BOUNDARY LAYERS AND HYDRODYNAMIC LIMITS OF BOLTZMANN EQUATION (I): INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT

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1 BOUNDARY LAYERS AND HYDRODYNAMIC LIMITS OF BOLTZMANN EQUATION (I: INCOMPRESSIBLE NAVIER-STOKES-FOURIER LIMIT NING JIANG AND NADER MASMOUDI Abstract. We establish an incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation (with a general collision kernel considered over a bounded domain. Appropriately scaled families of DiPerna-Lions renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to zero. Every limit point is a weak solution to a Navier- Stokes-Fourier system with different types of fluid boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately which means that they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. These acoustic waves are known to oscillate faster and faster when the Knudsen number goes to zero and to prevent strong convergence in the periodic or the whole space case. This extends the work of Golse and Saint-Raymond [8, 9] and Levermore and Masmoudi [7] to the case of a bounded domain. The case of a bounded domain was already considered by Masmoudi and Saint-Raymond [33] for the (linear Stokes-Fourier limit and without studying the damping of the acoustic waves. Contents. Introduction. Boltzmann Equation in Bounded Domain 5.. Maxwell Boundary Condition 5.. Nondimensionalized Form of the Boltzmann Equation 6.3. Navier-Stokes Scaling 9.4. A Priori Estimates 0.5. DiPerna-Lions Solutions 3. Statement of the Main Results 3.. Dirichlet Boundary Condition 3.. Navier Boundary Condition 3 4. Geometry of the boundary 4 5. Acoustic Modes Acoustic Operator A Eigenspaces of A Conditions on Φ k the operator A iλ τ,k 7 6. Analysis of the Kinetic Boundary Layer Equation 9 7. Approximate Eigenfunctions-Eigenvalues Date: March, 00.

2 N. JIANG AND N. MASMOUDI 7.. Motivation 7.. Statement of the Proposition Main Idea of the Proof 4 8. Proof of Proposition 7.: the case 0 < β < Double Induction (m, n = (0, (m, n = (, (m, 0 for all m General Process Proof of Proposition 7.: the case β = Proof of Proposition 7.: the case < β < (0, n for n (, n for n General process 44. Proof of Proposition 7.: the case β = 45. Proof of Proposition 7.: Truncation Error Estimates 46.. Estimates of R τ,k,n 46.. Estimates of g τ,k,n gτ,k,int 0, Boundary error estimate Proof of the Weak Convergence Proof of the Strong Convergence Strong Convergence in L Proof of Proposition Estimates of c ± k, Estimates of b ± k, Proof of Lemma Conclusion 6 References 63. Introduction The hydrodynamic limit of the Boltzmann equation got a lot of interest in the previous decade. The main contribution is the rigorous derivation of the Navier-Stokes-Fourier system from the Boltzmann equation by Golse and Saint-Raymond [8, 9] for the case of hard potentials. This was then extended by Levermore and Masmoudi [7] to include soft potentials. One of the ingredients of the proof of convergence is the treatment of the acoustic waves which are highly oscillating. A compensated compactness type argument was used by Lions and Masmoudi [30] to prove that they have no contribution on the equation satisfied by the weak limit. This argument was previously used in the compressible incompressible limit [9]. In this paper, we prove that in a bounded domain and with an accommodation coefficient bigger than the Knudsen number, there is no need for the argument in [9] since we can prove that the acoustic waves are damped instantaneously. Our work is based on the construction of viscous and kinetic (Knudsen boundary layers of size and. A similar property was used by Desjardins, Grenier, Lions, and Masmoudi [0] in the compressible

3 NAVIER-STOKES-FOURIER 3 incompressible limit. However, there are two main differences with [0]. The first one is that a generic assumption on the domain has to be made in [0] (in particular there are modes which are not damped in the disc. Here, we do not need this assumption. The reason is that we are dealing with the whole acoustic system, namely including the temperature and that we are including a kinetic effect. The second difference, is that we need to build a kinetic boundary layer. The solvability condition of this layer gives the boundary data for the viscous boundary layer. We consider a bounded domain R D, D, with boundary C. The Navier-Stokes-Fourier system governs the fluctuations of mass density, bulk velocity, and temperature (ρ, u, θ about their spatially homogeneous equilibrium values in a Boussinesq regime. Specifically, after a suitable choice of units, these fluctuations satisfy the incompressibility and Boussinesq relations x u = 0, ρ + θ = 0; (. while their evolution is determined by the Navier-Stokes and heat equations t u + u x u + x p = ν x u, u(x, 0 = u 0 (x, t θ + u x θ = κ (. D+ xθ, θ(x, 0 = θ 0 (x, where ν > 0 is the kinematic viscosity and κ > 0 is the heat thermal conductivity. We will derive two types of natural physical boundary conditions for the incompressible Navier-Stokes-Fourier system (.. The first is the (homogeneous Dirichlet boundary condition, namely, u = 0, θ = 0 on R +. (.3 The other is the so-called Navier slip boundary condition, which was proposed by Navier [38]: [νd(u n + χu] tan = 0, u n = 0 on R +, κ n θ + χ D+θ = 0 on (.4 D+ R+, where d(u denotes the symmetric part of the stress tensor d ij (u = ( iu j + j u i, and n denotes the directional derivative along the outer normal n. In the above Navier boundary condition, χ > 0 is the reciprocal of the slip length which depends on the material of the container. Fluid regimes are those where the mean free path is small compared to the macroscopic length scales i.e. where the Knudsen number is small. The incompressible Navier- Stokes-Fourier system can be formally derived from the Boltzmann equation through a scaling in which the fluctuations of the density F about an absolute Maxwellian M are scaled to be on an order, see []. The program that justifies the fluid dynamic limits from the Boltzmann equation in the framework of DiPerna-Lions [] was initiated by Bardos-Golse-Levermore [, 3]. Since then, there has been lots of contributions to this program [4,, 8, 9, 3, 7, 30, 3, 33, 39] among which we particularly mention the result of Golse and Saint-Raymond [8] which is the first complete justification of Navier-Stokes-Fourier limit from Boltzmann equation in a class of bounded collision kernels, without making any nonlinear weak compactness hypothesis. They have recently extended their result to the case of hard potentials [9]. With some new nonlinear estimates, Levermore and Masmoudi [7] treated a broader class of collision kernels which includes all hard potential cases and, for the first time in this program, soft potential cases.

4 4 N. JIANG AND N. MASMOUDI However, all of the above works are carried out in either the periodic spatial domain or the whole space, except for [33] in which the (linear Stokes-Fourier is recovered with the same collision kernels assumption as in []. In [33], the renormalized solutions to the Boltzmann equation in bounded domain (see [37] was proved to pass to the limit in the different types of kinetic boundary conditions and recovered different fluid boundary conditions, either Dirichlet condition (.3, or Navier slip boundary condition (.4, depending on the relative sizes of the accommodation coefficient and Knudsen number. Y. Sone and his collaborators observed that formally the boundary conditions of the fluid equations derived from the kinetic equation with Maxwell type boundary conditions, depend on the relative importance of the accommodation coefficient and the Knudsen number, see [43] Chapter 3 and 4. The current work, generalizes the previous work of Masmoudi and Saint-Raymond [33] in three ways: First, it works for general collision kernels, namely in the framework of [7]. Second, it extends the result of [33] from linear Stokes-Fourier system to nonlinear Navier-Stokes-Fourier system. The third extension which is the main novelty of the current work is in the treatment of the Dirichlet boundary condition case. Indeed, we prove, in the Dirichlet boundary condition case, that the convergence is strong. We point out that in all the previous works mentioned above, the convergence is in w-l, unless the initial data is well-prepared, i.e. is hydrodynamic and satisfies the Boussinesq and incompressibility relations. This weak convergence is caused by the persistence of fast acoustic waves. In the Navier-Stokes regime, the Reynold number Re is unit order, then the von Kámán relation = Ma Re implies that in the fluid limit 0, the Mach number Ma must go to zero. As is well know physically, one expects that as Ma 0, fast acoustic waves are generated carrying the energy of the potential part of the flow. For the periodic flows, or for some particular boundary conditions such as Navier condition (.4, these waves subsist forever and their frequency grows with. Mathematically, this means that the convergence is only weak. This phenomena happens in many cases among which we only mention [8, 9]. In [0], a striking phenomena (damping of acoustic waves caused by the Dirichlet boundary condition was found when Desjardins, Grenier, Lions, and Masmoudi considered the incompressible limit of the isentropic Navier-Stokes equations. In the case of a viscous flow in a bounded domain with Dirichlet boundary condition, and under a generic assumption on the domain (related to the so-called Schiffer s conjecture and the Pompeiu problem [9], they showed that the acoustic waves are instantaneously (asymptotically damped, due to the formation of a thin boundary layer in time. This layer is caused by a boundary layer in space and dissipates the energy carried by the acoustic waves. From a mathematical point of view, strong convergence was obtained. The current work can be seen as an extension of [0] to the kinetic case. The main idea is to use a family of test functions which solve the linearized Boltzmann equation and which can capture the propagation of the fast acoustic waves. These test functions are constructed through considering a family of approximate eigenfunctions of a dual operator with a dual kinetic boundary condition with respect to the original Boltzmann equation. The leading term of the approximate eigenvalue is purely imaginary describing the acoustic mode, the real part of the next order term is strictly negative which gives the strict dissipation when applying the test functions to the renormalized Boltzmann equation.

5 NAVIER-STOKES-FOURIER 5 In contrast to [0], the family of the approximate eigenfunctions includes two boundary layers: viscous fluid boundary layer and kinetic Knudsen layer, while in [0], only a fluid boundary layer was necessary. An other important difference is the fact that here, we do not need any assumption on the domain. The reason behind it is that the Navier-Stokes- Fourier system has also some dissipation in the temperature equation which is ignored in the isentropic model. (in particular this dissipation property holds in the case of the ball. In the current paper, the kinetic-fluid coupled boundary layers are constructed for the case that the accommodation coefficient is of the form α ɛ = χ β with 0 β <. We found that β = is a threshold. A key role is played by the linearized kinetic boundary layer equation which has been studied extensively (see [, 8, 5, 4, 44] and recently extended by the authors to a broader class of the collision kernel [4]. Applying the linearized kinetic boundary layer equation to construct the two layer eigenfunctions is the main novelty of the current paper. To the best of our knowledge these two layer eigenfunctions are new even in the applied literature. For instance, Y. Sone, only construct Knudsen layers relative to the incompressible flow (which then does not require the fluid boundary layer. The paper is organized as follows: the next section contains preliminary material regarding the Boltzmann equation in a bounded domain. We state the main theorems in Section 3 which include the weak convergence for the Navier slip boundary and strong convergence for the Dirichlet boundary. In Section 4, we list some differential geometry properties of the boundary as a submanifold of R D. Section 5 provides an introduction to the acoustic modes while Section 6 is about the analysis of the kinetic boundary layer equation whose solvability provides the boundary conditions of the fluid variables. In Section 7, we presents the constructions of the test functions used in the proof of the Main Theorem. The proof of the main proposition on the boundary layers is given in Section 8, 9 and 0. In Section, we establish the weak convergence results in the main theorem. Section contains a proof of the strong convergence for the Dirichlet boundary case. In the last section, we make some concluding remarks.. Boltzmann Equation in Bounded Domain Here we will introduce the Boltzmann equation in a bounded domain, only so far as to set our notations, which is essentially that of [3] and [33]. More complete introduction to the Boltzmann equation can be found in [6, 7, 6, 43]... Maxwell Boundary Condition. We take, a smooth bounded domain of R D, and O = R D, the space-velocity domain. Let n(x be the outward unit normal vector at x. We denote by dσ x the Lebesgue measure on the boundary. We define the outgoing and incoming sets Σ + and Σ at the boundary by Σ ± = {(x, v Σ : ±n(x v > 0} where Σ = R D. (. Denoted by γf the trace of F over Σ. The boundary condition takes the form of a balance between the values of the outgoing and incoming parts of the trace of F, namely γ ± F = Σ± γf. In order to describe the interaction between particles and the wall, J.-C. Maxwell [34] proposed in 879 the following phenomenological law which splits into a local reflection and a diffuse (or Maxwell reflection γ F = ( αl(γ + F + αk(γ + F on Σ (.

6 6 N. JIANG AND N. MASMOUDI where α [0, ] is a constant, also called the accommodation coefficient. The local reflection operator L is given by Lφ(x, v = φ(x, R x v, (.3 where R x v = v [n(x v] n(x is the velocity before the collision with the wall. The diffuse reflection operator K is given by where φ is the outgoing mass flux Kφ(x, v = π φ(xm(v (.4 φ(x = v n(x>0 and M is the following absolute Maxwellian φ(x, vv n(x dv, (.5 M(v = (π D/ exp ( v, (.6 that corresponds to the spatially homogeneous fluid state with density and temperature equal to and bulk velocity equals to 0. We notice that v n(x πm(v dv = v n(x πm(v dv =, (.7 v n(x>0 v n(x<0 which expresses the conservation of mass at the boundary. Here we take the temperature of the wall to be constant and equal to... Nondimensionalized Form of the Boltzmann Equation. We consider a sequence of renormalized solutions to the rescaled Boltzmann equation t F ɛ + v x F ɛ = B(F ɛ, F ɛ on R + O, F ɛ (0, x, v = Fɛ in (x, v on O, γ F ɛ = ( αl(γ + F ɛ + αk(γ + F ɛ on R + Σ, (.8 where the measurable function Fɛ in 0 is the initial distrubution funtion. The positive, nondimensional parameter is the Knudsen number that is the ratio of the mean free path to the macroscopic length scale. it is supposed to be small, which justifies studying the limit going to 0. The Boltzmann collision operator B acts only on the v argument of F and is formally given by B(F, F = (F F F F b(ω, v v dω dv, (.9 S D R D where v ranges over R D endowed with its Lebesgue measure dv, while ω ranges over the unit sphere S D = {ω R D : ω = } endowed with its rotationally invariant unit measure dω. The F, F, F, and F appearing in the integrand designate F (t, x, evaluated at the velocities v, v, v and v, respectively, where the primed velocities are defined by v = v ω[ω (v v], v = v + ω[ω (v v], (.0

7 NAVIER-STOKES-FOURIER 7 for any given (ω, v, v S D R D R D. This expresses the conservation of momentum and energy for particle pairs after a collision, namely, v + v = v + v, v + v = v + v. (. The collision kernel b is a positive, locally integrable function. The Galilean invariance of the collision kernel implies that b has the classical form b(ω, v = v Σ( ω ˆv, v, (. where ˆv = v/ v and Σ is the specific differential cross section. This symmetry implies that the quantity b(ω, v dω is a function of v only. The DiPerna-Lions theory that provides the global existence of renormalized solutions requires that b satisfies lim v + v S D K b(ω, v v dω dv = 0 (.3 for any compact set K R D. There are some additional assumptions on b needed in the work of Levermore-Masmoudi [7]. For the convenience of the reader, we list these assumptions here. A major role will be played by the attenuation coefficient a, which is defined as a(v = b(v vm dv = b(ω, v v dωm dv. (.4 R D S D R D A few facts about a are readily evident from what we have already assumed. Because (.3 holds, one can show that lim v a(v + v = 0. (.5 Our second assumption regarding the collision kernel b is that a satisfies a lower bound of the form C a ( + v α a(v, (.6 for some constant C a > 0 and α R. The third assumption is that there exists s (, ] and C b (0, such that s s b(v v a(v a(v a(v M dv C b. (.7 R D Another major role in what follows will be played by the linearized around the global Maxwellian M collision operator L, which is defined by L g = ( g + g g g b(ω, v v dωm dv. (.8 S D R D One has the decomposition a L = I + K K +, (.9 where the loss operator K and the gain operator K + are defined by K g = g b(v vm dv, (.0 a R D

8 8 N. JIANG AND N. MASMOUDI K + g = ( g + g a b(ω, v v dωm dv. (. R D The fourth assumption regarding the collision kernel b is that K + : L (amdv L (amdv is compact. (. Combining the gain operator assumption (. and the loss operator assumption (.7, we conclude that a L : Lp (amdv L p (amdv is Fredholm (.3 for every p (,. From this Fredholm property we can define the psuedo-inverse of L, called L : L : L p (a p Mdv Null (L L p (amdv. (.4 Moreover, L is a bounded operator. The fifth assumption regarding b is that for every δ > 0 there exists C δ such that b satisfies b(v v + δ b(v C v δ ( + a(v ( + a(v for every v, v R D. (.5 + v v It is well known that the null space of the linearized Boltzmann operator L is given by Null(L span{, v,, v D, v }. Define P as the orthogonal projection from L (Mdv onto Null(L, which is given by P g = g + v g + ( v D ( D v g. (.6 Futhermore, we define P = I P. The matrix-valued function A(v and the vector-valued function B(v are defined by A(v = v v D v I, B(v = v v D+ v. (.7 It is easy to see that each entry of A and B is in L (a M dv Null (L. introduce  L (amdv; R D D and B L (amdv; R D which are given by We also  = L A, B = L B. (.8 For the sake of simplicity, we take the following normalizations b(ω, v v dωm dv dv =, S D R D R D dω =, Mdv =, dx =, S D R D (.9 associated with the domains S D R D R D, S D, R D and respectively, and R D F in dx dv =, (.30 associated with the initial data F in.

9 NAVIER-STOKES-FOURIER 9 Because Mdv is a positive unit measure on R D, we denote by ξ the average over this measure of any integrable function ξ = ξ(v, ξ = ξ(vmdv. (.3 R D Moreover, we also use the following average on the boundary ξ = ξ(v [n(x v] πmdv. (.3 Notice that R D Σ+ = Σ =. (.33 Because dµ = b(ω, v v dωm dv M dv is a positive unit measure on S D R D R D, we denote by Ξ the average over this measure of any integrable function Ξ = Ξ(ω, v, v Ξ = Ξ(ω, v, v dµ. (.34 S D R D R D The measure dµ is invariant under the coordinate transformations (ω, v, v (ω, v, v, (ω, v, v (ω, v, v. (.35 These, and compositions of these, are called dµ-symmetries..3. Navier-Stokes Scaling. The Navier-Stokes-Fourier system can be formally derived from the Boltzmann equation through a scaling in which the fluctuations of the kinetic densities F about the absolute Maxwellian M are scaled to be of order. More precisely, we take Rewriting equation (.8 for G yields F = MG = M( + g. (.36 t G ɛ + v x G ɛ = Q(G ɛ, G ɛ on R + O, G ɛ (0, x, v = G in ɛ (x, v on O, γ G ɛ = ( αl(γ + G ɛ + α γ + G on R + Σ, (.37 where the collision kernel Q is now given by Q(G, G = (G G G Gb(ω, v vdωm dv. (.38 S D R D In terms of g and with the previous notations, the system (.8 finally reads t g + v x g + Lg = Q(g, g on R + O, g (0, x, v = g in (x, v on O, γ g = ( αl(γ + g + α γ + g on R + Σ. (.39

10 0 N. JIANG AND N. MASMOUDI.4. A Priori Estimates. Due to the presence of the boundary, the classical a priori estimates for the Boltzmann equation, namely the entropy and energy bounds, are modified. First, because all particles arriving at the boundary are reflected or diffused, we have conservation of mass, which can be written as G ɛ dx = G in ɛ dx =. (.40 Mulplying the equation (.37 by log(g ɛ and integrating in x and v, we get formally t G ɛ log(g ɛ G ɛ + dx + (G ɛ log(g ɛ G ɛ + v n(x dσ x Mdv Σ = (.4 log(g ɛ Q(G ɛ, G ɛ dx. By denoting h(z = (+z log(+z z, for z > and using that it is a convex function, we can compute the boundary term in the following way: Ẽ (γ + G ɛ = [G ɛ log(g ɛ G ɛ + ]v n(x dσ x Mdv Σ = [h(γ + g ɛ h (( α ɛ γ + g ɛ + α ɛ γ + g ɛ ] v n(x Mdvdσ x Σ + [h(γ + g ɛ ( α ɛ h(γ + g ɛ α ɛ h( γ + g ɛ ] v n(x Mdvdσ x (.4 Σ + = α ɛ E(γ + G ɛ, π where E(γ + G ɛ, the so-called Darrozès-Guiraud information, is given by E(γ + G ɛ = [ h(γ + g ɛ h( γ + g ɛ ] dσ x. (.43 Jensen s inequality implies that E(γ + G ɛ 0. Noticing that Ẽ(γ + G ɛ αɛ π E(γ + G ɛ, we get the entropy inequality t ( H(G ɛ (t + R(G ɛ(s + Ẽ(γ + G ɛ (s ds H(G in ɛ, (.44 0 where H(G is the relative entropy functional H(G = G log(g G + dx, (.45 and R(G is the entropy dissipation rate functional ( G R(G = 4 log G (G G G G G G dx. (.46

11 NAVIER-STOKES-FOURIER.5. DiPerna-Lions Solutions. We will work in the setting of DiPerna-Lions renormalized solutions. These solutions were initially constructed by DiPerna and Lions [] over the whole space R D for any initial data satisfying natural physical bounds. Their result was then extended to the case of a bounded domain by Hamdache [] for the case of specular reflection at the boundary. Mischler [35, 36, 37] developed a general theory to treat other boundary conditions, namely the case of general Maxwell boundary conditions (.. The DiPerna-Lions theory does not yield solutions that are known to solve the Boltzmann equation in the usual weak sense. Rather, it gives the existence of a global weak solution to a class of formally equivalent initial value problems that are obtained by multiplying (.37 by Γ (G, where Γ is the derivative of an admissible function Γ: ( t + v x Γ(G ɛ = Γ (G ɛ Q(G ɛ, G ɛ on R + O, G ɛ (0,, = G in ɛ 0 on O. (.47 A function Γ : [0, R is called admissible if it is continuously differentiable and for some constant C Γ < its derivative satisfies Γ (z + z C Γ. (.48 Mischler [37] extended DiPerna-Lions theory to domains with a boundary on which the Maxwell reflection boundary condition is imposed. This required the proof of a so-called trace theorem that shows thatγg makes sense. In particular, Mischler showed that γg lies in the set of all v n Mdv dσ x dt-measurable functions over R + that are finite almost everywhere, which we denote L 0 ( v n Mdv dσ x dt. The weak formulation of the renormalized Boltzmann equation (.47 is given by Γ(G ɛ (t Y dx Γ(G ɛ (t Y dx = t t t t Γ(G ɛ v x Y dx dt + t t Γ (G ɛ Q(G ɛ, G ɛ Y dx dt, Γ(γG ɛ Y (v n(x dσ x dt (.49 for every Y C L ( R D and every [t, t ] [0, ]. Moreover, the boundary condition is also understood in the renormalized sense: Γ(γ G ɛ = Γ (( αl(γ + G ɛ + α F ɛ on R + Σ, (.50 where the equality holds almost everywhere and in the sense of distribution. Proposition.. (Renormalized solutions [37] Let b satisfy the condition (.3. Given any initial data G in ɛ satisfying G in ( + v + log G in Mdv dx < +, (.5 O there exists at least one G ɛ 0 in C([0, ; L ( Mdv dx such that (.49 and (.50 hold for all admissible functions Γ. Moreover, G ɛ satisfies the following global entropy

12 N. JIANG AND N. MASMOUDI inequality for all t > 0: H(G ɛ (t + t R(G ɛ (s ds + 0 t 0 Ẽ (γ + G ɛ (s H(G in ɛ. (.5 DiPerna-Lions renormalized solutions are not known to satisfy many properties that one would formally expect to be satisfied by solutions to the Boltzmannn equation. In particular, the theory does not assert either the local conservation of momentum, the global conservation of energy, the global entropy equality, or even a local entropy inequality; none does it assert the uniqueness of the solution. Nevertheless, as will be stated below, it provides enough control to establish the Navier-Stokes limit. 3. Statement of the Main Results In this section we state our main results. Specifically, we derive the incompressible Navier-Stokes-Fourier limits with different boundary conditions depending on the quotient between the accommodation coefficients α and the Knudsen number. More importantly, the limit in the case of Dirichlet boundary condition is strong because of the instantaneous damping from the boundary layers. 3.. Dirichlet Boundary Condition. The main theorem of this paper is the following strong convergence to the Navier-Stokes-Fourier system with Dirichlet boundary condition when the accommodation coefficient α ɛ is much larger than the Knudsen number, i.e. α ɛ as 0. Theorem 3.. (Dirichlet Boundary Condition Let b be a collision kernel that satisfies conditions (.5, (.6, (.7, and (.. Let G in be any family satisfying (.5 and the renormalization (.30. Let g in be the associated family of fluctuations given by G in = + g in. Assume that the family g in satisfies H(G in C in, (3. and ( lim g in, vg in, ( v 0 D gin = (ρ in, u in, θ in, (3. in the sense of distributions for some (ρ in, u in, θ in L (dx ; R R D R. Let G be any family of DiPerna-Lions renormalized solutions to the Boltzmann equation (.37 that have G in as initial values, and the accommodation coefficient α ɛ satisfies α ɛ as 0. (3.3 Then the family of fluctuations g given by (.36 is relatively compact in L loc (dt; L (σmdvdx. Every limit point g of g has the infinitesimal Maxwellian form g = v u + ( v D+ θ, (3.4 where (u, θ C([0, ; L (dx ; R D R L (dt ; H (dx ; R D R with mean zero over, and it satisfies the Navier-Stokes-Fourier system with Dirichlet boundary condition (., (., and (.3, where kinematic viscosity ν and thermal conductivity κ are given by ν = (D (D+ κ = B L B. D (3.5 The initial data is given by u 0 = Pu in, θ 0 = D D+ θin D+ ρin. (3.6

13 Moreover, every subsequence g k NAVIER-STOKES-FOURIER 3 of g that converges to g as k 0 also satisfies vg k u in L p loc (dt; L (dx; R D, ( D v g k θ in L p loc (dt; L (dx; R for every p <. (3.7 Furthermore, P g is relatively compact in w-l loc (dt; w-l (σmdvdx. For every subsequence k so that g k converges to g, P g A : u u + B uθ + Cθ Â : xu B (3.8 x θ, in w-l loc(dt; w-l (σmdvdx, as k 0. Remark: In previous works [8, 9, 7], under the assumptions (3. and (3., the convergence to (3.4 and (3.7 are only in w-l. As a consequence, the convergence to the quadratic term (3.8, which is the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with the Navier-Stokes scaling, could not be obtained. In Theorem 3., by showing the acoustic waves are instantaneously damped, we justify not only the strong convergence to the leading order term g, but also weak convergence to the next order corrector ( Navier Boundary Condition. The second result treats the case of Navier boundary condition. We will make the same assumption as in the previous theorem, but instead of αɛ, we assume that α ɛ = χ with χ 0 or, more generally, α ɛ χ, as 0. (3.9 For this case, although the coupled viscous boundary layer and the Knudsen layer still have dissipative effect, however, the damping happen in a longer time scale O(. So, unlike the Dirichlet boundary condition case, the fast acoustic waves can be damped, but not instantaneously. Nevertheless, we can show the weak convergence result, thus generalize the Stokes-Fourier limit of [33] to Navier-Stokes-Fourier limit with slip Navier boundary condition. Theorem 3.. (Navier Boundary Condition Let b be a collision kernel that satisfies conditions (.5, (.6, (.7, and (.. Let G in be any family satisfying (.5 and the renormalization (.30 and g in be the associated family of fluctuations given by G in = + g in. Assume that H(G in C in, (3.0 and ( lim g in, vg in, ( v 0 D gin = (ρ in, u in, θ in (3. in the sense of distributions for some (ρ in, u in, θ in L (dx ; R R D R. Let G be any corresponding family of renormalized solutions to the Boltzmann equation (.37, where the accommodation coefficients satisfy α ɛ χ, as 0. (3. Then the family g is relatively compact in w-l loc (dt; w-l (σmdvdx. Every limit point g of g in w-l loc (dt; w-l (σmdvdx has the infinitesimal Maxwellian form g = v u + ( v D+ θ, (3.3

14 4 N. JIANG AND N. MASMOUDI where (u, θ C([0, ; L (dx ; R D R L (dt; H (dx ; R D R and satisfies the Navier- Stokes-Fourier system with Navier boundary condition (., (., and (.4, where kinematic viscosity ν and thermal conductivity κ are given by (3.5, the initial data is given by (3.6. Moreover, every subsequence g k of g that converges to g as k 0 also satisfies P vg k u in L loc(dt ; L (dx ; R D, ( D+ v g k θ in L loc(dt ; L (dx ; R. 4. Geometry of the boundary (3.4 In this section, we collect some differential geometry properties related to the boundary which can be considered as a (D dimension compact Riemannian manifold with a metric induced from the standard Euclidian metric of R D. Let T ( and T ( denote the tangent and normal bundles of in R D respectively. Furthermore, let v be the covariant derivative along the vector v R D, and A n : T ( T ( be the Wengarten map which is defined as: v n = A n (v + ( v n, for any v T (. (4. Define h : T ( T ( T ( the second fundamental form of : v u = ( v u tan + h(v, u, for any v, u T (. (4. The Wengarten map and the second fundamental form are related by the following formula: A n (u, v = h(u, v, n, for any v, u T (. (4.3 For the vector field [( x u+( x u T n] tan, we can give a more geometric representation. We have the following formula: [( x u + ( x u T n] tan = [ n u] tan + A n (u. (4.4 Proof. Take any a = a i i T (, where i = x i is the coordinate base of R D. Then a i ui x j n j = n j j u, a i i = n u, a. (4.5 a i uj x i n j = a i i u, n = h(a, u, u = A n (u, a. (4.6 There is a tubular neighborhood U = {x : dist(x, < δ} of such that the nearest point projection map is well defined. More precisely, we have the following lemma: Lemma 4.. If is a compact C k submanifold of dimension D embedded in R D, then there is δ = δ > 0 and a map π C k ({x : dist(x, < δ} ; R D such that the following properties hold: (i: for all x R D with dist(x, < δ ; (ii: π(x, x π(x T Π(x(, x π(x = dist(x,, and z x > dist(x, for any z \ {π(x} ; π(x + z x, for x, z T x (, z < δ,

15 (iii: Let HessΠ x denote the Hessian of π at x, then NAVIER-STOKES-FOURIER 5 Hessπ x (V, V = h x (V, V, for x V, V T x (, where h x is the second fundamental form of at x. For the proof of this lemma for even more general submanifold, see [4]. Because is a (D dimensional manifold, so locally π(x can be represented as π(x = (π (x,, π D (x. (4.7 Let d(x be the distance function to the boundary, i.e. d(x = dist(x, = x π(x. (4.8 It is easy to see that x d is perpendicular to the level surface of the distance function d, i.e. the set S z = {x : d(x = z}. In particular, on the boundary, x d is perpendicular to S 0 =. Without loss of generality, we can normalize the distance function so that x d(x = n(x when x. By the definition of the projection Π, we have π(x + t x d(x = π(x for t small, (4.9 and consequently, x π α x d = 0, for α =,, D. In particular, for t small enough, x π α (x T x ( when x. Next, we calculate the induced Riemannian metric from R D on. In a local coordinate system, these Riemannian metric can be represented as where g αβ =,. Noticing that π α π β can be determined by g = g αβ dπ α dπ β, (4.0 x i = πα x i, and, = δ π α x i x j ij, the metric g αβ g αβ π α x i π β x i =. (4. 5. Acoustic Modes 5.. Acoustic Operator A. Recall that the Leray s projection P on the space of divergencefree vector fields and Q on the space of gradients are defined by where Qu = x q and q solves We define Hilbert spaces x q = x u in, x q n = u n on, and P = I Q, (5. q dx = 0. H = { U = (ρ, u, θ L (dx; C C D C }, { } V = U H : x U dx <, (5. (5.3 endowed with inner product U U H = (ρ ρ + u u + Dθ θ dx, (5.4

16 6 N. JIANG AND N. MASMOUDI where f means the complex conjugate of f. Next, we define the acoustic operator A: A ρ u = x u x (ρ + θ, (5.5 θ D x u over the domain D(A = {U = (ρ, u, θ V : u n = 0 on }. (5.6 The null space of A and its orthogonal with respect to the inner product (5.4 are characterized as and Null(A = {( ϕ, w, ϕ V : x w = 0 and w n = 0 on }, (5.7 Null(A = {(ρ, u, θ V : θ = D ρ, u = xφ, for some φ H (}, (5.8 respectively. Because Null(A includes the incompressibility and Boussinesq relations, we call it incompressible regime. We will see in the next subsection that Null(A is spanned by the eigenspaces of the acoustic operator A, so we call it acoustic regime. For any U = (ρ, u, θ H, we can define Π and Π the projections to the incompressible regime Null(A and acoustic regime Null(A respectively as follows: ΠU = ( ρ D θ, Pu, D θ ρ, D+ D+ D+ D+ Π U = ( D (ρ + θ, Qu, (ρ + θ. D+ D+ ( Eigenspaces of A. The eigenvalues and eigenvectors of the acoustic operator A in a bounded domain can be constructed from those of the Laplace operator with Neumann boundary condition in the following way. D Let D+ [λk ], λ k > 0, k N be the nondecreasing sequence of eigenvalues and Φ k the orthonormal basis of L ( functions with eigenvectors of the Laplace operator N with homogeneous Neumann boundary condition: More specifically, x Φ k = D D+ [λk ] Φ k in, x Φ k n = 0 on. (5.0 0 < λ λ λ k +, as k. (5. The (non-zero eigenvalues of A read as follows ( T U τ,k D = D+ Φk, xφk, iλ τ,k D+ Φk, (5. and corresponding eigenvectors are U τ,k such that AU τ,k = iλ τ,k U τ,k, (5.3 where τ denotes either + or, and λ τ,k = τλ k. Note that U τ,k have a norm D under the inner product (5.4, then we have an orthonormal basis of the acoustic modes, i.e. D+ { ±}L Null(A D+ = Span U D τ,k k N, τ =. (5.4

17 NAVIER-STOKES-FOURIER 7 Furthermore, from U τ,k we can construct infinitesimal Maxwellians g τ,k which are in the null space of L: ( g τ,k = D D+ Φk + v xφk + v iλ τ,k D+ Φk D. (5.5 These infinitesimal Maxwellians will be the building blocks of the approximate eigenfunctions of L = L v x Conditions on Φ k. Note that Φ k, k are determined up to a constant. Some orthogonality condition is required for eigenvalues with multiplicity greater than. If λ k = λ l and k l, we require that Φ k and Φ l satisfy one of the following two conditions: Condition : ( ν x Φ k x Φ l + D+ [λk ] κφ k Φ l dσ x = 0, (5.6 and ( ν x Φ k + D+ [λk ] κ Φ k dσ x ( ν x Φ l + D+ [λk ] κ Φ l dσ x (5.7 Condition : ( x Φ k x Φ l + (D+ (D+ [λ k ] Φ k Φ l dσ x = 0, (5.8 and ( ( x Φ k + (D+ [λ k ] Φ k dσ (D+ x x Φ l + (D+ [λ k ] Φ l dσ (D+ x. (5.9 The above two conditions should be compared with condition (0 in [0]. When the eigenvalue of N is not simple, these orthogonality conditions will be needed in the construction of the kinetic-fluid coupled boundary layers. That Φ k satisfies conditions (5.6-(5.7 or (5.8-(5.9 depends on the values of β which appears in the accommodation coefficients α ɛ = χ β. When 0 β <, the condition (5.6 is required, while when β <, the condition (5.8 is required the operator A iλ τ,k. Next, for each acoustic mode k and τ = + or, we consider operator A iλ τ,k. This operator, especially its pseudo inverse, will be important in the construction of the boundary layers. Then main difficulty is that the kernel is nontrivial when then eigenvalues have multiplicity greater than. More specifically, Ker ( A iλ τ,k = Span { U τ,l : λ l = λ k}, (5.0 while its orthogonal space with respect to the inner product (5.4 is by Ker ( A iλ τ,k =Span { U +,l, U,l : λ l λ k} Span { U τ,l : λ l = λ k} Null(A. (5. We can define a bounded pseudo inverse of A iλ τ,k by ( A iλ τ,k ( 0,0 : Ker A iλ τ,k ( 0,0 Ker A iλ τ,k 0,0, (5. ( A iλ τ,k U δ,l = iλ δ,l iλ τ,k U δ,l, for any U δ,l with λ l λ k, (5.3 ( A iλ τ,k U τ,l = iλ τ,k U τ,l, for any U τ,l with λ l = λ k, (5.4

18 8 N. JIANG AND N. MASMOUDI and ( A iλ τ,k (ρ, v, ρ T = iλ τ,k (ρ, v, ρ T, (5.5 for any (ρ, v, ρ T Null(A and τ, δ {+, }. It is obvious that this pseudo inverse is a bounded operator. Lemma 5.. For each acoustic modes k and τ {+, }, let (Φ k, D D+ [λk ] be the eigenfunctions and eigenvalues of the Laplace operator N with homogeneous Neumann boundary condition defined in (5.0. Let U τ,k be defined in (5.. For given µ τ,k, F τ,k and g τ,k, consider the system for V τ,k = (ρ k, v τ,k, θ k T : (A iλ τ,k V τ,k = iµ τ,k U τ,k + F τ,k, v τ,k n = g τ,k on. V τ,k can be determined modulo Ker ( A iλ0,0 τ,k under the following two conditions: (i iµ τ,k must satisfy iµ τ,k = D+ D (5.6 g τ,k Φ k dσ x D+ D F τ,k U τ,k. (5.7 (ii F τ,k must satisfy the compatibility condition: g τ,k Φ l dσ x = F τ,k U τ,l, for λ l = λ k with k l. (5.8 Remark: Strictly speaking, (5.6 is not rigorous because v τ,k n is non-zero, so V τ,k is not in the domain of A. For notational simplicity, we still use A in (5.6 and later, just mean the expression of A in (5.5 regardless of the domain. Proof. For any g τ,k H (, there exists ṽ τ,k H ( ; R D, such that γṽ τ,k n = g τ,k, where γ is the usual trace operator from H ( ; R D to H (. We define Ṽ τ,k = V τ,k (0, ṽ τ,k, 0 T. (5.9 Then Ṽ τ,k has zero the normal velocity on the boundary, thus is in the domain of A. From (5.6, Ṽ τ,k satisfies (A iλ τ,k Ṽ τ,k = (A iλ τ,k (0, ṽ τ,k, 0 + iµ τ,k U τ,k + F τ,k. (5.30 The solvability of (5.30 is that the righthand side is in Ker ( A iλ τ,k. Inner product with U τ,k gives (5.7 while inner product with U τ,l with λ k = λ l, k l gives (5.8. Under these conditions, by applying the psedu inverse operator ( A iλ τ,k, we can uniquely solve Ṽ τ,k in Ker ( A iλ τ,k, denoted by Ṽ τ,k. Finally, V τ,k can be represented as V τ,k = D D+ λ k =λ l V τ,k U τ,l U τ,l + Ṽ (5.3 τ,k + (0, ṽ τ,k, 0 T D (0, ṽ τ,k, 0 T U τ,l U τ,l. D+ λ k =λ l At this stage, the projection of V τ,k on Ker ( A iλ τ,k, i.e. the first term in the righthand side of (5.3, can not be determined. It is easy to see that the rest part of V τ,k, which is the last three terms in (5.3, is uniquely determined, although the lifting of the trace g τ,k is not unique.

19 NAVIER-STOKES-FOURIER 9 6. Analysis of the Kinetic Boundary Layer Equation In this section, we collect three results in kinetic equations which will be frequently used later in this paper. The first one is a standard result in kinetic theory: Lemma 6.. The solvability condition for the linear kinetic equation Lg = f is f, ζ(v = 0, for ζ Span{, v, v }. (6. This result will be used to derive the equations and algebraic relations satisfied by the fluid variables. The second result we will use is quoted from [3] Lemma 4.4. Lemma 6.. The components of A Â and B B satisfy the following identities: where ν and κ are given by (3.5. A ij Âkl = ν ( δ ik δ jl + δ il δ jk D δ ijδ kl, B i Bj = D+ κδ ij, (6. The next result is about the linear boundary layer equation which will be used to determine the boundary conditions of the fluid variables. Lemma 6.3. Considering the following linear kinetic boundary layer equation in half space: (v n ξ g bb + Lg bb = K bb, in ξ > 0, (6.3 g bb 0, as ξ, with boundary condition at ξ = 0 γ + g bb = Lγ g bb + h bb, on ξ = 0, v n > 0, (6.4 In the above equations, the boundary source term h bb is taken the following form: where g and f are of the forms: and h bb = [γ + g Lγ g] + χ [ γ f Lγ f], (6.5 g =ρ g + u g v + θ g ( v D ( ζ u b n : Â + ζθ b n B + ( x u int : Â + xθ int B + S g, f = ρ f + u f v + θ f ( v D (6.6 + S f, (6.7 and where S g Null(L and S f are source terms. Then there exists a solution g bb of the equation (6.3 if the following the boundary conditions are satisfied by the fluid variables: (i On the boundary, the normal components of velocity is u g n = 0 K bb dξ. (6.8

20 0 N. JIANG AND N. MASMOUDI (ii On the boundary, the tangential components of velocities and temperature satisfy ν [ ζ u b] tan =χ [uf ] tan + [ νd(u int n ] [ ] tan tan χ [ γ S f Lγ S f ] (v nvm dv + (v nvs g tan 0 K bb v tan dξ, κ ζ θ b = D+χθ D+ f + κ n θ int χ u (D+ f n D+ 0 v n>0 v n>0 [ γ S f Lγ S f ] (v n v M dv + D+ (v n v S g K bb ( v D+ dξ, (6.9 where kinematic viscosity ν and thermal conductivity κ are given by (3.5, u tan denotes the tangential components of the vector u. Proof. The solvability conditions of the linear boundary layer equation (6.3 with boundary condition (6.4 are given by h bb η(v nm dv = K bb η dξ, (6.0 v n>0 for all η(v Null(L satisfying the condition: 0 η(r x v = η(v. (6. It is obvious that and v satisfy (6.. We assume η(v = D ai v i with v n > 0 also satisfy (6. which requires that (v n D a i n i = 0, (6. i= which implies that the vector a = (a,, a D T is perpendicular to the outer normal vector n. The formula (6.8 can be derived by taking η = in (6.0. Simple calculations show that h bb (v(v nm dv = vγg n v n>0 (6.3 = u g n vs g n. Note that S g Null(L, hence (6.8 follows. To prove (6.9, take η = D a iv i in (6.0. h bb (a i v i (n j v j χm dv = v n>0 0 K bb (a i v i dξ, (6.4 which implies that [γ + g Lγ g] (a i v i (n j v j M dv = γga ij a i n j. (6.5 v n>0

21 NAVIER-STOKES-FOURIER Using the definition of the viscosity ν in (3.5 and Lemma 6., we have [γ + g Lγ g] (a i v i (n j v j M dv v n>0 = ν[ ζ u b ] a + ν [ ( x u int + ( x u int T n ] a + S g An a. (6.6 Take η = v, we have [γ + g Lγ g] (v n v M dv = γgb n + (D + (v nγg. (6.7 v n>0 Using the definition of the thermal conductivity κ in (3.5 and Lemma 6., we have [γ + g Lγ g] (v n v M dv v n>0 (6.8 = (D + κ ζ θ b + (D + κ x θ int n + (D + u g n + (v n v S g. It can be calculated that γ f Lγ f = ( D+ θ f π u f n v (R x u f θ f v + γ S f Lγ S f, (6.9 Where R x u f = u f (u f nn is the reflection of u f with respect to the normal n(x. Thus [ γ f Lγ f] (a i v i (v nm dv v n>0 [ ] (6.0 and = π (R x u f a + [ γ S f Lγ S f ] A nm dv v n>0 [ γ f Lγ f] v v M dv v n>0 = u f n D+ π θ f + [ γ S f Lγ S f ] (v n v M dv. v n>0 a, (6. As showed in (4., the vector a is perpendicular to n. Thus the resulting vector of inner product with a is the tangential part. Finally, noticing [R x u f ] tan = [u f ] tan, we then finish the proof of the Lemma 6.3. Corollary 6.. Considering the same linear kinetic boundary layer equation (6.3, with the boundary condition: γ + g bb Lγ g bb = χ [ γ g bb Lγ g bb] + h bb, on ξ = 0, v n > 0, (6. where h bb is given the same as in (6.5, (6.6 and (6.7. Then to solve g bb, only one solvability condition is needed: On the boundary, the normal components of velocity is u g n = 0 K bb dξ, (6.3 which is the same as (6.8. Furthermore, under this condition, on the boundary, the tangential components of velocities and temperature satisfy (6.9 with replacing f by f + g bb.

22 N. JIANG AND N. MASMOUDI Proof. For equation (6.3 with boundary condition (6. we need only one solvability condition, i.e. η = in (6.0. Thus, the boundary condition for the normal direction (6.8 is derived as before. Under this solvability condition, g bb is solved. Note that the term χ [ γ g bb Lγ g bb] has the same form as in the source term χ [ γ f Lγ f]. Thus, we can put g bb into the term S f. Thus, for this case, the fluid boundary conditions are the same as before, except that the source term S f includes g bb. 7. Approximate Eigenfunctions-Eigenvalues 7.. Motivation. We define the operators L and L as L = L v x, L = L + v x. (7. Formally, L and L are dual in the following sense: L g g = g L g, (7. provided that g satisfies the Maxwell reflect boundary condition γ g = ( αl(γ + g + α γ + g on Σ, (7.3 and g satisfies the dual boundary condition γ + g = ( αl(γ g + α γ g on Σ +. (7.4 If g is the fluctuation defined in (.36, then g obeys the scaled Boltzmann equation (.39 in which L g appears with g satisfies the boundary condition (7.3. Then from (7., L g BL appears in the weak formulation of the Boltzmann equation if g BL is a test function. Thus, it is natural to construct approximate eigenfunctions and eigenvalues of L, satisfying the dual boundary condition (7.4. Specifically, we consider the kinetic eigenvalue problem: L g BL = iλ BL g BL, (7.5 with g BL satisfying the dual Maxwell boundary condition (7.4, where the accommodation coefficient α takes the value α ɛ = χ β, with β [0,. To solve the eigenvalue problem (7.5 and (7.4, we have two key observations. First, the solutions must include interior term and two boundary layer terms: the fluid viscous boundary layer terms with thickness, and the kinetic Knudsen layer term with thickness. Second, the fluid-kinetic coupled boundary layers depend on the values of β. Specifically, the constructions of the boundary layers are different for the cases 0 β < and β <. Thus, we make the ansatz for gbl and λ BL : g BL (x, v = [ ] gm,n(x, int v + gm,n(π(x, b d(x, v + gm,n(π(x, bb d(x, v m + β n, (7.6 m,n and λ BL = m,n λ m,n m + β n, (7.7 where the summations are taken over the values of m and n such that m + β n 0. In the above ansatz, g BL is consisted of three types of terms: the interior terms gm,n int, the fluid viscous boundary layer terms g b m,n, and the kinetic Knudsen layer terms g bb m,n. They are coupled through the boundary condition (7.4, which depends on the value of β, and β = is the threshold.

23 NAVIER-STOKES-FOURIER 3 Each g b m,n and g bb m,n are defined in U, the δ tubular neighborhood of in, where δ is the small number defined in Lemma 4., the projection π is defined in (4.7. After rescaling by and respectively, g b m,n, g bb m,n : R + R D R. (7.8 Both gm,n b and gm,n bb will vanish in the outside of U. Thus gm,n b and gm,n bb are required to be rapidly decreasing to 0 in the ζ and ξ respectively, which are defined by ζ = d(x and ξ = d(x. 7.. Statement of the Proposition. Now we state the proposition which is the key part of this paper. It can be considered as a kinetic analogue of the Proposition in [0]. Proposition 7.. Let be a C bounded domain of R D. Then, for every α ɛ = χ β, β [0,, every acoustic mode k, non-negative integer N, and each τ {+, }, there exists approximate eigenfunctions g τ,k,n and eigenvalues iλτ,k,n of L, and error terms R τ,k,n and rτ,k,n respectively, such that and g τ,k,n Moreover, iλ τ,k,n when 0 β <, L g τ,k,n = iλτ,k,n gτ,k,n + Rτ,k,N, (7.9 satisfy the approximate dual Maxwell boundary condition: [ ] γ + g τ,k,n Lγ g τ,k,n = α ɛ γ g τ,k,n Lγ g τ,k,n + r τ,k,n on Σ +. (7.0 has the following expansions (depending on the value of β: iλ τ,k,n = iλτ,k + iλ τ,k + O( β, with Re(iλ τ,k < 0 (7. iλ τ,k,n = iλτ,k + iλ τ,k β β + O(, with Re(iλ τ,k β < 0 (7. when β <. Furthermore, for all < r, p, we have error estimates: R τ,k,n L r (dx,l p (a p Mdv = O ( N, (7.3 and g τ,k,n gτ,k,int ( 0,0 L r (dx,l p (a p Mdv = O g τ,k,n gτ,k,int 0,0 L r (dx,l p (a p Mdv = O r, for 0 β <, ( β + r, for β <, (7.4 where g τ,k,int 0,0 is defined in (5.5. We also have the boundary error estimates: r τ,k,n L r (dσ x,l p (a p Mdv = O ( N + β, for 0 < β <, (7.5 and r τ,k,n = 0, for β = 0.

24 4 N. JIANG AND N. MASMOUDI 7.3. Main Idea of the Proof. For each (m, n, gm,n int is the summation of the hydrodynamic parts Pgm,n, int i.e. their projections on Null(L, and the kinetic parts P gm,n, int the projections on Null(L. The hydrodynamic part is denoted by ( Pgm,n int = ρ int m,n + v u int m,n + v θm,n int, (7.6 D and we call Um,n int = (ρ int m,n, u int m,n, θm,n int T the fluid variables of gm,n. int It can be shown that the coefficients of P gm,n int are in terms of Um int,n and their derivatives for (m, n < (m, n which means m m, n n and m + n < m + n. Thus, we need only to solve Um,n int for all (m, n to determine gm,n. int Similar notations will be used for gm,n, b and also we need only to solve Um,n. b We put the ansatz (7.6 and (7.7 into the equation (7.5, then collect the same order terms. For all 0 β <, the leading order of the kinetic-fluid boundary layer g int hydrodynamic, which means g0,0 int is completely determined by U int U int 0,0 satisfies the equation 0,0 is 0,0. we can derive that AU0,0 int = iλ 0,0 U0,0 int. (7.7 For (7.7 we can consider the following two cases: Case : λ 0,0 = 0, which implies that ρ int 0,0 + θ0,0 int = 0 and x u int 0,0 = 0. Starting from here, we can construct the boundary layer g BL in the incompressible modes; which we call the boundary layer Case : by comparing the equations (7.7 and (5.3, iλ 0,0 is an eigenvalue of the acoustic operator A, i.e. λ 0,0 = λ τ,k and U0,0 int = U τ,k, where k 0 is the acoustic modes, and τ denotes either + or. Starting from here, we can construct the boundary layer g BL which we call the boundary layer in the acoustic modes. Because the main goal of the current paper is about the interaction between the acoustic waves and the boundary layers, we only consider the kinetic-fluid boundary layers in the acoustic modes for each k 0 and each τ {+, }. Consequently, we add superscript τ, k for each term in the ansatz. We will consider the boundary layers in the incompressible modes in the forthcoming paper [5]. The basic strategy to solve all terms in the ansatz (7.6 and (7.7 is the following: ( g τ,k,bb m,n satisfies the linear kinetic boundary layer equation (6.3. Applying Lemma 6.3, the solvability conditions for gm,n τ,k,bb give the boundary conditions (both normal and tangential directions of U τ,k,int m,n (m, n, (m, n and (m, n depend on the values of β; and U τ,k,b m,n. The relations between the pairs ( Um,n τ,k,int and iλ τ,k m,n can be solved by applying Lemma 5., where only the boundary condition in the normal direction u τ,k,int m,n n is needed; (3 If the multiplicity of the eigenvalue λ τ,k 0,0 is greater than, applying Lemma 5., Um,n τ,k,int can be only solved modulo Ker(A iλ τ,k 0,0. The components of Um,n τ,k,int in Ker(A iλ τ,k 0,0 will be solved by applying Lemma (5. again in later steps; (4 u τ,k,int m,n n = u τ,k,b m,n n + some known terms, while u τ,k,b m,n n can be derived from the ordinary differential system of Um,n; τ,k,b (5 To solve the ordinary differential system in (3, the boundary conditions on the tangent direction for U τ,k,b m,n are needed. These boundary conditions are coupled with U τ,k,int m,n, and these coupling relations are given in (;