a moment method analogous to the one used in [START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF]. As far as we know, the first derivation of this slip boundary condition, in the steady, linearized regime (i.e. leading to the Stokes equations in the fluid limit), is due to K. Aoki, T. Inamuro and Y. Onishi [START_REF] Aoki | Slightly rarefied gas flow over a body with small accommodation coefficient[END_REF] (see especially formula [START_REF] Shnirelman | On the nonuniqueness of weak solution of the Euler equation[END_REF] in that reference). That derivation uses a Hilbert expansion method (formal series expansion of the solution of the Boltzmann equation in powers of the Knudsen number ǫ). The interested reader is referred to the recent book by Y. Sone [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] (in particular to §3.7 there) for a systematic study of boundary conditions in the context of the fluid dynamic limit of the Boltzmann equation.

For a complete proof of the derivation of the same slip boundary condition as in Theorem 3.1 in the linearized regime -i.e. in a situation where the limiting equation is the Stokes, instead of the Navier-Stokes equations -the reader is referred to the work of N. Masmoudi and L. Saint-Raymond [START_REF] Masmoudi | Saint-Raymond From the Boltzmann equation to the Stokes-Fourier system in a bounded domain[END_REF].

From the Boltzmann Equation with Accomodation Boundary Condition to the Incompressible Euler Equations

In this section, we consider the Boltzmann equation in the scaling leading to the incompressible Euler equations in the fluid dynamic limit. We recall from [START_REF] Bardos | Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles[END_REF][START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF] that this scaling is [START_REF] Shnirelman | On the nonuniqueness of weak solution of the Euler equation[END_REF]

ǫ∂ t F ǫ + v • ∇ x F ǫ = 1 ǫ 1+q B(F ǫ , F ǫ ) , (x, v) ∈ Ω × R N ,
with q > 0, while the distribution function F ǫ is sought in the same form [START_REF] Mischler | Kinetic equations with Maxwell boundary ocnditions[END_REF] as in the Navier-Stokes limit. This scaled Boltzmann equation is supplemented with Maxwell's accommodation condition on ∂Ω, with accommodation coefficient α ≡ α(ǫ) driven by the small parameter ǫ:

(34) F ǫ (t, x, v) = (1 -α(ǫ))R x F ǫ (t, x, v) + α(ǫ)Λ F ǫ M (t, x)M(v) , x ∈ ∂Ω , v • n x < 0 ,
and with the initial condition

(35) F ǫ (0, x, v) = F in ǫ (x, v) , (x, v) ∈ Ω × R N .
The formal result presented in Theorem 3.1 suggests that, in the limit as ǫ → 0, the velocity field

lim ǫ→0 1 ǫ R N vF ǫ dv
should behave like the solution of the incompressible Navier-Stokes equations with kinematic viscosity of order ǫ q and with slip boundary condition with slip coefficient of the order of α(ǫ)/ǫ. Thus, if α(ǫ) = o(ǫ), Theorem 2.2 suggests that this velocity field should satisfy the incompressible Euler equations [START_REF] Beirão Da Veiga | Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory[END_REF]. In fact, the formal result in Theorem 3.1 is only a guide for our intuition, and we shall give a direct proof of the Euler limit starting from the Boltzmann equation with accommodation boundary condition without using the Navier-Stokes limit.

4.1. Renormalized solutions and a priori estimates. Global solutions of the Cauchy problem for the Boltzmann equation for initial data of arbitrary size have been constructed by R. DiPerna and P.-L.

Lions [START_REF] Diperna | On the Cauchy problem for the Boltzmann equation: global existence and weak stability results[END_REF]. Their theory of renormalized solutions was extended to the initial boundary value problem by S. Mischler [START_REF] Mischler | Kinetic equations with Maxwell boundary ocnditions[END_REF]. His result is summarized below -see also section 2 in [START_REF] Masmoudi | Saint-Raymond From the Boltzmann equation to the Stokes-Fourier system in a bounded domain[END_REF] and section 2.3.2 of [START_REF] Saint-Raymond | Hydrodynamic Limits of the Boltzmann Equation[END_REF].

Theorem 4.1 (Mischler). Let F in ǫ ≡ F in ǫ (x, v) ≥ 0 a.e. on Ω × R N be a measurable function satisfying

Ω×R N (1 + |v| 2 + | ln F in ǫ (x, v)|)F in ǫ (x, v)dxdv < +∞ .
There exists

F ǫ ∈ C(R + ; L 1 (Ω × R N ))
satisfying the initial condition [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF], and the Boltzmann equation [START_REF] Shnirelman | On the nonuniqueness of weak solution of the Euler equation[END_REF] together with the boundary condition [START_REF] Solonnikov | On a boundary value problem for a stationary system of Navier-Stokes equations[END_REF] in the renormalized sense, meaning that, for each

Γ ∈ C 1 (R + ) such that Z → √ 1 + ZΓ ′ (Z) is bounded on R + , the func- tion Γ ′ F ǫ M B(F ǫ , F ǫ ) ∈ L 1 loc (R + × Ω × R N ) and ∞ 0 Ω×R N Γ F ǫ M (ǫ∂ t + v • ∇ x )φMdvdxdt + 1 ǫ 1+q ∞ 0 Ω×R N Γ ′ F ǫ M B(F ǫ , F ǫ )φdvdxdt = ∞ 0 ∂Ω×R N Γ F ǫ M φv • n x MdvdS(x)dt -ǫ Ω×R N Γ F in ǫ M φ t=0 Mdvdx for each φ ∈ C 1 c (R + × Ω × R N ).

Moreover a) the trace of F ǫ on ∂Ω satisfies the accommodation boundary condition

F ǫ ∂Ω (t, x, v) = (1 -α)R x (F ǫ ∂Ω )(t, x, v) + Λ x F ǫ ∂Ω M (t, x)M(v) for a.e. (t, x, v) ∈ R + × ∂Ω × R N such that v • n x > 0; b) the distribution function F ǫ satisfies the local conservation law of mass ǫ∂ t R N F ǫ dv + div x R N vF ǫ dv = 0 with boundary condition R N F ǫ (t, x, v)v • n x dv = 0 , x ∈ ∂Ω , t > 0 ;
c) the distribution function F ǫ satisfies the relative entropy inequality

H(F ǫ |M)(t) -H(F in ǫ |M) ≤ - 1 ǫ 2+q t 0 Ω P ǫ (s, x)dxds - α ǫ t 0 ∂Ω DG ǫ (s, x)dxds
for each t > 0, where the following notations have been used: for each f, g measurable on Ω×R N such that f ≥ 0 and g > 0 a.e. , the relative entropy is

H(f |g) := Ω×R N h f g -1 gdvdx
with h(z) := (1 + z) ln(1 + z) -z , while the entropy production rate per unit volume is

P ǫ := R N ×R N ×S N-1 r F ′ ǫ F ′ ǫ * F ǫ F ǫ * -1 F ǫ F ǫ * b(v -v * , ω)dvdv * dω
with r(z) := z ln(1 + z) ≥ 0 , and the Darrozes-Guiraud information is

DG ǫ := 1 √ 2π Λ x h F ǫ M -1 -h Λ x F ǫ M -1 .
In particular H(f |g) ≥ 0 since h ≥ 0 on [-1, +∞) and P ǫ ≥ 0 a.e. on R + × Ω since r ≥ 0 on (-1, +∞), while DG ǫ ≥ 0 a.e. on R + × ∂Ω since h is convex and Λ x is the average with respect to a probability measure; d) for each T > 0 and each compact K ⊂ ∂Ω, there exists C K,T > 0 such that, for each ǫ > 0, one has

T 0 K×R N F ǫ (t, x, v)(v • n x ) 2 M(v)dvdS(x)dt ≤ C K,T , ǫ > 0 .
Statement d) appears in [START_REF] Masmoudi | Saint-Raymond From the Boltzmann equation to the Stokes-Fourier system in a bounded domain[END_REF], without proof. We give a brief justification for this estimate below.

Notice that in general, renormalized solutions of the initial-boundary value problem ( 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] are not known to satisfy the local conservation law of momentum -see equation (2.35) in [START_REF] Saint-Raymond | Hydrodynamic Limits of the Boltzmann Equation[END_REF] for a variant involving a defect measure, following an earlier remark due to P.-L. Lions and N. Masmoudi.

At variance, any classical solution

F ǫ ∈ C(R + × Ω × R N
) is continuously differentiable in (t, x) and such that [START_REF] Villani | Limites hydrodynamiques de l'équation de Boltzmann[END_REF] v → sup

0≤t≤T |x|≤R (|F ǫ (t, x, v)| + |∂ t F ǫ (t, x, v)| + |∇ x F ǫ (t, x, v)|)
is rapidly decaying as |v| → +∞ , one has

ǫ∂ t R N vF ǫ dv + div x R N v ⊗2 F ǫ dv = 0 , x ∈ Ω , t > 0 .
Moreover, for each w ∈ C 1 c (R + × Ω), Green's formula implies that (37)

t 0 Ω×R N (ǫv • ∂ t w(s, x) + v ⊗2 : ∇ x w(s, x))F ǫ (s, x, v)dvdxds = t 0 ∂Ω×R N v ⊗2 : w(s, x) ⊗ n x F ǫ (s, x, v)dvdS(x)ds +ǫ Ω×R N w(t, x) • vF ǫ (t, x, v)dvdx -ǫ Ω×R N w(0, x) • vF in ǫ (x, v)dvdx .
Let us use the accommodation condition [START_REF] Solonnikov | On a boundary value problem for a stationary system of Navier-Stokes equations[END_REF] to reduce the boundary integral:

R N v ⊗2 : w(t, x) ⊗ n x F ǫ (s, x, v)dv = R N (w(s, x)•v)(v•n x ) + F ǫ (s, x, v)dv - R N (w(s, x)•v)(v•n x ) -F ǫ (s, x, v)dv
and, whenever w is tangential on ∂Ω, one has

R N (w(s, x)••n x ) -F ǫ (s, x, v)dv = R N (w(s, x)•v)(v•n x ) -(1 -α)R x F ǫ + αΛ x F ǫ M M (s, x, v)dv = R N (w(s, x)•v)(v•n x ) + (1 -α)F ǫ + αΛ x F ǫ M M (s, x, v)dv . Therefore R N v ⊗2 : w(s, x) ⊗ n x F ǫ (s, x, v)dv = α R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dv -αΛ x F ǫ M R N (w(s, x) • v)(v • n x ) + Mdv
and the last integral vanishes since the integrand is odd in the tangential component of v.

Finally, whenever w ∈ C 1 c (R + × Ω) is tangential on ∂Ω, one has R N v ⊗2 : w(s, x) ⊗ n x F ǫ (s, x, v)dv = α R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dv .
Therefore, each classical solution F ǫ ∈ C(R + × Ω × R N ) of the initialboundary value problem ( 33)-( 34)-( 35) that is continuously differentiable in (t, x) and satisfies (36) also verifies ( 38)

t 0 Ω×R N (ǫv • ∂ t w(s, x) + v ⊗2 : ∇ x w(s, x))F ǫ (s, x, v)dvdxds = α t 0 ∂Ω×R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dvdS(x)ds +ǫ Ω×R N w(t, x) • vF ǫ (t, x, v)dvdx -ǫ Ω×R N w(0, x) • vF in ǫ (x, v)dvdx .
Henceforth, we shall consider exclusivey renormalized solutions of the initial-boundary value problem ( 33)-( 34)-( 35) satisfying the identity [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF] for each vector field w ∈ C1 c (R + × Ω) tangential on ∂Ω and such that div x w = 0 on Ω. Now for estimate d) in Theorem 4.1.

Proof of estimate d).

A renormalized solution of the initial-boundary value problem ( 33)-( 34)-( 35) can be constructed as the limit for ǫ > 0 fixed and m → +∞, of solutions F ǫ,m of the approximating equation

(39) ǫ∂ t F ǫ,m +v•∇ x F ǫ,m = 1 ǫ 1+q B m (F ǫ,m , F ǫ,m ) 1 + 1 m R N F ǫ,m dv , (x, v) ∈ Ω×R N ,
with the same initial and boundary conditions ( 34)-( 35) satisfied by F ǫ,m , where the approximate collision integral is given by the same expression as Boltzmann's collision integral with collision kernel b replaced with its truncated variant b m defined as

1 b m (v -v * , ω) := m ∧ b(v -v * , ω) .
Let ξ be a compactly supported C 1 vector field satisfying ξ(x) = a(x)n x for each x ∈ ∂Ω , with a ≥ 0 on ∂Ω and a = 1 on K .

Since the approximate collision integral in (39) is normalized with an average of F ǫ,m with respect to v, all solutions of that equation satisfy equality [START_REF] Xiao | On the vanishing viscosity limit for the 3-D Navier-Stokes equations with a slip boundary condition[END_REF] for any w ∈ C 1 c (R + × R N ), i.e. the local conservation of momentum. In other words,

t 0 Ω×R N (v ⊗2 : ∇ x ξ(x))F ǫ,m (s, x, v)dvdxds = t 0 ∂Ω×R N (v • ξ(x))(v • n x )F ǫ,m (s, x, v)dvdS(x)ds +ǫ Ω×R N ξ(x) • vF ǫ,m (t, x, v)dvdx -ǫ Ω×R N ξ(x) • vF in ǫ,m (x, v)dvdx .
Therefore, since F ǫ,m ≥ 0 a.e. and (v

• ξ(x))(v • n x ) = a(x)(v • n x ) 2 ≥ 0 for each x ∈ ∂Ω and v ∈ R N , one has (40) 0 ≤ t 0 K×R N (v • n x ) 2 F ǫ,m (s, x, v)dvdS(x)ds
≤ t 0 ∂Ω×R N (v • ξ(x))(v • n x )F ǫ,m (s, x, v)dvdS(x)ds
= t 0 Ω×R N (v ⊗2 : ∇ x ξ(x))F ǫ,m (s, x, v)dvdxds +ǫ Ω×R N ξ(x) • vF in ǫ,m (x, v)dvdx -ǫ Ω×R N ξ(x) • vF ǫ,m (t, x, v)dvdx .
At this point, we recall that the function h

: z → (1 + z) ln(1 + z) -z introduced in Theorem 4.1 has Legendre dual h * (ζ) = sup z>-1 (ζz -h(z)) = e ζ -ζ -1 .
By Young's inequality -or equivalently, by definition of h * (see for instance [START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF]) -one has

1 4 (1 + |v| 2 )ǫ|g ǫ,n | ≤ h(ǫ|g ǫ,n |) + h * ( 1 4 (1 + |v| 2 )) so that, for each nonnegative χ ∈ C c (R N ), (41)
Ω×R N χ(x)(1 + |v| 2 )F ǫ,m (t, x, v)dvdx = Ω×R N χ(x)(1 + |v| 2 )(1 + ǫg ǫ,m )(t, x, v)M(v)dvdx ≤ Ω×R N χ(x)(1 + h * ( 1 4 (1 + |v| 2 ))M(v)dvdx + χ L ∞ H(F ǫ,m |M)(t) ≤ Ω×R N χ(x)(1 + h * ( 1 4 (1 + |v| 2 ))M(v)dvdx + χ L ∞ H(F in ǫ |M
) . since the relative entropy estimate c) in Theorem 4.1 is also satisfied by the approximate solution F ǫ,m .

Since ξ is compactly supported, putting together (40), (41), and letting m → +∞ leads to estimate d).

4.2. The Euler limit. Let u in ∈ H(Ω), and pick initial data F in ǫ for the Boltzmann equation satisfying

(42) 1 ǫ 2 H(F in ǫ |M 1,ǫu in ,1 ) → 0 Theorem 4.2.
For each ǫ > 0, let F ǫ be a renormalized solution of ( 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] satisfying the local momentum conservation law [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF] for each w ∈ C 1 c (R + × Ω) satisfying div x w = 0 and w • n x ∂Ω = 0. Assume that the accommodation parameter α in the accommodation condition [START_REF] Solonnikov | On a boundary value problem for a stationary system of Navier-Stokes equations[END_REF] at the boundary depends on the scaling parameter ǫ in such a way that α(ǫ) = o(ǫ) as ǫ → 0 .

Then, for each compact K ⊂ Ω, the family

1 ǫ R N vF ǫ dv is relatively compact in L ∞ (R + ; L 1 (K)
), and each of its limit points as ǫ → 0 is a dissipative solution of the Euler equations [START_REF] Beirão Da Veiga | Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory[END_REF].

Assume that u in ∈ H(Ω) is smooth enough so that the initialboundary value problem for the Euler equations ( 8) has a classical solution on some finite time interval [0, T ] -for instance u in ∈ H s (Ω) with s > N 2 + 1, or u in ∈ C 1,θ with 0 < θ < 1. In that case, the convergence result above can be strengthened with the notion of entropic convergence, invented by Dave Levermore specifically to handle such problems. Definition 4.3 (C. Bardos, F. Golse, C.D. Levermore [START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF]). A family

g ǫ ≡ g ǫ (x, v) of L 1 loc (Ω × R N ; Mdxdv) is said to converge entropically at order ǫ to g ≡ g(x, v) as ǫ → 0 if the following conditions hold (i) 1 + ǫg ǫ ≥ 0 a.e. on Ω × R N for each ǫ, (ii) g ǫ → g weakly in L 1 loc (Ω × R N ; Mdxdv) as ǫ → 0, (iii) and 1 ǫ 2 H(M(1 + ǫg ǫ )|M) → 1 2 Ω×R N g(x, v) 2 M(v)dxdv as ǫ → 0.
We recall that, if g ǫ → g at order ǫ, then, for each compact K ⊂ Ω, one has

K R N (1 + |v| 2 )|g ǫ (x, v) -g(x, v)|M(v)dvdx → 0 as ǫ → 0 .
In other words, entropic convergence implies strong L 1 convergence with the weight (1 + |v| 2 )M(v) (see Proposition 4.11 in [START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF].) Whenever the incompressible Euler equations ( 8) have a classical solution u on [0, T ] × Ω, using the weak-strong uniqueness property of dissipative solutions and the conservation of energy satisfied by classical solutions of (8), we arrive at the following stronger convergence result, which is a straightforward consequence of Theorem 4.2. The interested reader is referred to the proof of Theorem 6.2 in [START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF] where the squeezing argument leading from weak compactness to entropic convergence is explained in detail.

Corollary 4.4. Consider a family

F in ǫ ≡ F in ǫ (x, v) ≥ 0 a.e. of mea- surable functions on Ω × R N such that F in ǫ (x, v) -M(v) ǫM(v) → u in (x) • v
entropically of order ǫ as ǫ → 0, where u in ∈ H(Ω) is smooth enough so that the initial-boundary value problem ( 8) has a classical solution u defined on the time interval [0, T ] with T > 0.

For each ǫ > 0, let F ǫ be a renormalized solution of ( 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] satisfying the local momentum conservation law [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF] for each w ∈ C 1 c (R + × Ω) that satisfies div x w = 0 and w • n x ∂Ω = 0. Assume that the accommodation parameter α in the accommodation condition [START_REF] Solonnikov | On a boundary value problem for a stationary system of Navier-Stokes equations[END_REF] at the boundary depends on the scaling parameter ǫ in such a way that

α(ǫ) = o(ǫ) as ǫ → 0 . Then F ǫ (t, x, v) -M(v) ǫM(v) → u(t, x) • v entropically of order ǫ as ǫ → 0, for a.e. t ∈ [0, T ].
The proof of Theorem 4.2 above occupies the remaining part of the present section. 4.3. The relative entropy inequality. Statement c) in Theorem 4.1 bears on the evolution of the relative entropy of the distribution F ǫ with respect to the uniform Maxwellian M = M 1,0,1 . In the next proposition, we consider the evolution of the relative entropy of the distribution F ǫ with respect to a local Maxwellian of the form M 1,ǫw,1 , where w is a solenoidal velocity field on Ω that is tangential to ∂Ω. Then, for each ǫ > 0, renormalized solution F ǫ of the initial-boundary value problem ( 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] satisfying the momentum conservation identity [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF], also satisy the relative entropy inequality

(43) 1 ǫ 2 H(F ǫ |M 1,ǫw,1 )(t) - 1 ǫ 2 H(F in ǫ |M 1,ǫw(0,•),1 ) ≤ - 1 ǫ 4+q t 0 Ω P ǫ (s, x)dxds - α ǫ 3 t 0 ∂Ω DG ǫ (s, x)dxds - 1 ǫ 2 t 0 Ω×R N (v -ǫw(s, x)) ⊗2 : ∇ x w(s, x)F ǫ (s, x, v)dxdvds - 1 ǫ t 0 Ω×R N (v -ǫw(s, x)) • E(w)(s, x)F ǫ (s, x, v)dvdxds + α ǫ 2 t 0 ∂Ω×R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dvdS(x)ds
for each t > 0.

Proof. We begin with the straightforward identity

H(F ǫ |M 1,ǫw,1 ) = H(F ǫ |M) + Ω×R N F ǫ ln M M 1,ǫw,1 dxdv = H(F ǫ |M) + Ω×R N F ǫ ( 1 2 |v -ǫw| 2 -1 2 |v| 2 )dxdv = H(F ǫ |M) + Ω×R N F ǫ ( 1 2 ǫ 2 |w| 2 -ǫv • w)dxdv . Thus (44) H(F ǫ |M 1,ǫw,1 )(t) -H(F ǫ |M 1,ǫw,1 )(0) = H(F ǫ |M)(t) -H(F ǫ |M)(0) + 1 2 ǫ 2 Ω×R N F ǫ (t, x, v)|w(t, x)| 2 dxdv -1 2 ǫ 2 Ω×R N F in ǫ (x, v)|w(0, x)| 2 dxdv -ǫ Ω×R N F ǫ (t, x, v)v • w(t, x)dxdv +ǫ Ω×R N F in ǫ (x, v)v • w(0, x)dxdv .
According to the continuity equation in statement a) in Theorem 4.1

(45)

+ 1 2 ǫ 2 Ω×R N F ǫ (t, x, v)|w(t, x)| 2 dxdv -1 2 ǫ 2 Ω×R N F in ǫ (x, v)|w(0, x)| 2 dxdv = t 0 Ω×R N F ǫ (ǫ 2 ∂ t + ǫv • ∇ x ) 1 2 |w| 2 dxdvds = t 0 Ω×R N F ǫ w • (ǫ 2 ∂ t w + ǫv • ∇ x w)dxdvds .
In

+ t 0 Ω×R N w • (ǫ 2 ∂ t w + ǫv • ∇ x w)(s, x)F ǫ (s, x, v)dxdvds - t 0 Ω×R N (ǫv • ∂ t w(s, x) + v ⊗2 : ∇ x w(s, x))F ǫ (s, x, v)dvdxds + α t 0 ∂Ω×R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dvdS(x)ds .
Next we express ∂ t w in terms of E(w) = ∂ t w + w • ∇ x w and ∇ x w: thus

(ǫv • ∂ t w(s, x) + v ⊗2 : ∇ x w(s, x)) -w • (ǫ 2 ∂ t w + ǫv • ∇ x w) = (v -ǫw) ⊗2 : ∇ x w + ǫ(v -ǫw) • E(w) .
In the right hand side of ( 46), we substitute

t 0 Ω×R N w • (ǫ 2 ∂ t w + ǫv • ∇ x w)(s, x)F ǫ (s, x, v)dxdvds - t 0 Ω×R N (ǫv • ∂ t w(s, x) + v ⊗2 : ∇ x w(s, x))F ǫ (s, x, v)dvdxds = - t 0 Ω×R N (v -ǫw(s, x)) ⊗2 : ∇ x w(s, x)F ǫ (s, x, v)dxdvds ǫ t 0 Ω×R N (v -ǫw(s, x)) • E(w)(s, x)F ǫ (s, x, v)dvdxds
and arrive at the relative entropy inequality (43).

4.4. Control of the boundary term. The relative entropy inequality (43) is the same as in the one considered in [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF], [START_REF] Lions | From Boltzmann Equation to the Navier-Stokes and Euler Equations II[END_REF] and [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF], except for the boundary term -i.e. the last term on the right hand side, which is in general not nonpositive. Since the effect of the boundary is our main interest in this paper, and the only difference with the case of the Cauchy problem treated in [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF], the core of our argument is to obtain a control of that term. 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] satisfies the inequality

(47) α ǫ 2 R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dv = α ǫ R N (w(s, x) • v)(v • n x ) + g ǫ (s, x, v)M(v)dv ≤ α 2ǫ 3 DG ǫ (s, x) + α ǫ C(w)1 w(s,x) =0 Λ x (F ǫ )(s, x)
a.e. in (s, x) ∈ R + × ∂Ω, where

C(w) := 1 2 R N (e 2 w L ∞ |v| -2 w L ∞ |v| -1)(v 1 ) + M(v)dv .
We use Young's inequality for a translate of the function h defined in Theorem 4.1, much in the same way as in the proof of Theorem 6.2 in [START_REF] Bardos | Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation[END_REF] (see especially pp. 738-739 there).

Proof. Let z 0 > -1; for each z > -1, set l(z -z 0 ) := h(z) -h(z 0 ) -h ′ (z 0 )(z -z 0 ) .
We recall that the Legendre dual of the function h defined in Theorem 4.1 is h * (p) := sup z>-1

(pz -h(z)) = e p -p -1 , p ∈ R .

A straightforward computation shows that

l * (p) = sup z>-1 (p(z -z 0 ) -l(z -z 0 )) = sup z>-1 (p(z -z 0 ) -h(z) + h(z 0 ) + h ′ (z 0 )(z -z 0 )) =h(z 0 ) -(h ′ (z 0 ) + p)z 0 + sup z>-1 ((h ′ (z 0 ) + p)z -h(z)) =h(z 0 ) -(h ′ (z 0 ) + p)z 0 + h * (h ′ (z 0 ) + p) =h(z 0 ) -(h ′ (z 0 ) + p)z 0 + e h ′ (z 0 ) e p -h ′ (z 0 ) -p -1 =(1 + z 0 ) ln(1 + z 0 ) -z 0 -(ln(1 + z 0 ) + p)z 0 + (1 + z 0 )e p -ln(1 + z 0 ) -p -1 =(1 + z 0 )(e p -p -1) = (1 + z 0 )h * (p) .
Writing F ǫ = M(1 + ǫg ǫ ) and observing that

R N (w(s, x) • v)(v • n x ) + M(v)dv = 0 , (s, x) ∈ R + × ∂Ω since w is tangential on ∂Ω, one has (48) R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dv = ǫ R N (w(s, x) • v)(v • n x ) + g ǫ (s, x, v)M(v)dv . Then R N (w(s, x) • v)(v • n x ) + g ǫ (s, x, v)M(v)dv = R N (w(s, x) • v)(v • n x ) + (g ǫ (s, x, v) -Λ x (g ǫ ))M(v)dv by (48) since Λ x (g ǫ ) is independent of v.
By definition Young's inequality (or equivalently, by definition of the Legendre dual of l),

2ǫ 2 (g ǫ -Λ x (g ǫ ))(w • v) ≤ l(ǫ(g ǫ -Λ x (g ǫ ))) + l * (ǫ(2w • v)) , so that 2 R N (w • v)(v • n x ) + g ǫ Mdv ≤ 1 ǫ 2 R N l(ǫ(g ǫ -Λ x (ǫ)))(v • n x ) + Mdv + 1 ǫ 2 R N l * (ǫ(2w • v))(v • n x ) + Mdv .
First, since Λ x is the average under a probability measure, one has

R N l(ǫ(g ǫ -Λ x (ǫ)))(v • n x ) + Mdv = 1 √ 2π Λ x (l(ǫ(g ǫ -Λ x (ǫ)))) = 1 √ 2π Λ x (h(ǫg ǫ ) -h(ǫΛ x (g ǫ )) -h ′ (ǫΛ x (g ǫ ))(ǫg ǫ -ǫΛ x (g ǫ ))) = 1 √ 2π Λ x (h(ǫg ǫ ) -h(ǫΛ x (g ǫ )) -h ′ (ǫΛ x (g ǫ )) 1 √ 2π Λ x (ǫg ǫ -ǫΛ x (g ǫ ))) = 1 √ 2π (Λ x (h(ǫg ǫ ) -h(ǫΛ x (g ǫ ))) = DG ǫ On the other hand 1 ǫ 2 R N l * (ǫ(2w • v))(v • n x ) + Mdv = (1 + ǫΛ(g ǫ )) R N e 2ǫ|w||v| -2ǫ|w||v| -1 ǫ 2 (v • n x ) + Mdv ≤ Λ(F ǫ ) R N (e 2|w||v| -2|w||v| -1)(v • n x ) + Mdv -since, for each a > 0, the map ǫ → e aǫ -aǫ -1 ǫ 2 = a 2 n≥2 (aǫ) n-2 n! is increasing. Finally α ǫ 2 R N (w(s, x) • v)(v • n x ) + F ǫ (s, x, v)dv ≤ α ǫ R N (w(s, x) • v)(v • n x ) + g ǫ (s, x, v)M(v)dv ≤ α 2ǫ 3 DG ǫ (s, x) + α ǫ C(w)1 w(s,x) =0 Λ(F ǫ )(s, x)
where

C(w) := 1 2 R N (e 2 w L ∞ |v| -2 w L ∞ |v| -1)(v 1 ) + M(v)dv .
After integrating in (s, x) both sides of (47), the first term on the right hand side of (47) will be absorbed by the Darrozes-Guiraud information on the right hand side of (43), so that, with Lemma 4.6, the inequality (43) is transformed into (49)

1 ǫ 2 H(F ǫ |M 1,ǫw,1 )(t) - 1 ǫ 2 H(F in ǫ |M 1,ǫw(0,•),1 ) ≤ - 1 ǫ 4+q t 0 Ω P ǫ (s, x)dxds - α 2ǫ 3 t 0 ∂Ω DG ǫ (s, x)dxds - 1 ǫ 2 t 0 Ω×R N (v -ǫw(s, x)) ⊗2 : ∇ x w(s, x)F ǫ (s, x, v)dxdvds - 1 ǫ t 0 Ω×R N (v -ǫw(s, x)) • E(w)(s, x)F ǫ (s, x, v)dvdxds + α ǫ C(w) t 0 ∂Ω∩supp(w) Λ x (F ǫ )(s, x)dS(x)ds ,
for all t ≥ 0.

4.5. Control of the outgoing mass flux. In the lemma below, we shall prove that the outgoing mass flux Λ x (F ǫ ) is uniformly bounded in L 1 loc (R + × ∂Ω), so that the last term on the right hand side of (49) vanishes under the assumption α(ǫ) = o(ǫ) as ǫ → 0. Lemma 4.7. With the notations of Theorem 4.1, for each ǫ > 0, each renormalized solution F ǫ of the initial-boundary value problem ( 33)-( 34)- [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] satisfies the inequality

(50) R N F ǫ (t, x, v)(v • n x ) + dv ≤ 1 h(η) DG ǫ (t, x) + 1 √ 2π(1 -η) R N F ǫ (v • n x ) 2 dv
for all η ∈ (0, 1), a.e. in (t, x) ∈ R + × ∂Ω.

Proof. First we recast the Darrozes-Guiraud information in the form (51)

DG ǫ = 1 √ 2π Λ x (h(ǫg ǫ ) -h(ǫΛ x (g ǫ ))) = 1 √ 2π Λ x (G ǫ ln G ǫ -G ǫ -Λ x (G ǫ ) ln Λ x (G ǫ ) + Λ x (G ǫ )) = 1 √ 2π Λ x G ǫ ln G ǫ Λ x (G ǫ ) -G ǫ + Λ x (G ǫ ) , since Λ x ((G ǫ -Λ x (G ǫ )) ln Λ x (G ǫ )) = Λ x ((G ǫ -Λ x (G ǫ ))) ln Λ x (G ǫ ) = 0 .
Then we consider the integral

I : = Λ x G ǫ R N (v • n x ) 2 + ∧ 1Mdv = Λ x G ǫ R N 1 |Gǫ/ΛxGǫ-1|>η (v • n x ) 2 + ∧ 1Mdv + Λ x G ǫ R N 1 |Gǫ/ΛxGǫ-1|≤η (v • n x ) 2 + ∧ 1Mdv =: I 1 + I 2 avec η ∈]0, 1[.
The first term is estimated in terms of the Darrozes-Guiraud information on the boundary, in view of (51):

I 1 ≤ 1 h(η) Λ x G ǫ R N 1 |Gǫ/ΛxGǫ-1|>η h G ǫ Λ x G ǫ -1 (v • n x ) 2 + ∧ 1Mdv ≤ 1 h(η) Λ x G ǫ R N h G ǫ Λ x G ǫ -1 (v • n x ) 2 + ∧ 1Mdv ≤ 1 h(η) R N G ǫ ln G ǫ Λ x G ǫ -G ǫ + Λ x G ǫ (v • n x ) + Mdv ≤ √ 2π h(η) DG ǫ (t, x) .
As for the second term, one has

I 2 = Λ x G ǫ R N 1 |Gǫ/ΛxGǫ-1|≤η (v • n x ) 2 + ∧ 1Mdv ≤ 1 1 -η R N G ǫ 1 |Gǫ/ΛxGǫ-1|≤η (v • n x ) 2 + ∧ 1Mdv ≤ 1 1 -η R N G ǫ (v • n x ) 2 Mdv .
Putting together both estimates gives

I = I 1 + I 2 ≤ √ 2π h(η) DG ǫ (t, x) + 1 1 -η R N F ǫ (v • n x ) 2 dv . Since R N (v • n x ) 2 + ∧ 1Mdv = 1 0 z 2 e -z 2 /2 dz √ 2π =: J > 0 is independent of x, we conclude that R N F ǫ (t, x, v)(v • n x ) + dv = 1 √ 2π Λ x (G ǫ )(t, x) = 1 √ 2π I J
which, together with the previous inequality, leads to the announced estimate.

4.6. Convergence to the incompressible Euler equations. At this point bring together the relative entropy inequality (43) and the boundary control (47), thereby arriving at the estimate With (53), the relative entropy inequality (52) is precisely of the same form as the inequality stated as Theorem 5 in [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF]. One then concludes by the same argument as in [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF].

1 ǫ 2 H(F ǫ |M 1,ǫw,1 )(t) - 1 ǫ 2 H(F in ǫ |M 1,ǫw(0,•),1 ) ≤ - 1 ǫ 4+q t 0 Ω P ǫ (s, x)dxds - α 2ǫ 3 t 0 ∂Ω DG ǫ (s, x)dxds - 1 ǫ 2 t 0 Ω×R N (v -ǫw(s, x)) ⊗2 : ∇ x w(s, x)F ǫ (s, x, v)dxdvds

Conclusion and final remarks

As recalled above, the convergence of solutions of the boundary value problem for the Navier-Stokes equations in the large Reynolds number regime is an open problem, as well as the validity of the Prandlt equation for the boundary layer. Convergence to solutions of the Euler Equation, for general boundary conditions (including the Dirichlet boundary condition) are proven under only the most stringent regularity assumptions.

On the other hand the onset of von Karman vortex streets and the Kolmogorov hypothesis on turbulence based on a non zero energy dissipation in the large Reynolds (see chapter 5 in [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]) suggests that, in general, the limit is not a solution of the Euler equation. This is in agreement with Kato's criterion [START_REF] Kato | Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary[END_REF] relating the convergence of the solutions of the Navier-Stokes equations with Dirichlet boundary condition to a solution of the Euler equations with the vanishing of the viscous energy dissipation in a boundary layer with thickness O(Re -1 ).

For the Navier slip boundary condition ( 14) or (13), the inviscid limit is established in the present paper is proven under the only assumption λ → 0 as ν → 0.

The following remarks are in order • at variance with previous results [START_REF] Beirão Da Veiga | Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory[END_REF][START_REF] Xiao | On the vanishing viscosity limit for the 3-D Navier-Stokes equations with a slip boundary condition[END_REF], no regularity assumption is required for all the results in the present paper, as only estimates are used in the proof; • what is proved here is the convergence to a dissipative solution (hence to the unique classical whenever it exists); therefore, this convergence is also true even if no classical solution exists, or even if the L 2 initial data corresponds to a "wild" solution à la C. DeLellis and L. Szkelyhidi (see [START_REF] De Lellis | The Euler equations as a differential inclusion[END_REF]) for which there is no uniqueness of the Euler solution;

• therefore the main goal of the present paper is to show the strong similarity between the inviscid limit for the Navier-Stokes equations and the fluid dynamic limit for the Boltzmann equation; the accommodation coefficient α, the Mach Ma and Strouhal Sh numbers, the Reynolds number Re (see [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF], §1.9), and the slip coefficient λ are related by Ma = Sh = ǫ , 1 Re = o(1) , α ǫ = λ , as ǫ → 0, so that the conditions α = o(ǫ) as ǫ → 0 and λ → 0 as ν → 0 are consistent. In all case the convergence of the solution of the Navier-Stokes equations to a classical solution of the Euler equations will imples that the energy dissipation vanishes in the limit, as observed by Kato [START_REF] Kato | Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary[END_REF]. Likewise, the entropic convergence obtained in the present paper in the case of renormalized solutions of the Boltzmann equation implies that the sum of the entropy dissipation and of the Darrozes-Guiraud information at the boundary vanishes with ǫ.

By analogy with the work of Kato [START_REF] Kato | Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary[END_REF], a possible conjecture is that, whenever lim ǫ→0 α(ǫ) ǫ > 0 , the vanishing of both the Darrozes-Guiraud information at the boundary and of the entropy production implies that the inviscid fluid dynamic limit of the Boltzmann equation is described by a solution of the Euler equations.

Other, perhaps less delicate problems could be analyzed by the methods used in the present paper. For instance, Maxwell's accommodation is but one example in a wide class of nonlocal boundary conditions for the Boltzmann equation; the Navier-Stokes and Euler hydrodynamic limits of the Boltzmann equation should be considered also for such boundary conditions (see [START_REF] Sone | Molecular Gas Dynamics: Theory, Techniques and Applications[END_REF] §1.6). Likewise, the condition [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF], which may not be verified for all renormalized solutions of the initial boundary value problem for the Boltzmann equation should be removed on principle (as in [START_REF] Saint-Raymond | Convergence of solutions to the Boltzmann equation in the incompressible Euler limit[END_REF]).

In other words, there remain many open problems related to the issues discussed in the present paper, to which we shall return in future publications.

  or in the larger class allowed by the growth at infinity of the collision kernel b. As a result, whenever F is a classical solution of the scaled Boltzmann equation[START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] with appropriate decay as |v| → ∞,

  and these relations are the local conservation laws of mass, momentum and energy respectively.Whenever b(v -v * , ω) has polynomial growth as |v -v * | → ∞, for each positive, rapidly decaying φ ∈ C(R N ) such that ln φ has polynomial growth as |v| → ∞, Boltzmann's H Theorem states that[START_REF] Leray | Essai sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF]

Proposition 4 . 5 .

 45 Let w ∈ C 1 c (R + × Ω) be such that div x w = 0 and w • n ∂Ω = 0 .

Lemma 4 . 6 .

 46 With the notations of Theorem 4.1, for each ǫ > 0 and each w ∈ C 1 c (R + × Ω) that is tangential on the boundary ∂Ω, each renormalized solution F ǫ of the initial-boundary value problem (

ΛF

  ǫw(s, x)) • E(w)(s, x)F ǫ (s, x, v)dvdxds x (F ǫ )(s, x)dS(x)ds .Next, we use the pointwise inequality (50) with, say, η = 1 2 , to control the last integral on the right-hand side above:(52) 1 ǫ 2 H(F ǫ |M 1,ǫw,1 )(t) -1 ǫ 2 H(F in ǫ |M 1,ǫw(0,•),1 ) ǫw(s, x)) ⊗2 : ∇ x w(s, x)F ǫ (s, x, v)dxdvds ǫw(s, x)) • E(w)(s, x)F ǫ (s, x, v)dvdxds + 2α ǫ C(w) t 0 ∂Ω∩supp(w) R N F ǫ (v • n x ) 2 dvdS(x)ds .Now, statement d) in Theorem 4.1 and the scaling assumption on the accommodation parameter, i.e. α(ǫ) = o(ǫ), show that, for each T > 0 ǫ (v • n x ) 2 dvdS(x)ds → 0 uniformly in t ∈ [0, T ] as ǫ → 0.

For all x, y ∈ R, the notation x ∧ y designates min(x, y).