J. Math. Pures Appl. 9 009 508 55 www.elsevier.com/locate/matpur The incompressible Navier Stokes limit of the Boltzmann equation for hard cutoff potentials François Golse a,b,, Laure Saint-Raymond c a Ecole polytechnique, Centre de mathématiques L. Schwartz, F98 Palaiseau cedex, France b Université Paris Diderot - Paris 7, Laboratoire J.-L. Lions, 4 place Jussieu, Boîte courrier 87, F755 Paris cedex 05, France c Ecole Normale Supérieure, Département de Mathématiques et Applications, 45 rue d Ulm, F7530 Paris cedex 05, France Received October 008 Available online 0 April 009 Abstract The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 55 004 8 6] for Maxwell molecules. 009 Elsevier Masson SAS. All rights reserved. Résumé On montre dans cet article qu à extraction de sous-suites près, les suites de solutions renormalisées de l équation de Boltzmann convergent vers des limites décrites par les solutions de Leray des équations de Navier Stokes dans la limite où les nombres de Mach et de Knudsen sont petits et asymptotiquement équivalents. Cette convergence est établie ici dans le cas de potentiels durs avec troncature angulaire au sens de H. Grad, ce qui complète les résultats antérieurs des mêmes auteurs [Invent. Math. 55 004 8 6] pour le cas des molécules Maxwelliennes. 009 Elsevier Masson SAS. All rights reserved. MSC: 35Q35; 35Q30; 8C40 Keywords: Hydrodynamic limit; Boltzmann equation; Incompressible Navier Stokes equations; Renormalized solutions; Leray solutions. Introduction The subject matter of this paper is the derivation of the Navier Stokes equations for incompressible fluids from the Boltzmann equation, which is the governing equation in the kinetic theory of rarefied, monatomic gases. In the kinetic theory of gases founded by Maxwell and Boltzmann, the state of a monatomic gas is described by the molecular number density in the single-body phase space, f f t, x, v 0 that is the density with respect to the Lebesgue measure dx dv of molecules with velocity v and position x at time t 0. Henceforth, we restrict * Corresponding author at: Ecole polytechnique, Centre de mathématiques L. Schwartz, F98 Palaiseau cedex, France. E-mail addresses: golse@math.polytechnique.fr F. Golse, saintray@dma.ens.fr L. Saint-Raymond. 00-784/$ see front matter 009 Elsevier Masson SAS. All rights reserved. doi:0.06/j.matpur.009.0.03
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 509 our attention to the case where the gas fills the Euclidean space. For a perfect gas, the number density f satisfies the Boltzmann equation: t f + v x f = Bf, f, x, v,. where Bf, f is the Boltzmann collision integral. The Boltzmann collision integral acts only on the v variable in the number density f. In other words, B is a bilinear operator defined on functions of the single variable v, and it is understood that the notation Bf, f t, x, v designates B f t, x,, f t, x, v.. For each continuous f f v rapidly decaying at infinity, the collision integral is given by: Bf, f v = f v f v f vf v bv v, ω dv dω,.3 where S v v v, v, ω = v v v ωω, v v v, v, ω = v + v v ωω..4 The collision integral is then extended by continuity to wider classes of densities f, depending on the specifics of the function b. The function b bv v, ω, called the collision kernel, is measurable, a.e. positive, and satisfies the symmetry: bv v, ω = bv v, ω = bv v, ω a.e. in v, v, ω..5 Throughout the present paper, we assume that b satisfies: β 0 < bz, ω C b + z cosẑ, ω a.e. on S, z bz, ω dω a.e. on,.6 + z S C b for some C b > 0 and β [0, ]. The bounds.6 are verified by all collision kernels coming from a repulsive, binary intermolecular potential of the form Ur = U 0 /r s with Grad s angular cutoff see [5] and s 4. Such power-law potentials are said to be hard if s 4 and soft otherwise: in other words, we shall be dealing with hard cutoff potentials. The case of a hard-sphere interaction binary elastic collisions between spherical particles corresponds with bz, ω = z ω ;.7 it is a limiting case of hard potentials that obviously satisfies.6, even without Grad s cutoff. At the time of this writing, the Boltzmann equation has been derived from molecular dynamics i.e. Newton s equations of classical mechanics applied to a large number of spherical particles in the case of hard sphere collisions, by O.E. Lanford [6], see also [9] for the case of compactly supported potentials. Thus the collision kernel b given by.7 plays an important role in the mathematical theory of the Boltzmann equation. The only nonnegative, measurable number densities f such that Bf, f = 0 are Maxwellian densities, i.e. densities of the form: f v = R v U e Θ =: M πθ 3/ R,U,Θ v.8 for some R 0, Θ> 0 and U. Maxwellian densities whose parameters R, U, Θ are constants are called uniform Maxwellians, whereas Maxwellian densities whose parameters R, U, Θ are functions of t and x are referred to as local Maxwellians. Uniform Maxwellians are solutions of.; however, local Maxwellians are not solutions of. in general. The incompressible Navier Stokes limit of the Boltzmann equation can be stated as follows.
50 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Navier Stokes limit of the Boltzmann equation. Let u in u in x be a divergence-free vector field on. For each > 0, consider the initial number density: f in x, v = M,u in x,v..9 Notice that the number density f in is a slowly varying perturbation of order of the uniform Maxwellian M,0,. Let f solve the Boltzmann equation. with initial data.9, and define: u t, x := t vf, x,v dv..0 Then, in the limit as 0 + and possibly after extracting a converging subsequence, the velocity field u satisfies u u in D R +, where u is a solution of the incompressible Navier Stokes equations: with initial data t u + div x u u + x p = ν x u, x, t > 0, div x u = 0,. u t=0 = u in.. The viscosity ν is defined in terms of the collision kernel b, by some implicit formula, that will be given below. More general initial data than.9 can actually be handled with our method: see below for a precise statement of the Navier Stokes limit theorem. Hydrodynamic limits of the Boltzmann equation leading to incompressible fluid equations have been extensively studied by many authors. See in particular [] for formal computations, and [,3] for a general program of deriving global solutions of incompressible fluid models from global solutions of the Boltzmann equation. The derivation of global weak Leray solutions of the Navier Stokes equations from global weak renormalized à la DiPerna Lions solutions of the Boltzmann equation is presented in [3], under additional assumptions on the Boltzmann solutions which remained unverified. In a series of later publications [0,,4,0] some of these assumptions have been removed, except one that involved controlling the build-up of particles with large kinetic energy and possible concentrations in the x-variable. This last assumption was removed by the second author in the case of the model BGK equation [3,4], by a kind of dispersion argument based on the fact that relaxation to local equilibrium improves the regularity in v of number density fluctuations. Finally, a complete proof of the Navier Stokes limit of the Boltzmann equation was proposed in [3]. In this paper, the regularization in v was obtained by a rather different argument specifically, by the smoothing properties of the gain part of Boltzmann s collision integral since not much is known about relaxation to local equilibrium for weak solutions of the Boltzmann equation. While the results above holds for global solutions of the Boltzmann equation without restriction on the size or symmetries of its initial data, earlier results had been obtained in the regime of smooth solutions [7,5]. Since the regularity of Leray solutions of the Navier Stokes equations in 3 space dimensions is not known at the time of this writing, such results are limited to either local in time solutions, or to solutions with initial data that are small in some appropriate norm. The present paper extends the result of [3] to the case of hard cutoff potentials in the sense of Grad i.e. assuming that the collision kernel satisfies.6. Indeed, [3] only treated the case of Maxwell molecules, for which the collision kernel is of the form: bz, ω = cosz, ω b cosz, ω with C b C. The method used in the present paper also significantly simplifies the original proof in [3] in the case of Maxwell molecules. Independently, C.D. Levermore and N. Masmoudi have extended the analysis of [3] to a wider class of collision kernels that includes soft potentials with a weak angular cutoff in the sense of DiPerna Lions: see [7]. Their proof is written in the case where the spatial domain is the 3-torus /Z 3.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 5 In the present paper, we handle the case of the Euclidean space, which involves additional technical difficulties concerning truncations at infinity and the Leray projection on divergence-free vector fields see Appendix C below.. Formulation of the problem and main results.. Global solutions of the Boltzmann equation The only global existence theory for the Boltzmann equation without extra smallness assumption on the size of the initial data known to this date is the DiPerna Lions theory of renormalized solutions [8,8]. We shall present their theory in the setting best adapted to the hydrodynamic limit considered in the present paper. All incompressible hydrodynamic limits of the Boltzmann equation involve some background, uniform Maxwellian equilibrium state whose role from a physical viewpoint is to set the scale of the speed of sound. Without loss of generality, we assume this uniform equilibrium state to be the centered, reduced Gaussian density: Mv := M,0, v = π 3/ e v /.. Our statement of the Navier Stokes limit of the Boltzmann equation given above suggests that one has to handle the scaled number density: t F t, x, v = f, x,v,. where f is a solution of the Boltzmann equation.. This scaled number density is a solution of the scaled Boltzmann equation: Throughout the present section, is any fixed, positive number. t F + v x F = BF,F, x, v, t > 0..3 Definition.. A renormalized solution of the scaled Boltzmann equation.3 relatively to the global equilibrium M is a function, F C R +,L loc such that Γ F M BF, F L loc R+, and which satisfies, M t + v x Γ F M F = Γ BF, F,.4 M for each normalizing nonlinearity: Γ C R + such that Γ z C + z, z 0. The DiPerna Lions theory is based on the only a priori estimates that have natural physical interpretation. In particular, the distance between any number density F F x, v and the uniform equilibrium M is measured in terms of the relative entropy: F H F M := F ln F + M dx dv..5 M Introducing hz = + z ln + z z 0, z >,.6
5 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 we see that H F M = h F M M dv dx 0, with equality if and only if F = M a.e. in x, v. While the relative entropy measures the distance of a number density F to the particular equilibrium M, the local entropy production rate measures the distance of F to the set of all Maxwellian densities. Its expression is as follows: EF = F F 4 FF ln F bv v, ω dv dv dω..7 FF S F The DiPerna Lions existence theorem is the following statement [8,8]. Theorem.. Assume that the collision kernel b satisfies Grad s cutoff assumption.6 for some β [0, ]. Let F in F in x, v be any measurable, a.e. nonnegative function on such that Then, for each > 0, there exists a renormalized solution, H F in M < +..8 F C R +,L loc R R 3, relatively to M of the scaled Boltzmann equation.3 such that Moreover, F satisfies: a the continuity equation and b the entropy inequality t H F Mt + t 0 F t=0 = F in. F dv + div x vf dv = 0,.9 EF s, x ds dx H F in M, t > 0..0 Besides the continuity equation.9, classical solutions of the scaled Boltzmann equation.3 with fast enough decay as v would satisfy the local conservation of momentum, t vf dv + div x v vf dv = 0,. as well as the local conservation of energy, t v F dv + div x v v F dv = 0.. Renormalized solutions of the Boltzmann equation.3 are not known to satisfy any of these conservation laws except that of mass i.e. the continuity equation.9. Since these local conservation laws are the fundamental objects in every fluid theory, we expect to recover them somehow in the hydrodynamic limit 0 +.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 53.. The convergence theorem It will be more convenient to replace the number density F by its ratio to the uniform Maxwellian equilibrium M; also we shall be dealing mostly with perturbations of order of the uniform Maxwellian state M. Thus we define: G = F M, g = G..3 Likewise, the Lebesgue measure dv will be replaced with the unit measure M dv, and we shall systematically use the notation: φ = φvmv dv, for each φ L M dv..4 For the same reason, quantities like the local entropy production rate involve the measure: dµv, v, ω = bv v, ωm dv M dv dω, dµv, v, ω =,.5 S whose normalization can be assumed without loss of generality, by some appropriate choice of physical units for the collision kernel b. We shall also use the notation: ψ = ψv, v, ω dµv, v, ω for ψ L S, dµ..6 S From now on, we consider solutions of the scaled Boltzmann equation.3 that are perturbations of order about the uniform Maxwellian M. This is conveniently expressed in terms of the relative entropy. F in x, v be a family of measurable, a.e. nonnegative func- Proposition.3 Uniform a priori estimates. Let F in tions such that sup >0 H F in M = C in < +..7 Consider a family F of renormalized solutions of the scaled Boltzmann equation.3 with initial data, Then F t=0 = F in..8 a the family of relative number density fluctuations g satisfies h g t, x, dx C in,.9 where h is the function defined in.6; b the family G is bounded in L R + ; L M dv dx: G dx C in ;.0 c hence the family g is relatively compact in L loc dt dx; L M dv; d the family of relative number densities G satisfies the entropy production or dissipation estimate: 0 G G G G dx dt C in..
54 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Proof. The entropy inequality implies that H F Mt = which is the estimate a. The estimate b follows from a and the elementary identity: From the identity, hg t, x dx H F in M C in, hz z = z ln z z z + z = z ln z z z = z z ln z z + 0. and the bound b, we deduce the weak compactness statement c. Finally, the entropy inequality implies that Observing that EF = 4 = 4 and using the elementary inequality, G G g = +,. 0 F S G G G G ln EF s, x dx ds C in 4. F F F F ln F F F G G G G, 4 X Yln X Y X Y, X, Y > 0, leads to the dissipation estimate d. bv v, ω dv dv dω Our main result in the present paper is a description of all limit points of the family of number density fluctuations g. Theorem.4. Let F in be a family of measurable, a.e. nonnegative functions defined on satisfying the scaling condition.7. Let F be a family of renormalized solutions relative to M of the scaled Boltzmann equation.3 with initial data.8, for a hard cutoff collision kernel b that satisfies.6 with β [0, ]. Define the relative number density G and the number density fluctuation g by the formulas.3. Then, any limit point g in L loc dt dx; L M dv of the family of number density fluctuations g is an infinitesimal Maxwellian of the form, gt, x, v = ut, x v + θt, x v 5, where the vector field u and the function θ are solutions of the Navier Stokes Fourier system: t u + div x u u + x p = ν x u, div x u = 0, t θ + div x uθ = κ x θ,.3
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 55 with initial data u in = w lim θ in = w lim 0 P 0 5 v vf in, dv F in M dv,.4 where P is the Leray orthogonal projection in L on the space of divergence-free vector fields and the weak limits above are taken along converging subsequences. Finally, the weak solution u, θ of.3 so obtained satisfies the energy inequality: ut, x + 5 θt, x dx + 4 t 0 ν x u + 5 κ xθ dx lim 0 + H F in M..5 The viscosity ν and thermal conductivity κ are defined implicitly in terms of the collision kernel b by the formulas.7 below. There are several ways of stating the formulas giving ν and κ. Perhaps the quickest route to arrive at these formulas is as follows. Consider the Dirichlet form associated to the Boltzmann collision integral linearized at the uniform equilibrium M: D M Φ := 8 Φ + Φ Φ Φ..6 The notation designates the Euclidean norm on when Φ is vector-valued, or the Frobenius norm on M 3 R defined by A =tracea A / when Φ is matrix-valued. Let D be the Legendre dual of D, defined by the formula D Ψ := sup Ψ Φ DΦ, Φ where the notation Φv Ψ v designates the Euclidean inner product in whenever Φ, Ψ are vector valued, or the Frobenius inner product in M 3 R whenever Φ, Ψ are matrix-valued the Frobenius inner product being defined by A B = tracea B. With these notations, one has: ν := v 5 D v 3 v I, κ := 4 5 D v v 5..7 The weak solutions of the Navier Stokes Fourier system obtained in Theorem.4 satisfy the energy inequality.5 and thus are strikingly similar to Leray solutions of the Navier Stokes equations in 3 space dimensions of which they are a generalization. The reader is invited to check that, whenever the initial data F in is chosen so that H F in M u in x dx as 0 +, then the vector field u obtained in Theorem.4 is indeed a Leray solution of the Navier Stokes equations. More information on this kind of issues can be found in [3]. See in particular the statements of Corollary.8 and Theorem.9 in [3], which hold verbatim in the case of hard cutoff potentials considered in the present paper, and which are deduced from Theorem.4 as explained in [3]..3. Mathematical tools and notations for the hydrodynamic limit An important feature of the Boltzmann collision integral is the following symmetry relations the collision symmetries. These collision symmetries are straightforward, but fundamental consequences of the identities.5
56 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 verified by the collision kernel, and can be formulated in the following manner. Let Φ Φv, v be such that Φ L S, dµ. Then Φv, v dµv, v, ω = Φv, v dµv, v, ω S S = v v, v, ω, v v, v, ω dµv, v, ω,.8 S Φ where v and v are defined in terms of v, v,ω by the formulas.4. Since the Navier Stokes limit of the Boltzmann equation is a statement on number density fluctuations about the uniform Maxwellian M, it is fairly natural to consider the linearization at M of the collision integral. First, the quadratic collision integral is polarized into a symmetric bilinear operator, by the formula The linearized collision integral is defined as BF, G := BF + G, F + G BF, F BG, G. Lf = M BM, Mf..9 Assuming that the collision kernel b comes from a hard cutoff potential in the sense of Grad.6, one can show see [5] for instance that L is a possibly unbounded, self-adjoint, nonnegative Fredholm operator on the Hilbert space L, M dv with domain, and nullspace, and that L can be decomposed as DL = L,a v M dv, Ker L = span {,v,v,v 3, v },.30 Lgv = a v gv Kgv, where K is a compact integral operator on L M dv and a = a v is a scalar function called the collision frequency that satisfies, for some C>, 0 <a a v a + + v β. In particular, L has a spectral gap, meaning that there exists C>0 such that f Lf C f Πf L Ma dv,.3 for each f DL, where Π is the orthogonal projection on Ker L in L, M dv, i.e. Πf = f + vf v + 3 v f v 3..3 The bilinear collision integral intertwined with the multiplication by M is defined by: Qf, g = M BMf, Mg..33 Under the only assumption that the collision kernel satisfies.5 together with the bound, β, bz, ω dω a + + z.34 S Q maps continuously L, M + v β dv into L,a M dv. Indeed, by using the Cauchy Schwarz inequality and the collision symmetries.8 entailed by.5:
Qg, h L a M dv = F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 57 a v S g h + g h gh g hbv v, ωm dv dω M dv a v bv v, ωm dv dω 4 S h + g h gh g h bv v, ωm dv dω M dv S g sup v a v bv v, ωm dv dω S g h + g h + gh + g h dµv, v, ω C S gh + g h bv v, ω dω MM dv dv S 4C g L + v β M dv h L + v β M dv..35 Another important property of the bilinear operator Q is the following relation: which follows from differentiating twice both sides of the equality, Qf, f = L f for each f Ker L,.36 BM R,U,Θ, M R,U,Θ = 0, with respect to R 0, Θ> 0 and U see for instance [], formulas 59 60 for a quick proof of this identity. Young s inequality. Since the family of number density fluctuations g satisfies the uniform bound a in Proposition.3 and the measure M dv has total mass, the fluctuation g can be integrated against functions of v with at most quadratic growth at infinity, by an argument analogous to the Hölder inequality. This argument will be used in various places in the proof, and we present it here for the reader s convenience. To the function h in.6, we associate its Legendre dual h defined by: h ζ := sup z> Thus, for each ζ> 0 and each z>, one has: since ζz hz = e ζ ζ. ζ z h z + h ζ hz + h ζ,.37 h z hz, z >. The inequality.37 is referred to as the Young inequality by analogy with the classical Young inequality: which holds whenever < p, q < satisfy p + q =. ζz zp p + ζ q, z, ζ > 0, q Notations regarding functional spaces. Finally, we shall systematically use the following notations. First, Lebesgue spaces without mention of the domain of integration always designate that Lebesgue space on the largest domain of integration on which the measure is defined. For instance:
58 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 L p M dv designates L p ; M dv, L p M dv dx designates L p ; M dv dx, L p dµ designates L p S ; dµ. When the measure is the Lebesgue measure, we shall simply denote: L p x := L p ; dx, L p t,x := L p R + ; dt dx. Whenever E is a normed space, the notations Oδ E and oδ E designate a family of elements of E whose norms are Oδ or oδ. For instance O E designates a bounded family in E, while o E designates a sequence that converges to 0 in E. Although L p loc spaces are not normed spaces, we designate by the notation Oδ L p loc Ω a family f L p loc Ω such that, for each compact K Ω, The notation oδ p L loc Ω is defined similarly..4. Outline of the proof of Theorem.4 f L p K = Oδ. In terms of the fluctuation g, the scaled Boltzmann equation.3 with initial condition.8 can be put in the form: t g + v x g = Lg + Qg,g, g t=0 = g in..38 Step : We first prove that any limit point g of the family of fluctuations g as 0 + satisfies, g = Πg, where Π is the orthogonal projection on the nullspace of L defined in.3. Hence, the limiting fluctuation g is an infinitesimal Maxwellian, i.e. of the form: gt, x, v = ρt, x + ut, x v + θt, x v 3..39 The limiting form of the continuity equation.9 is equivalent to the incompressibility condition on u: Step : In order to obtain equations for the moments, div x u = 0. ρ = g, u= vg, and θ = 3 v g, we pass to the limit in approximate local conservation laws deduced from the Boltzmann equation in the following manner. Besides the square-root renormalization, we use a renormalization of the scaled Boltzmann equation.3 based on a smooth truncation γ such that Define: γ C R +, [0, ], γ [0, 3 ], γ [,+ 0..40 ˆγ z = d dz z γ z..4 Notice that supp ˆγ [0, ], ˆγ [0, 3 ], and ˆγ L + γ L..4 We use below the notation γ and ˆγ to denote respectively γ G and ˆγ G.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 59 We also use a truncation of high velocities, defined as follows: given k > 6, we set: K = k ln..43 For each continuous scalar function, or vector- or tensor-field ξ ξv, we denote by ξ K the following truncation of ξ: ξ K v = ξv v K..44 Renormalizing the scaled Boltzmann equation.3 with the nonlinearity Γ Z = Z γ Z, we arrive at the following form of.38: t g γ + v xg γ = 3 ˆγ QG,G. Multiplying each side of the equation above by ξ K, and averaging in the variable v leads to with t ξ K g γ + div x vξ K g γ = 3 ξk ˆγ G G G G..45 Henceforth we use the following notations for the fluxes of momentum or energy: Likewise, we use the notation, F ζ = ζ K g γ,.46 ζv = Av := v 3 v I, or ζv = Bv := v v 5. D ξ = 3 ξk ˆγ G G G G,.47 for the conservation defect corresponding with the truncated quantity ξ ξv, where ξ span{,v,v,v 3, v }. The Navier Stokes motion equation is obtained by passing to the limit as 0 modulo gradient fields in Eq..45 for ξv = v j, j =,, 3, recast as t v K g γ + div x F A + x 3 v K g γ = D v,.48 while the temperature equation is obtained by passing to the limit in that same equation with ξv = v 5, i.e. in t v 5 g K γ + div x F B = D v 5..49 For the mathematical study of that limiting process, the uniform a priori estimates obtained from the scaled entropy inequality are not sufficient. Our first task is therefore to improve these estimates using both: a the properties of the collision operator see Section 3, namely a suitable control on the relaxation based on the coercivity estimate.3: φlφ C φ Πφ L Ma dv, b and the properties of the free transport operator see Section 4, namely dispersion and velocity averaging. With the estimates obtained in Sections 3 4, we first prove in Section 5 that the conservation defects vanish asymptotically: D ξ 0 in L loc dt dx, ξ span{ v,v,v 3, v }. Next we analyze the asymptotic behavior of the flux terms. This requires splitting these flux terms into a convection and a diffusion part Section 6, G F ζ ζ Π + ˆζ Q G, G 0 in L loc dt dx,
50 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 where ˆζ is the unique solution in Ker L of the Fredholm integral equation, Lˆζ = ζ. For instance, the tensor field A and the vector field B defined by, satisfy Av := v v 3 v I, Bv := v 5 v.50 A Ker L, B Ker L.5 componentwise, so that there exists a unique tensor field  and a unique vector field ˆB such that L = A, L ˆB = B,  and ˆB Ker L,.5 The diffusion terms are easily proved to converge towards the dissipation terms in the Navier Stokes Fourier system: ˆζ Q G, G ˆζ v x g in L loc dt dx. The formulas.7 for the viscosity ν and heat conduction κ are easily shown to be equivalent to ν =  : A, κ = 0 5 ˆB B..53 The nonlinear convection terms require a more careful treatment, involving in particular some spatial regularity argument and the filtering of acoustic waves see Section 7. 3. Controls on the velocity dependence of the number density fluctuations The goal of this section is to prove that the square number density fluctuation or more precisely the following variant thereof, G, is uniformly integrable in v with the weight + v p for each p<. In our previous work [3], we obtained this type of control for p = 0 only, by a fairly technical argument see Section 6 of [3]. Basically, we used the entropy production bound to estimate some notion of distance between the number density and the gain part of a fictitious collision integral. The conclusion followed from earlier results by Grad and Caflisch on the v-regularity of the gain term in Boltzmann s collision integral linearized at some uniform Maxwellian state. Unfortunately, this method seems to provide only estimates without the weight + v β with β as in.6 that is crucial for treating hard potentials other than the case of Maxwell molecules. Obtaining the weighted estimates requires some new ideas presented in this section. The first such idea is to use the spectral gap estimate.3 for the linearized collision integral. Instead of comparing the number density to some variant of the local Maxwellian equilibrium as in the case of the BGK model equation, treated in [3,4], or in the case of the Boltzmann equation with Maxwell molecules as in [3] we directly compare the number density fluctuation to the infinitesimal Maxwellian that is its projection on hydrodynamic modes. The lemma below provides the basic argument for arriving at such estimates. Lemma 3.. Under the assumptions of Theorem.4, one has: G G Π O L + O G L t,x M dv L M dv. 3.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 5 Proof. In order to simplify the presentation we first define some fictitious collision integrals L and Q, Lg = g + g g g M bv v, ω dv dω, S Qg, h = h + g h gh g hm bv v, ω dv dω, S g obtained from L and Q by replacing the original collision kernel b with bz, ω = Start from the elementary formula: G G L = Q, bz, ω + S bz, ω dω. G Q G, G. 3. Multiplying both sides of this equation by I Π G and using the spectral gap estimate.3 leads to G G Π L M dv Q G G, + Q G, G 3.3 L M dv. Denote By definition of b, one has: L M dv d µv, v, ω = MM bv v, ω dω dv dv. S bv v, ω dω. Hence Q is continuous on L M dv: by.35 Qg, h L M dv g L M dv h L M dv. Notice that b verifies.5 as does b. Plugging this estimate in 3.3 leads to G G Π C G L M dv L M dv + Q G, G L M dv Finally, applying the Cauchy Schwarz inequality as in the proof of.35, one finds that Q G, G sup bv v, ωm dv dω L M dv v 4 G G G G d µv, v, ω S S 4 G G G G dµv, v, ω, S since 0 b b. By the entropy production estimate d in Proposition.3, the inequality above implies that Q G, G = O L. L t,x M dv This estimate and 3.4 entail the inequality 3.. 3.4
5 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Notice that we could have used directly L and Q instead of their truncated analogues L and Q, obtaining bounds in weighted L spaces by some loop argument, unfortunately much more technical than the proof above. The main result in this section and one of the key new estimate in this paper is: Proposition 3.. Under the assumptions of Theorem.4, for each T>0, each compact K, and each p<, the family p G + v is uniformly integrable in v on [0,T] K with respect to the measure dt dx M dv. This means that, for each η> 0, there exists α> 0 such that, for each measurable ϕ ϕx, v verifying: one has: T 0 K Proof. Start from the decomposition: J := + v p G G pπ G = + v + + v p ϕ L x,v and ϕ L x L v α, ϕ p G + v M dv dx dt η. G + v p We recall from the entropy bound b in Proposition.3 that G = O L t L dx M dv G Π G so that, by definition.3 of the hydrodynamic projection Π G Π = O L t L x Lq M dv, 3.6 for all q<+. Therefore the first term in the right-hand side of 3.5 satisfies, I = G p G + v Π = O L t L x Lr M dv, 3.7 for all 0 p<+ and r<. In order to estimate the second term in the right-hand side of 3.5, we first remark that, for each δ> 0, each p< and each q<+, there exists some C = Cp, q, δ such that + v p/ G Indeed, by Young s inequality and Proposition.3a, + v p G = Oδ L t L dx M dv + O Cp, q, δ δ G + v p δ δ hg + δ h + v p δ = O δ L t L dx M dv + δ exp. 3.5. 3.8 L t,x Lq M dv + v p δ.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 53 We next use 3.8 with the two following observations: first, the obvious continuity statement 3.6. Also, because of 3. and the entropy bound b in Proposition.3, one has: G G Π = O L L loc dt dx. 3.9 M dv Hence Now p + v G p + v G G Π δ hg / + v p/ G G Π + δ p/ + v p G G + v exp δ Π =: II + III. II δ hg + δ + v p Π G + δ + v p G = Oδ L t L M dv dx + Oδ L t L M dv dx + δj. On the other hand III L loc dt dx;l r M dv δ p/ + v p + v exp δ G G Π with r = q q+. Putting all these controls together shows that = O δcp, q, δ, L q M dv L loc dt dx;l M dv J I + II + III = O L t L x Lr M dv + Oδ L t L M dv dx + Oδ L t L M dv dx + δj + O δcp, q, δ L loc dt dx;lr M dv, 3.0 i.e. δ + v p G O L t L x Lr M dv + O δcp, q, δ L loc dt dx;lr M dv + Oδ L t L M dv dx, which entails the uniform integrability in v stated in Proposition 3.. Remark. Replacing the estimate for II above with II 8δ hg + δ p G + v Π 8 and choosing δ = 4 in 3.0 shows that + δ 8 = Oδ L t L M dv dx + Oδ L t L M dv dx + δ 8 J, + v G + v p G is bounded in L loc dt dx; L M dv.
54 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 In [3], the Navier Stokes limit of the Boltzmann equation is established assuming the uniform integrability in [0,T] K for the measure dt dx M dv of a quantity analogous to the one considered in this bound. As we shall see, the Navier Stokes Fourier limit of the Boltzmann equation is derived in the present paper by using only the weaker information in Proposition 3.. 4. Compactness results for the number density fluctuations The following result is the main technical step in the present paper. Proposition 4.. Under the assumptions in Theorem.4, for each T>0, each compact K and each p<, the family of functions, G p, + v is uniformly integrable on [0,T] K for the measure dt dx M dv. This proposition is based on the uniform integrability in v of that same quantity, established in Proposition 3., together with a bound on the streaming operator applied to a variant of the number density fluctuation stated in Lemma 4.. Except for some additional truncations, the basic principle of the proof is essentially the same as explained in Lemma 3.6 of [3] which is recalled in Appendix B. In other words, while the result of Proposition 3. provides some kind of regularity in v only for the number density fluctuation, the bound on the free transport part of the Boltzmann equation gives the missing regularity in the x-variable. The technical difficulty comes from the fact that the square-root renormalization Γ Z = Z is not admissible for the Boltzmann equation due to the singularity at Z = 0. We will therefore use an approximation of the square-root, namely z z + α for some α ], [. Lemma 4.. Under the assumptions in Theorem.4, for each α> 0, one has: t + v x α + G = O α/ L M dv dx dt + O L + v β M dv dx dt + O L loc dt dx;l + v β M dv. Proof. Start from the renormalized form of the scaled Boltzmann equation.3, with normalizing function: Γ Z = α + Z. This equation can be written as t + v x α + G = α QG,G = Q + G + Q, 4. with Q = α G + G G bv G G v, ω dω M dv, Q = G α G G + G G G G bv v, ω dω M dv. 4. The entropy production estimate d in Proposition.3 and the obvious inequality α + G α/ imply that Q L M dv dx dt Cin α/. 4.3
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 55 On the other hand Q = G G G G G G α + G bv v, ω dω M dv. Write G G = +. Apply the Cauchy Schwarz inequality as in the proof of.35, then G G G G G bv v, ωm dv dω Therefore β sup + v v β + sup + v v L + v β M dv / G G G G / bv v, ωm dv dω G M bv v, ω dv dω G G G G G C + G bv v, ωm dv L M + v β dv / G G G G /. L + v β M dv G G G G /, because of the upper bound in Grad s cut-off assumption.6. Hence, on account of Proposition 3. and the entropy production estimate d in Proposition.3 Q = O L + v β M dv dx dt + O L loc dt dx;l + v β M dv. 4.4 Both estimates 4.3 and 4.4 together with 4. entail the control in Lemma 4.. With Lemma 4. at our disposal, we next proceed to the: Proof of Proposition 4.. Step. We claim that, for α>, α + G G = O α L t L loc dx;l M dv + O α/ L t L M dv dx. 4.5 Indeed, α + G G α G >/ α + G + G + α/ G / O α L + G α/ t,x,v = O α L + O α/ t,x,v L t L M dv dx, 4.6 and we conclude with the decomposition, α + G G = O α L + O α/ t,x,v L t L M dv dx O α L + O α/ t,x,v L t L M dv dx + G, together with the fluctuation control b in Proposition.3.
56 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Step. Let γ be a smooth truncation as in.40, and set We claim that, for each fixed δ> 0, Indeed, φ δ = α + G γ δ t + v x φ δ = O δ α + G. L loc M dv dx dt. 4.7 t + v x φ δ δ = γ α + G α + G Q + Q, where γ Z = γ Z + Zγ Z, while Q and Q are defined in 4.. Clearly, γ has support in [0, ], so that γ δ α + G α + G = O. δ L t,x,v On the other hand, the fluctuation control b in Proposition.3 and the estimate 4.6 imply that γ δ α + G α + G = O L t L loc dx;l M dv. Together with Lemma 4., these last two estimates lead to the following bound: t + v x φ δ α/ = O + O δ L L t L loc M dv dx dt dx;l + v β/ M dv + O δ. L loc dt dx;l + v β M dv Pick then α, ; the last estimate implies that 4.7 holds for each δ> 0, as announced. Step 3. On the other hand, we already know from the fluctuation control b in Proposition.3 and 4.5 that Moreover K φ δ Indeed, for each ϕ L x,v L x L v, one has: φ δ ϕm dv dx G ϕ M dx dv φ δ = O L t L loc M dv dx. 4.8 is locally uniformly integrable in the v-variable. 4.9 + K α + G G ϕ M dv dx. The second term is O α φ L. Hence this term can be made smaller than any given η whenever < 0 η. Since denotes an extracted subsequence converging to 0, there remain only finitely many terms, say N Nη that can also be made smaller that η, this time by choosing φ L x L v smaller than c cnη, η. As for the first term, it can be made less than η whenever φ L x L v c η, by Proposition 3.. Therefore φ δ ϕm dv dx η for each and δ> 0, K whenever φ L x L v mincnη, η, c η, which establishes 4.9.
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 57 Applying Theorem B. taken from [3] in Appendix B, we conclude from 4.8, 4.9 and 4.7 that for the measure M dv dx dt. Step 4. But, for each, δ 0,, one has: α + G so that for each δ> 0, φ δ is locally uniformly integrable on R +, 4.0 φ δ = α + G α + G α + G G G >/δ γ δ G>/δ φ δ = O ln δ L t α + G C ln δ hg G >/δ,, L M dv dx by the fluctuation control a in Proposition.3. This and 4.0 imply that α + G is also locally uniformly integrable on R +, 4. for the measure M dv dx dt. Because of the estimate 4.5 in Step, we finally conclude that G is locally uniformly integrable on R +, 4. for the measure M dv dx dt. Together with the control of large velocities in Proposition 3., the statement 4. entails Proposition 4.. Here is a first consequence of Proposition 4., bearing on the relaxation to infinitesimal Maxwellians. Proposition 4.3. Under the assumptions of Theorem.4, one has: G G Π 0 in L loc dt dx; L + v p M dv, for each p< as 0. Proof. By Proposition 4., the family, + v p G G Π, is uniformly integrable on [0,T] K for the measure M dv dx dt, for each T>0and each compact K. On the other hand, 3. and the fluctuation control b in Proposition.3 imply that G G Π 0 in L loc M dv dx dt, and therefore in M dv dx dt-measure locally on R +. Therefore p G G + v Π 0 in L loc dt dx; L M dv, which implies the convergence stated above.
58 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 We conclude this section with the following variant of the classical velocity averaging theorem [,], stated as Theorem B. in [3]. This result is needed in order to handle the nonlinear terms appearing in the hydrodynamic limit. Proposition 4.4. Under the assumptions of Theorem.4, for each ξ L M dv, each T>0 and each compact K, as y 0 +, uniformly in > 0. Proof. Observe that since, up to extraction, T 0 K ξg γ t, x + y ξg γ t, x dx dt 0, G G g γ = G + γ, G + γ 0 a.e. and G + γ 3 +, it follows from Proposition 4. and Theorem A. in Appendix A, referred to as the Product Limit Theorem, that G g γ 0 in L loc dt dx; L M dv, 4.3 as 0. This estimate, and Step in the proof of Proposition 4. and especially the estimate 4.5 there shows that one can replace g γ with α +G with α> in the equicontinuity statement of Proposition 4.4. Using 4. shows that, for each α,, the family, α + G is locally uniformly integrable on R +, for the measure M dv dx dt. In view of the estimate 4.5 and Proposition 3., we also control the contribution of large velocities in the above term, so that, for each T>0and each compact K, α + G is uniformly integrable on [0,T] K, for the measure M dv dx dt. On the other hand, Lemma 4. shows that the family, t + v x α + G is bounded in L loc M dv dx dt. Applying then Theorem B. taken from [3] in Appendix B shows that, for each T>0and each compact K, one has: T 0 K ξ α + G t, x + y ξ as y 0 uniformly in, which concludes the proof of Proposition 4.4. 5. Vanishing of conservation defects α + G t, x dx dt 0, Conservation defects appear in the renormalized form of the Boltzmann equation precisely because the natural symmetries of the collision integral are broken by the renormalization procedure. However, these conservation defects vanish in the hydrodynamic limit, as shown by the following:
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 59 Proposition 5.. Under the same assumptions as in Theorem.4, for each ξ span{,v,v,v 3, v }, one has the following convergence for the conservation defects D ξ defined by.47: D ξ 0 in L loc dt dx as 0. Proof. For ξ span{,v,v,v 3, v }, the associated defect D ξ is split as follows: with and D ξ = D ξ + D ξ, 5. D ξ = 3 ξk ˆγ G G G G, D ξ = 3 ξk ˆγ G G G G G G, with the notation.5 and.6. That the term D ξ vanishes for ξv = O v as v + is easily seen as follows: G D ξ G G G L ξ K ˆγ L t,x t,x,v OK O = O ln, 5. because of the entropy production estimate in Proposition.3d and the choice of K in.43. We further decompose D ξ in the following manner: with D ξ = D ξ + D L t,x,µ G D ξ = G G G ξ v >K ˆγ D ξ = ξ ˆγ ˆγ ˆγ ˆγ ξ + D3 ξ, 5.3 G G, G G G G G G and, by symmetry in the v and v variables, D 3 ξ = G ξ + ξ ˆγ ˆγ ˆγ ˆγ G G G G G. The terms D ξ and D3 ξ are easily mastered by the following classical estimate on the tail of Gaussian distributions see for instance [3] on p. 03 for a proof. Lemma 5.. Let G N z be the centered, reduced Gaussian density in R N, i.e. Then as R +. z >R G N z = π N/ e z. z p G N z dz π N/ S N R p+n e R,,
530 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Indeed, because of the upper bound on the collision cross-section in.6, for each T>0and each compact K, D ξ L [0,T ] K ξ G G G G v >K ˆγ G G L [0,T ] K,L µ L [0,T ] K,L M dv L t,x,µ C/ b ξ β v >K + v / L M dv ˆγ G L t,x,v G + v β / G G G G. L t,x,µ In the last right-hand side of the above chain of inequalities, one has obviously: ˆγ G L t,x,v = O. From Young s inequality and the entropy bound.9, we deduce that G + v + v + v + 4 hg + h = O 4 L [0,T ] K,L M dv. Lemma 5. and the condition ξv = O v as v + imply that on account of.43. Thus ξ β v >K + v / L M dv = O K β+5 e K = O k/ ln β+5, D ξ L [0,T ] K = O k/ ln β+5 0, 5.4 for all ξv = O v as v + as soon as k >. Next we handle D 3 ξ. Whenever ξ is a collision invariant i.e. whenever ξ belongs to the linear span of {,v,v,v 3, v } then ξ + ξ = ξ + ξ, and using the v, v v,v symmetry.8 in the integral defining ξ leads to D 3 where and Then so that G D 3 ξ = ξ + ξ ˆγ ˆγ ˆγ ˆγ G G G = D 3 ξ D 3 ξ, D 3 ξ = ξ + ξ v + v K ˆγ ˆγ ˆγ ˆγ D 3 ξ = ξ + ξ v + v >K ˆγ ˆγ ˆγ ˆγ D 3 ξ G L G G G t,x = O OK ˆγ 4 L, L t,x,µ G G G G 4 G G G G 4,. ξ + ξ v + v K ˆγ ˆγ ˆγ ˆγ L t,x,v,v,ω D 3 ξ L t,x = OK 0 as 0. 5.5
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 53 On the other hand, since G [0, ] whenever ˆγ G = 0, D 3 ξ L 6 ˆγ 4 t,x L 3 ξ + ξ v + v >K L µ O + v 3 + v + v v β v + v >K = O + v 3 + v +β/ v + v >K = O O e K/ K β+6, so that 3 L MM dv dv L MM dv dv D 3 ξ L = O k/ 3 ln β+6 0 as 0, 5.6 t,x for k > 6, by a direct application of Lemma 5. in v R3 v i.e. with N = 6. Whereas the terms D ξ, D ξ, and D3 ξ are shown to vanish by means of only the entropy and entropy production bounds in Proposition.3a d and Lemma 5., the term D ξ is much less elementary to handle. First, we split D ξ as D ξ = ξ ˆγ ˆγ G G G G G G + ξ G ˆγ ˆγ ˆγ +ˆγ ˆγ ˆγ ˆγ G G G G G = D ξ + D ξ. For each T>0and each compact K, the first term satisfies, D ξ L [0,T ] K C b ˆγ G + v β/ β/ G ξ + v L G G G M dv = O ˆγ G + v β/ provided that ξv = O v m for some m N. Since supp ˆγ [ 3, +, then Furthermore, as G G 3/ ˆγ G L [0,T ] K;L M dv L t,x,µ L [0,T ] K;L M dv, 3/ whenever ˆγ =, and one has: 3 ˆγ G. 3 ˆγ G L t,x,v ˆγ + γ L and ˆγ 0 a.e., 5.7 the uniform integrability stated in Proposition 4. and the Product Limit Theorem see Appendix A imply that Thus ˆγ G 0 in L [0,T] K, L M + v β dv. 5.8 D ξ L 0 as 0. 5.9 [0,T ] K
53 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Finally, we consider the term D ξ, one has: D ξ L [0,T ] K ˆγ ξ L [0,T ] K;L µ + ˆγ ξ L [0,T ] K;L µ ˆγ G G G G G L t,x,v L t,x,µ = O ˆγ + v + v L[0,T ] K;L µ = O ˆγ +β/ + v, L[0,T ] K;LM dv where the first equality uses the vv v v symmetry in.8. Since supp ˆγ [ 3, +, G 3/ whenever ˆγ =, one has: By 5.7 and 5.8, ˆγ ˆγ 3 ˆγ 3 ˆγ + γ L and G G Π ˆγ G + Π G. 0 in L loc dt dx, L M dv, 5.0 since G > 3/ whenever ˆγ =, whereas by Proposition.3b and Lemma 3., G Π = O L t L x Lq M dv, G G Π = O L loc dt dx,l M dv, for all q>+. Then, ˆγ = O L loc dt dx,l q M dv, for all q<. In particular, for each r<+, ˆγ + v r is uniformly bounded in L loc dt dx, L M dv. By interpolation with 5.0 we conclude that ˆγ r + v 0 as 0, 5. L [0,T ] K;L M dv and consequently D ξ 0 in L loc dt dx as 0. 5. The convergences 5., 5.4, 5.9, 5., 5.5 and 5.6 eventually imply Proposition 5.. Remark. The same arguments leading to 5.8 and to 5. imply that, for all r R, γ r + v 0 as 0. 5.3 L [0,T ] K;L M dv
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 533 6. Asymptotic behavior of the flux terms The purpose of the present section is to establish the following: Proposition 6.. Under the same assumptions as in Theorem.4, one has: G F ζ ζ Π + ˆζ Q G, G 0 in L loc dt dx, as 0, where ζ and ˆζ designate respectively either A and  or B and ˆB defined by.50 and.5. Proof. First, we decompose the flux term F ζ as follows: F ζ = ζ G K g γ = ζ K G = ζ K γ + = F ζ + F ζ. γ G ζ K γ We further split the term F ζ as F with ζ = F F ζ = G ζ K Π F ζ = ζ v K γ F 3 ζ ζ = Π ζ + F ζ + F3 ζ G G G + Π G Π, G γ,. 6. The term F ζ is easily disposed of. Indeed, the definition.3 of the hydrodynamic projection Π implies G that, Π + v p is, for each p 0, a finite linear combination of functions of v of order O v p+4 as v +, with coefficients that are quadratic in ξ G for ξ {,v,v,v 3, v }. Together with Proposition 4., this implies that, for each T>0and each compact K, G p Π + v is uniformly integrable on [0,T] K, 6. for the measure M dv dx dt. On the other hand, v K γ 0 and v K γ a.e. Since ζv = O v 3 as v +, this and the Product Limit Theorem imply that F ζ 0 in L loc dt dx. 6.3 The term F ζ requires a slightly more involved treatment. We start with the following decomposition: for each T>0and each compact K, F ζ L [0,T ] K G G ζ K γ + Π L [0,T ] K;L M dv G G Π 6.4 L [0,T ] K;L M dv.
534 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 Since γ = γ G = 0 whenever G >, one has for each q<+, G G G G G γ = γ Π + Π = O L t L dx M dv O L t L x Lq M dv + O O L loc dt dx,l M dv. In particular ζ G K γ = O, L [0,T ] K;L M dv since ζv = O v 3 as v +. This and 6. imply that ζ G G K γ + Π Using 6.4, 6.5 and Proposition 4.3 show that This and 6.3 imply that as 0. Next we handle the term F F ζ = = F F ζ ζ Π L loc dt dx;l M dv = O. 6.5 F ζ 0 in L loc dt dx. 6.6 G ζ. We first decompose it as follows: G ζ v >K γ + ζ γ G ζ + F 0 in L loc dt dx, 6.7 + ζ G ζ + F3 ζ. 6.8 Then, by.0 and Lemma 5., one has: F ζ L t L x γ L ζ v >K G L M dv On the other hand, for each T>0and each compact K, F ζ L [0,T ] K T / ζ γ L t L M dv dx O e K / K = O k/ ln. 6.9 L [0,T ] K;L M dv G 0 as 0, 6.0 L t L M dv dx because of.0 and of 5.3, since ζv = O v 3 as v +. Finally, we transform F 3 ζ as follows: F 3 ζ = ˆζ L G G G = ˆζ Q, Q G, G. Writing G Q, G G G = Q Π,Π G G G G + Q Π, + Π,
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 535 and using the classical relation see [] for instance, we arrive at Thus G Q, Qφ, φ = L φ for each φ Ker L, G = L G Π G G G G + Q Π, + Π F 3 ζ ζ = G Π ˆζ Q G, G G G G G + ˆζ Q Π, + Π. 6. By continuity of Q see.35, G G G G ˆζ Q Π, + Π C ˆζ G G L am dv Π G G + Π 0 L [0,T ] K;L + v β M dv L [0,T ] K L [0,T ] K;L + v β M dv as 0, for each T>0and each compact K, because of 6. and Proposition 4.3. Thus, by 6.9, 6.0 and 6. F ζ ζ G Π + ˆζ Q G, G 0 6. in L loc dt dx as 0. The convergences 6.7 and 6. eventually imply Proposition 6.. 7. Proof of Theorem.4 Throughout this section U Ux designates an arbitrary compactly supported, C, divergence-free vector field on. Taking the inner product with U of both sides of.48 gives t v K g γ U dx F A : x U dx = D v U dx 0 in L loc dt, 7. by Proposition 5.. Likewise, the energy equation.49 and Proposition 5. lead to t v 5 g K γ + div x F B = D v 5 0 in L loc dt dx. 7. By Proposition 6., one can decompose the fluxes as. where F A = F conv A + F diff A + o L loc dt dx, F B = F conv B + F diff B + o L loc dt dx, 7.3
536 F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 F conv A = A Π F diff A = G, Â Q G, G, 7.4 while F conv B = F diff B = B Π G, ˆB Q G, G. 7.5 Classical computations that can be found for instance in [3] using the fact that A is orthogonal in L M dv to Ker L as well as to odd functions of v and functions of v show that F conv G A = A A : v. In a similar way, B is orthogonal in L M dv to Ker L and to even functions of v, so that F conv G G B = B B v 3 v. 7.. Convergence of the diffusion terms The convergence of F diff A and F diff B comes only from weak compactness results, and from the following characterization of the weak limits. Proposition 7.. Under the same assumptions as in Theorem.4, one has, up to extraction of a subsequence n 0, G n G n G n G n g n g, and q, 7.6 in w L loc dt dx; L M dv and in w L dt dx dµ, respectively. Furthermore g L t L dx M dv is an infinitesimal Maxwellian of the form, n gt, x, v = ut, x v + θt, x v 5, div x u = 0, 7.7 and q L dt dx dµ satisfies: qbv v, ω dωm dv = v xg = A : xu + B x θ. 7.8 Proof. Proposition.3c shows that g is relatively compact in w L loc dt dx; L M dv while. implies that G G G G Pick then any sequence n 0 such that g n g, and is relatively compact in w L dt dx dµ. G n G n G n G n in w L loc dt dx; L M dv and in w L dt dx dµ respectively. n q,
F. Golse, L. Saint-Raymond / J. Math. Pures Appl. 9 009 508 55 537 Step : From. we deduce that n G n g in w L loc dt, L dx M dv. In particular, by Proposition 4.3, g is an infinitesimal Maxwellian, i.e. of the form: gt, x, v = ρt, x + ut, x v + θt, x v 3. Taking limits in the local conservation of mass leads then to or in other words div x vg = 0, div x u = 0, which is the incompressibility constraint. Multiplying the approximate momentum equation.48 by, t v K g γ + div x F A + 3 x 3 v K g γ = D v, using Propositions 5. and 6. to control D v and the remainder term in F A, G F A A Π + Â G G G G 0, and estimating F conv A and F diff A by the entropy and entropy production bounds.0., G A Π = O in L t L x, Â G G G G = O L, t,x we also obtain: x v g = 0, or equivalently, since v g = 3p + θ L R + ; L, ρ + θ = 0, which is the Boussinesq relation. One therefore has 7.7. Step : Start from 4. in the proof of Lemma 4.: t + v x α + G = α QG,G = Q + G + Q. Recall that Next observe that Q = Proposition 4. implies that Q 0 in L M dv dx dt. 7.9 G G G G G G α + G bv v, ω dω M dv. G this and the second limit in 7.6 imply that in L loc dt dx; L + v β M dv as 0;