The Boltzmann Equation and Its Hydrodynamic Limits 3

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1 1 2 2 3 CHAPTER 3 3 4 4 5 5 6 6 7 The Boltzmann Equation and 7 8 Its Hydrodynamic Limits 8 9 9 10 10 11 11 12 12 13 François Golse 13 14 Institut Universitaire de France, and Laboratoire Jacques-Louis Lions, Université Paris 7, 14 Boîte Courrier 187, F75252 Paris Cedex 05, France 15 15 E-mail: [email protected] 16 16 17 17 18 18 19 Contents 19 20 1. Introduction ...... 3 20 2. Fluid dynamics: A presentation of models ...... 5 21 21 2.1.ThecompressibleEulersystem...... 7 22 2.2.ThecompressibleNavier–Stokessystem...... 9 22 23 2.3.Theacousticsystem...... 10 23 24 2.4.TheincompressibleEulerequations...... 11 24 25 2.5.TheincompressibleNavier–Stokesequations...... 13 25 2.6.Thetemperatureequationforincompressibleflows...... 14 26 26 2.7. Coupling of the velocity and temperature fields by conservative forces ...... 14 27 27 3.TheBoltzmannequationanditsformalproperties...... 16 28 3.1.Conservationlaws...... 19 28 29 3.2. Boltzmann’s H -theorem...... 23 29 30 3.3. H -theoremandaprioriestimates...... 26 30 31 3.4. Further remarks on the H -theorem...... 34 31 3.5. The collision kernel ...... 37 32 32 3.6. The linearized collision integral ...... 44 33 4. Hydrodynamic scalings for the Boltzmann equation ...... 51 33 34 4.1.Notionofararefiedgas...... 51 34 35 4.2.ThedimensionlessBoltzmannequation...... 53 35 36 5.CompressiblelimitsoftheBoltzmannequation:Formalresults...... 55 36 5.1.ThecompressibleEulerlimit:TheHilbertexpansion...... 55 37 37 5.2. The compressible Navier–Stokes limit: The Chapman–Enskog expansion ...... 59 38 5.3.ThecompressibleEulerlimit:Themomentmethod...... 63 38 39 5.4.Theacousticlimit...... 66 39 40 6.IncompressiblelimitsoftheBoltzmannequation:Formalresults...... 68 40 41 6.1.TheincompressibleNavier–Stokeslimit...... 68 41 42 42 43 HANDBOOK OF DIFFERENTIAL EQUATIONS 43 44 Evolutionary Equations, volume 2 44 Edited by C.M. Dafermos and E. Feireisl 45 45 © 2005 Elsevier B.V. All rights reserved

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2 F. G o l s e

1 6.2.TheincompressibleStokesandEulerlimits...... 75 1 2 6.3.Otherincompressiblemodels...... 78 2 3 7. Mathematical theory of the Cauchy problem for hydrodynamic models ...... 80 3 7.1.TheStokesandacousticsystems...... 80 4 4 7.2.TheincompressibleNavier–Stokesequations...... 82 5 7.3.TheincompressibleNavier–Stokes–Fouriersystem...... 84 5 6 7.4.ThecompressibleEulersystem...... 85 6 7 7.5.TheincompressibleEulerequations...... 87 7 8 8. Mathematical theory of the Cauchy problem for the Boltzmann equation ...... 90 8 8.1.Globalclassicalsolutionsfor“small”data...... 90 9 9 8.2.TheDiPerna–Lionstheory...... 91 10 8.3.VariantsoftheDiPerna–Lionstheory...... 100 10 11 9. The Hilbert expansion method: Application to the compressible Euler limit ...... 103 11 12 10. The relative entropy method: Application to the incompressible Euler limit ...... 107 12 13 11.Applicationsofthemomentmethod...... 111 13 11.1.Theacousticlimit...... 111 14 14 11.2. The Stokes–Fourier limit ...... 113 15 11.3.TheNavier–Stokes–Fourierlimit...... 115 15 16 11.4. Sketch of the proof of the Navier–Stokes–Fourier limit by the moment method ...... 116 16 17 11.5.Thenonlinearcompactnessestimate...... 131 17 18 12.Conclusionsandopenproblems...... 137 18 References...... 139 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 3

The Boltzmann equation and its hydrodynamic limits 3

1 1. Introduction 1 2 2 3 3 The classical models of fluid dynamics, such as the Euler or Navier–Stokes equations, 4 4 were first established by applying Newton’s second law of motion to each infinitesimal 5 5 volume element of the fluid considered, see, for instance, Chapter 1 of [75]. While this 6 6 7 method has the advantage of being universal – indeed, all hydrodynamic models can be 7 8 obtained in this way – it has one major drawback: equations of state and transport co- 8 9 efficients (such as the viscosity or heat conductivity) are given as phenomenological or 9 10 experimental data, and are not related to microscopic data (essentially, to the laws govern- 10 11 ing molecular interactions). As a matter of fact, a microscopic theory of liquids is most 11 12 likely too complex to be of any use in deriving the macroscopic models of fluid mechanics. 12 13 In the case of gases or plasmas, however, molecular interactions are on principle much 13 14 more elementary, so that one can hope to express thermodynamic functions and transport 14 15 coefficients in terms of purely mechanical data concerning collisions between gas mole- 15 16 cules. 16 17 In fact, the subject of hydrodynamic limits goes back to the work of the founders J. Clerk 17 18 Maxwell and L. Boltzmann, of the kinetic theory of gases. Both checked the consistency of 18 19 their new – and, at the time, controversial – theory with the well-established laws of fluid 19 20 mechanics. Interestingly, while the very existence of atoms was subject to heated debates, 20 21 kinetic theory would provide estimates on the size of a gas molecule from macroscopic 21 22 data such as the viscosity of the gas. 22 23 Much later, D. Hilbert formulated the question of hydrodynamic limits as a mathemat- 23 24 ical problem, as an example in his 6th problem on the axiomatization of physics [68]. In 24 25 Hilbert’s own words “[...] Boltzmann’s work on the principles of mechanics suggests the 25 26 problem of developing mathematically the limiting processes [...] which lead from the 26 27 atomistic view to the laws of motion of continua”. Some years later, Hilbert himself at- 27 28 tacked the problem in [69], as an application of his own fundamental work on integral 28 29 29 equations. 30 30 There is an ambiguity in Hilbert’s formulation. Indeed, what is meant by “the atomistic 31 31 view” could designate two very different theories. One is molecular dynamics (i.e., the 32 32 N-body problem of classical mechanics with elastic collisions, assuming for simplicity 33 33 34 all bodies to be spherical and of equal mass). The other possibility is to start from the 34 35 kinetic theory of gases, and more precisely from the Boltzmann equation, which is what 35 36 Hilbert himself did in [69]. However, one should be aware that the Boltzmann equation is 36 37 not itself a “first principle” of physics, but a low density limit of molecular dynamics. In 37 38 the days of Maxwell and Boltzmann, and maybe even at the time of Hilbert’s own papers 38 39 on the subject, this may not have been so clear to everyone. In particular, much of the 39 40 controversy on irreversibility could perhaps have been avoided with a clear understanding 40 41 of the relations between molecular gas dynamics and the kinetic theory of gases. 41 42 In any case, the problem of hydrodynamic limits is to obtain rigorous derivations of 42 43 macroscopic models such as the fundamental partial differential equations (PDEs) of fluid 43 44 mechanics from a microscopic description of matter, be it molecular dynamics or the ki- 44 45 netic theory of gases. The situation can be illustrated by the following diagram. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 4

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1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 Throughout the present chapter, we are concerned with only the vertical arrow in the dia- 10 11 gram above. As a matter of fact, this is perhaps the part of the subject that is best understood 11 12 so far, at least according to the mathematical standards of rigor. 12 13 The other arrows in this diagram correspond with situations that are only partially un- 13 14 derstood, and where certain issues are still clouded with mystery. Before starting our dis- 14 15 cussion of the hydrodynamic limits of the kinetic theory of gases, let us say a few words 15 16 on these other limits and direct the interested reader at the related literature. 16 17 Although beyond the scope of this chapter, the horizontal arrow is of considerable inter- 17 18 est to our discussion, being a justification of the kinetic theory of gases on the basis of the 18 19 molecular gas dynamics (viewed as a first principle of classical, nonrelativistic physics). 19 20 A rigorous derivation of the Boltzmann equation from molecular dynamics on short time 20 21 intervals was obtained by Lanford [77]; see also the very nice rendition of Lanford’s work 21 22 in the book [28]. Hence, although not a first principle itself, the Boltzmann equation is rig- 22 23 orously derived from first principles and therefore has more physical legitimacy than phe- 23 24 nomenological models (such as lattice gases or stochastic Hamiltonian models). Besides, 24 25 the Boltzmann equation is currently used by engineers in aerospace industry, in vacuum 25 26 technology, in nuclear engineering, as well as several other applied fields, a more complete 26 27 list of those being available in the Proceedings of the Rarefied Gas Dynamics Symposia. 27 28 28 On the other hand, “formal” derivations of the Euler system for compressible fluids from 29 29 molecular dynamics were proposed by Morrey [98]. Later on, S.R.S. Varadhan and his col- 30 30 laborators studied the same limit, however with a different method. Instead of taking mole- 31 31 cular dynamics as their starting point, they modified slightly the N-body Hamiltonian by 32 32 33 adding an arbitrarily small noise term to the kinetic energy; they also cut off high velocities 33 34 at a threshold compatible with the maximum speed observed on the macroscopic system. 34 35 Starting from this stochastic variant of molecular gas dynamics, they derived the Euler sys- 35 36 tem of compressible fluids for short times (before the onset of singularities such as shock 36 37 waves); see for instance [120] and the references therein, notably [104], see also [33], and 37 38 the more recent reference [44]. The role of the extra noise term in their derivation is to 38 39 guarantee some form of the ergodic principle, i.e., that the only invariant measure for the 39 40 Hamiltonian in the limit of infinitely many particles is a local Gibbs state (parametrized 40 41 by macroscopic quantities). At the time of this writing, deriving the Euler system of com- 41 42 pressible fluids from molecular gas dynamics without additional noise terms as in [104] 42 43 and for all positive times seems beyond reach. 43 44 For these reasons, we have limited our discussion to only the derivation of hydrody- 44 45 namic models from the kinetic theory of gases, i.e., from the Boltzmann equation. For a 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 5

The Boltzmann equation and its hydrodynamic limits 5

1 more general view of the subject of hydrodynamic limits, the reader is advised to read the 1 2 excellent survey article by Esposito and Pulvirenti [41], whose selection of topics is quite 2 3 different from ours. 3 4 This chapter is organized as follows: in Section 2 we review the classical models of fluid 4 5 mechanics. Section 3 introduces the Boltzmann equation and discusses its structure and 5 6 main formal properties. In Section 4 we discuss the dimensionless form of the Boltzmann 6 7 equation and introduce its main scaling parameters. Sections 5 and 6 explain in detail 7 8 the formal derivation of the most classical PDEs of fluid mechanics from the Boltzmann 8 9 equation by several different methods. Section 7 recalls the known mathematical results 9 10 on the Cauchy problem for the PDEs of fluid mechanics. In Section 8 we review the state 10 11 of the art on the existence theory for the Boltzmann equation. Sections 9–11 sketch the 11 12 mathematical proofs of the formal derivations described in Sections 6 and 7; here again, 12 13 we present three different methods for establishing these hydrodynamic limits and discuss 13 14 their respective merits. 14 15 We have chosen to emphasize compactness methods, leading to global results, and espe- 15 16 cially the derivation of global weak solutions of the incompressible Navier–Stokes equa- 16 17 tions from renormalized solutions of the Boltzmann equation. There is more than a simple 17 18 matter of taste in this choice. Indeed, it is a nontrivial question to decide whether these 18 19 hydrodynamic limits are intrinsic properties of the microscopic versus macroscopic mod- 19 20 els governing the dynamics of gases, or simply an illustration of more or less standard 20 21 techniques in asymptotic analysis. The second viewpoint leads to derivations of hydrody- 21 22 namic models that fall short of describing any singular behavior beyond isolated shock 22 23 waves in compressible gas dynamics. The first viewpoint uses the specific structure of the 23 24 Boltzmann equation to design convergence proofs that are based on only the a priori esti- 24 25 mates on this equation that have an intrinsic physical meaning; these convergence proofs 25 26 are insensitive to whether singularities appear in finite time on the limiting hydrodynamic 26 27 model. 27 28 28 29 29 30 30 2. Fluid dynamics: A presentation of models 31 31 32 32 33 Usually, one thinks of a fluid – more generally, a continuous medium – as a set of material 33 RN 34 points which, at any given time t, fill a smooth domain in the Euclidean space , where 34 = 35 N 1, 2, 3 are the dimensions of physical interest. 35 36 The purpose of fluid dynamics is to describe the state of the fluid at any instant of time 36 37 with a small number of fields – such as the velocity or temperature fields – defined on the 37 38 domain filled by the fluid. 38 39 These fields are governed by several partial differential equations that share a common 39 40 structure which we briefly recall below. 40 41 Consider the motion of a continuous medium, and denote by 41 42 42 N 43 X(t,s; a) ∈ R 43 44 44 45 the position at time t of the material point which occupied position a at time s. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 6

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1 The kinematics of such a medium is based on the parallel transport along the family of 1 2 curves t → X(t,s; a) indexed by a (s, being the origin of times, is kept fixed). The infini- 2 3 tesimal description of this parallel transport involves the first-order differential operator 3 4 4 5 N 5 D 6 = ∂t + u(t, x) ·∇x = ∂t + uj (t, x) ∂x , 6 Dt j 7 j=1 7 8 8 9 where the velocity field u(t, x) is defined in terms of the particle paths X(t,s; a) by the 9 10 formula 10 11 11 12 d 12 X(t,s; a) = u t,X(t,s; a) . 13 dt 13 14 14 15 D 15 The operator Dt is usually called the material derivative, and using it allows one to elimi- 16 nate the trajectories X(t,s; a). In other words, instead of following the motion of each ma- 16 17 terial point, one looks at any fixed point in the Euclidean space RN ,sayx, and observes, 17 18 at any given time t, the velocity u(t, x) of the material point that is located at the posi- 18 19 tion x at time t. This is called the Eulerian description of a continuous medium, whereas 19 20 the description in terms of X(t,s; a) is called the Lagrangian description. Interestingly, 20 21 the connections between the kinetic theory of gases (or plasmas) and fluid dynamics are 21 22 always formulated in terms of the Eulerian, instead of the Lagrangian description, although 22 23 the latter may seem more natural when dealing with the motion of a gas at the atomic or 23 24 molecular level. 24 25 Fluid dynamics rests on three fundamental laws – or equations: 25 26 • the continuity equation, 26 27 • the motion equation, and 27 28 • the energy balance equation. 28 29 The continuity equation states that the density ρ of the fluid is transported by the flow, 29 30 i.e., that the measure ρ(t,x)dx is the image of the measure ρ(s,a)da under the map 30 31 a → X(t,s; a). The infinitesimal formulation of this fact is 31 32 32 33 Dρ 33 =−ρ divx u. (2.1) 34 Dt 34 35 35 36 The motion equation states that each portion of the fluid obeys Newton’s second law 36 37 d = 37 of motion (i.e., dt (momentum) force). The acceleration is computed in terms of the 38 material derivative, and the infinitesimal formulation of the motion equation is 38 39 39 40 Du 40 ρ = divx S + ρf, (2.2) 41 Dt 41 42 42 43 where f istheexternalforcefield(e.g.,gravity,Lorentzforceinthecaseofa plasma...) 43 44 and S is the stress tensor. The meaning of S is as follows: at any given time t, isolate a 44 45 smooth domain Ω in the fluid, denote by ∂Ω its boundary, by nx the unit normal field 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 7

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1 on ∂Ω pointing toward the outside of Ω, and by dσ(x) the surface element on ∂Ω. Then, 1 2 the force exerted by the fluid outside Ω on the fluid inside Ω is 2 3 3 4 4 S(t,x)nx dσ(x). 5 ∂Ω 5 6 6 7 Finally, the energy balance equation involves the internal energy of the fluid per unit of 7 1 | |2 + 8 mass E; the total energy per unit of mass is 2 u E (the sum of the kinetic energy and 8 9 the internal energy). It states that the material derivative of the total energy of any portion 9 10 of fluid is the sum of the works of the stresses and of the external force f, minus the heat 10 11 flux lost by that portion of fluid. Its infinitesimal formulation is 11 12 12 D 1 2 13 ρ |u| + E =−divx Q + divx(Su) + ρf · u, (2.3) 13 14 Dt 2 14 15 where Q is the heat flux. 15 16 In the motion and energy balance equations, f is a given vector field, while the density ρ, 16 17 the velocity field u, the internal energy E, the stress tensor S and the heat flux Q are un- 17 18 known. However, these quantities are usually not independent, but are related by equations 18 19 of state that depend on the fluid considered. 19 20 Equations (2.1)–(2.3) are Galilean invariant. Specifically, let v ∈ R3; define the Galilean 20 21 transformation 21 22 22 23 x = x + vt, u t,x = u(t, x) + v, φ t,x = φ(t,x) 23 24 24 25 for φ = ρ,S,f,E,Q. Then, setting 25 26 26 27 D 27 = ∂t + u ·∇ 28 Dt x 28 29 29 30 one deduces from (2.1)–(2.3) that 30 31 31 D 32 ρ =−ρ divx u , 32 33 Dt 33 34 D 34 ρ u = divx S + ρ f , 35 Dt 35 36 36 D 1 2 37 ρ u + E =−div Q + div S u + ρ f · u . 37 Dt 2 x x 38 38 39 39 40 2.1. The compressible Euler system 40 41 41 42 An ideal fluid is one where the effects of viscosity and thermal conductivity can be ne- 42 43 glected. In this case, Q = 0 and the stress tensor is of the form 43 44 44 45 S =−pI, 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 8

8 F. G o l s e

1 where the scalar p is the pressure. Hence the system consisting of the continuity equation 1 2 the motion equation and the energy balance equation becomes 2 3 3

4 ∂t ρ + divx(ρu) = 0, 4 5 5 ρ ∂t u + (u ·∇x)u =−∇xp + ρf, (2.4) 6 6 7 ρ ∂t E + (u ·∇x)E =−p divx u + ρf · u. 7 8 8 9 Thus, the unknowns are the density ρ, the velocity ﬁeld u, the pressure p and the internal 9 10 energy E. However, the quantities ρ, p and E are not independent, but are related by 10 11 equations of state. 11 12 Choosing the density ρ and the temperature θ as independent thermodynamic variables, 12 13 these equations of state are relations that express the pressure p and the internal energy E 13 14 in terms of ρ and θ 14 15 15 16 p ≡ p(ρ,θ), E ≡ E(ρ,θ). (2.5) 16 17 17 18 Hence (2.4) is a system of N +2 partial differential equations for the unknowns ρ, u and θ; 18 19 notice that there are in fact N + 2 scalar unknowns, ρ and θ,plustheN components of the 19 20 vector ﬁeld u. 20 21 The case of a perfect gas is of particular importance for the rest of this chapter. In this 21 22 case, the equations of state are 22 23 23 24 kθ 24 p(ρ,θ) = kρθ, e(ρ, θ ) = , (2.6) 25 γ − 1 25 26 26 27 where k is the Boltzmann constant (k = 1.38 · 10−23 JK−1) and γ>1 is a constant called 27 28 the adiabatic exponent. For a perfect gas whose molecules have n degrees of freedom 28 29 29 30 2 30 31 γ = 1 + . 31 n 32 32 33 For instance, in the case of a perfect monatomic gas, each molecule has 3 degrees of free- 33 34 dom (the coordinates of its center of mass); hence γ = 5/3. In the case of a diatomic gas, 34 35 each molecule has 5 degrees of freedom (the coordinates of its center of mass and the 35 36 direction of the line passing through the centers of both atoms); hence γ = 7/5. 36 37 From now on, we choose a temperature scale such that k = 1. 37 38 Adding the continuity equation to the motion equation and to the energy balance equa- 38 39 tion, one can recast (2.4) in the form 39 40 40 41 41 ∂t ρ + divx(ρu) = 0, 42 42 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) = ρf, (2.7) 43 43 44 44 1 2 1 1 2 γ ∂t ρ |u| + θ + divx ρu |u| + θ = ρf · u. 45 2 γ − 1 2 γ − 1 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 9

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1 In the absence of external force, i.e., when f = 0, (2.7) is a hyperbolic system of conserva- 1 2 tion laws. 2 3 3 4 4 5 2.2. The compressible Navier–Stokes system 5 6 6 7 If the fluid considered is not ideal, the viscous forces and heat conduction must be taken 7 8 into account. 8 9 In the case of moderate temperature gradients in the fluid, heat conduction is usually 9 10 modeled with Fourier’s law: the heat flux Q is proportional to the temperature gradient, 10 11 i.e., 11 12 12 13 13 Q =−κ∇xθ, 14 14 15 where the coefficient κ is called the heat conductivity. Usually, κ is a function of the pres- 15 16 sure and the temperature. Because of the equation of state for the pressure, one has equiv- 16 17 alently κ ≡ κ(ρ,θ) >0. 17 18 The viscous forces are modeled by adding a correction term to the pressure in the stress 18 19 tensor S. In the case where the gradient of the velocity field is not too large, this correction 19 20 term is linear in the gradient of the velocity field – by analogy with Fourier’s law. Usually, 20 21 the fluid under consideration is isotropic, and this implies that this correcting term is a lin- 21 22 22 ear combination of the scalar tensor (divx u)I and of the traceless part of the symmetrized 23 gradient of the velocity field 23 24 24 25 25 T 2 26 D(u) =∇ u +∇ u − (div u)I. 26 x x N x 27 27 28 In other words, the stress tensor takes the form 28 29 29 30 30 S =−pI + µ(divx u)I + λD(u), 31 31 32 where λ and µ are two positive scalar quantities referred to as the viscosity coefficients. 32 33 Again, λ and µ are functions of the pressure and temperature, which, by the equation of 33 34 state for the pressure, can be transformed into λ ≡ λ(ρ, θ) and µ ≡ µ(ρ, θ). 34 35 Inserting this form of the stress tensor in the motion and energy balance equation, one 35 36 finds the system of Navier–Stokes equations for compressible fluids 36 37 37 38 38 ∂t ρ + divx(ρu) = 0, 39 39 40 ρ ∂t u + (u ·∇x)u =−∇xp(ρ,θ) 40 41 41 + ρf + divx λ(ρ, θ)D(u) +∇x µ(ρ, θ) divx(u) , (2.8) 42 42 43 ρ ∂t E(ρ,θ) + (u ·∇x)E(ρ, θ) =−p(ρ,θ)divx u + divx κ(θ)∇xθ 43 44 44 1 2 45 + λ(ρ, θ)D(u) : D(u) + µ(ρ, θ)(divx u) . 45 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 10

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1 This is a degenerate parabolic system of partial differential equations in the unknowns 1 2 ρ, u and θ. Observe that there is no diffusion term in the first equation, which is clear 2 3 on physical grounds. Indeed, the meaning of the continuity equation is purely geomet- 3 4 ric – namely, the fact that the measure ρ dx is transported by the fluid flow – and cannot 4 5 be affected by physical assumptions on the fluid (such as whether the fluid is ideal or 5 6 not). 6 7 7 8 8 9 2.3. The acoustic system 9 10 10 11 The acoustic waves in an ideal fluid are small amplitude disturbances of a constant equi- 11 12 librium state. Therefore the propagation of acoustic waves is governed by the linearization 12 13 13 at a constant state (ρ,¯ u,¯ θ)¯ of the compressible Euler system. Without loss of generality, 14 14 one can assume by Galilean invariance that u¯ = 0. The density, velocity and temperature 15 15 fields are written as 16 16 17 17 ρ =¯ρ +˜ρ, u =˜u, θ = θ¯ + θ,˜ 18 18 19 19 20 where the letters adorned with tildes designate small disturbances of the background equi- 20 ¯ ¯ 21 librium state (ρ,0, θ). In the case of a perfect gas, and in the absence of external force (i.e., 21 = 22 for f 0), the acoustic system takes the form 22 23 23 24 ∂t ρ˜ +¯ρ divx u˜ = 0, 24 25 ¯ 25 θ ˜ 26 ∂ u˜ + ∇ ρ˜ +∇ θ = 0, (2.9) 26 t ρ¯ x x 27 27 28 1 ˜ ¯ 28 ∂t θ + θ divx u˜ = 0. 29 γ − 1 29 30 30 31 By combining the first and the last equation in the system above, one can put it in the form 31 32 32 ˜ 33 ρ˜ θ 33 ∂t + + γ divx u˜ = 0, 34 ρ¯ θ¯ 34 35 (2.10) 35 ρ˜ θ˜ 36 ˜ + ¯∇ + = 36 ∂t u θ x ¯ 0. 37 ρ¯ θ 37 38 38 39 Splitting the fluctuation of velocity field u˜ as the sum of a gradient field and of a solenoidal 39 40 (i.e., divergence-free) field 40 41 41 s s 42 u˜ =−∇xϕ +˜u , divx u = 0, 42 43 43 44 one deduces from the system (2.10) – together with boundary conditions, or conditions at 44 45 infinity, or else conditions on the mean value of the fields, whose detailed description does 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 11

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1 not belong here – that 1 2 2 3 ˜ ˜ 3 ¯ ρ θ ∂tt − γ θ x + = 0, 4 ρ¯ θ¯ 4 5 5 − ¯ = 6 ∂tt γ θ x ϕ 0, (2.11) 6 7 s 7 ∂t u = 0. 8 8 9 In other words, the acoustic system can be reduced to two independent wave equations 9 10 for (ρ/˜ ρ¯ + θ/˜ θ)¯ (the relative pressure fluctuation) and ϕ (the fluctuating stream function), 10 11 while the solenoidal part of the velocity fluctuation us is a constant of motion. 11 12 12 13 13 14 2.4. The incompressible Euler equations 14 15 15 16 Consider next the case of an incompressible, homogeneous ideal fluid. The evolution of 16 17 such a fluid is governed by the system (2.4) with ρ = const. The continuity and motion 17 18 equations in (2.4) reduce to 18 19 19 20 divx u = 0, 20 21 (2.12) 21 + ·∇ =−∇ + 22 ∂t u (u x)u xπ f, 22 23 = 23 24 where π p/ρ. At variance with the compressible Euler system, there is no need of an 24 π 25 equation of state to determine . Indeed, taking the divergence of both sides of the motion 25 26 equation leads to 26 27 27 − π = div (u ·∇ u) − div f = trace (∇ u)2 − div f, 28 x x x x x x 28 29 29 so that π can be expressed in terms of u by solving the Laplace equation. In other words, 30 30 π must be thought of as the Lagrange multiplier associated to the constraint divx u = 0. 31 31 The incompressible Euler equations arise in a different context, namely in the description 32 32 of incompressible flows of compressible fluids (such as perfect gases, for instance). 33 33 The dimensionless number that monitors the compressibility is the Mach number, i.e., 34 34 the ratio of the length of the velocity field to the speed of sound. In the case of a perfect 35 35 gas with adiabatic exponent γ , our discussion of the acoustic system above shows that the 36 √ 36 speed of sound in the gas at a temperature θ is c = γθ, so that the Mach number in that 37 37 case is 38 38 39 |u| 39 Ma = √ . (2.13) 40 γθ 40 41 41 42 With this definition, the Mach number is a local quantity, since u and θ are in general 42 43 functions of x and t. But one can replace |u| and θ in the definition above by constant 43 44 quantities of the same order of magnitude, for instance by averages of |u| and θ over large 44 45 spatial and temporal domains. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 12

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1 Flows√ of perfect gases are incompressible in the small Mach number limit. Setting 1 2 ε = Ma 1, consider the rescaled density, velocity and temperature fields defined by 2 3 3 4 t 4 ρ (t, x) = ρ ,x , 5 ε ε 5 6 6 1 t 7 u (t, x) = u ,x , (2.14) 7 ε ε ε 8 8 9 t 9 θ (t, x) = θ ,x , 10 ε ε 10 11 11 12 assuming (ρ,u,θ) is a solution of the compressible Euler system (2.7), with f ≡ 0for 12 13 13 simplicity. Hence (ρε,uε,θε) satisfies 14 14 15 15 ∂t ρε + divx(ρεuε) = 0, 16 16 1 17 ρ ∂ u + (u ·∇ )u + ∇ (ρ θ ) = , 17 ε t ε ε x ε 2 x ε ε 0 (2.15) 18 ε 18 19 ∂t θε + uε ·∇xθε + (γ − 1)θε divx uε = 0. 19 20 20 21 The leading-order term in the momentum equation is the gradient of the pressure field, 21 22 which suggests that, in the limit as ε → 0, ρεθε C(t); then, combining the continuity 22 23 and temperature equations above leads to 23 24 24 25 d 25 γ divx uε =−∂t ln(ρεθε) − uε ·∇x ln(ρεθε) ln C(t) . 26 dt 26 27 27 28 In many situations – for instance, if the spatial domain is a periodic box, or in the case of 28 29 a bounded domain Ω with the usual boundary condition uε · nx = 0on∂Ω – integrating 29 30 in x both sides of this equality leads to the incompressibility condition 30 31 31 32 divx uε 0 in the limit as ε → 0. 32 33 33 34 Hence the continuity equation reduces to 34 35 35 36 36 ∂t ρε + uε ·∇xρε 0 37 37 38 38 so that, if the initial data for ρε is a constant ρ¯, then 39 39 40 40 ρε(t, x) ¯ρ in the limit as ε → 0. 41 41 42 Then, the momentum equation reduces to 42 43 43 44 1 44 45 ∂ u + (u ·∇ )u − ∇ θ = gradient field 45 t ε ε x ε ε2 x ε dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 13

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1 in the limit as ε → 0. This discussion suggests that, in the small Mach number limit, flows 1 2 of a compressible fluid such as a perfect gas are well described by the incompressible Euler 2 3 equations. 3 4 4 5 5 6 6 7 2.5. The incompressible Navier–Stokes equations 7 8 8 9 Next, we start from the compressible Navier–Stokes system (2.8), and assume that the 9 10 density ρ is a constant. As above, the continuity equation in (2.8) reduces to the incom- 10 11 pressibility condition divx u = 0. Moreover, assuming that the viscosity λ is a constant, we 11 12 find that the momentum equation reduces to 12 13 13 14 14 ρ ∂t u + (u ·∇x)u +∇xp = ρf + λ xu. 15 15 16 16 17 Defining the kinematic viscosity to be 17 18 18 λ 19 ν = 19 20 ρ 20 21 21 22 and setting π = p/ρ, we arrive at the incompressible Navier–Stokes equations 22 23 23 24 24 divx u = 0, 25 (2.16) 25 26 ∂t u + (u ·∇x)u +∇xπ = f + ν xu. 26 27 27 28 We leave it to the reader to verify that the incompressible Navier–Stokes equations can be 28 29 viewed as the small Mach number limit of the compressible Navier–Stokes system, as was 29 30 30 done in the case of the incompressible√ Euler system. The scaling law is slightly different 31 from the Euler case: for ε = Ma,set 31 32 32 33 33 t x 34 ρ (t, x) = ρ , , 34 ε ε2 ε 35 35 36 1 t x 36 u (t, x) = u , , (2.17) 37 ε ε ε2 ε 37 38 38 t x 39 θ (t, x) = θ , , 39 ε 2 40 ε ε 40 41 41 42 where (ρ,u,θ)is a solution to the Navier–Stokes system (2.8), with f ≡ 0. Then, to leading 42 43 order as ε → 0, uε satisfies (2.16) with f ≡ 0. 43 44 So far, we have said nothing about the temperature field in incompressible flows; this 44 45 will be the subject matter of the next subsection. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 14

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1 2.6. The temperature equation for incompressible flows 1 2 2 3 Going back to the Navier–Stokes system (2.8) for a perfect gas with adiabatic exponent γ , 3 4 we see that, in the incompressible case where ρ = const, the third equation reduces to 4 5 5 6 1 1 6 ρ(∂t θ + u ·∇xθ)= divx κ(θ)∇xθ + λD(u) : D(u). (2.18) 7 γ − 1 2 7 8 8 9 On the right-hand side of (2.18), the first term represents the divergence of the heat flux 9 10 due to thermal conduction, as described by Fourier’s law, while the second term represents 10 11 the production of heat by intermolecular friction and is called the viscous heating term. 11 12 In some models that can be found in the literature, the viscous heating term is absent 12 13 from the temperature equation. Whether the viscous heating term should be taken into 13 14 account or not depends in fact on the relative size of the fluctuations of velocity field about 14 15 its average value, and of the fluctuations of temperature field about its average values. 15 16 If the fluctuations of velocity field are of a smaller order than the square-root of the 16 17 temperature fluctuations, then a straightforward scaling argument shows that the viscous 17 18 heating term can indeed be neglected in (2.18). If however, the fluctuations of velocity 18 19 field are at least of the same order of magnitude as the square-root of the temperature 19 20 fluctuations, then the viscous heating term cannot be neglected in (2.18). We shall discuss 20 21 this alternative further in the description of the incompressible hydrodynamic limits of the 21 22 Boltzmann equation. 22 23 23 24 24 25 2.7. Coupling of the velocity and temperature fields by conservative forces 25 26 26 27 In our discussion of the incompressible flows as low Mach number limits, we have ne- 27 28 glected so far the external force f. Split it as the sum of a gradient field (i.e., of a conserva- 28 29 tive force) and of a solenoidal field 29 30 30 s s 31 f =−∇xφ + f , divx f = 0. 31 32 32 33 Scale φ and f s as 33 34 34 35 1 t x s 1 s t x 35 φε(t, x) = φ , , f (t, x) = f , , (2.19) 36 ε ε2 ε ε ε3 ε2 ε 36 37 37 38 and assume that ρε and θε have fluctuations of order ε about their constant average values 38 39 39 ¯ ˜ 40 ρε =¯ρ + ερ˜ε,θε = θ + εθε. 40 41 41 42 In that case, the leading order in ε of the momentum equation in the Navier–Stokes sys- 42 43 tem (2.8) reduces to 43 44 44 ˜ ¯ 45 ∇x ρ¯θε + θρ˜ε +¯ρ∇xφε 0, 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 15

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1 or in other words, 1 2 2 3 ˜ 3 ρ˜ε θε φε 4 + + C(t), (2.20) 4 ρ¯ θ¯ θ¯ 5 5 6 an equality known as Boussinesq’s relation. In many cases, the boundary conditions (or 6 7 decay at infinity, or else periodicity conditions) entail that C(t) = 0. 7 8 The next order in ε of the momentum equation in the Navier–Stokes system is 8 9 9 10 10 ρ¯ ∂ u + (u ·∇ )u +˜ρ ∇ φ λ u +¯ρf + 11 t ε ε x ε ε x ε x ε ε gradient field 11 12 12 13 and one expresses the action of the conservative force ρ˜ε∇xφε as 13 14 14 ¯ 15 ˜ ∇ −ρ ˜ ∇ + 1∇ 2 15 ρε xφε θε xφε x φε 16 θ¯ 2 16 17 17 18 so that the momentum equation reduces to 18 19 19 20 θε 20 ∂t uε + (uε ·∇x)uε − ∇xφε ν xuε + fε + gradient field. (2.21) 21 θ¯ 21 22 22 23 As for the temperature equation, one should refrain from using directly (2.18). Indeed, this 23 24 equation has been derived from (2.8) in the purely incompressible case where ρ = const, 24 25 while in the present case ρ = const modulo terms of order ε. 25 26 In the present case, we must go back to the Navier–Stokes system (2.8) and write the 26 27 continuity and energy equation in terms of the fluctuations of density and temperature 27 28 28 29 29 ε(∂t ρ˜ε + uε ·∇xρ˜ε) +¯ρ divx uε = o(ε), 30 30 ¯ 31 1 ˜ ˜ ¯ κ(θ) ˜ 31 ε ∂t θε + uε ·∇xθε + θ divx uε = ε divx ∇xθε + o(ε). 32 γ − 1 ρ¯ 32 33 33 34 Next we must eliminate divx uε between both equations above; indeed, we only know that 34 35 divx uε 0 to leading order in ε, and it may not be true that divx uε = o(ε). Dividing the 35 36 first equation above by ρ¯ and the second by θ¯, one arrives at 36 37 37 38 ˜ ¯ ˜ 38 1 θε ρ˜ε κ(θ) θε 39 ε(∂t + uε ·∇x) − = ε x + o(ε). 39 γ − 1 θ¯ ρ¯ ρ¯ θ¯ 40 40 41 41 We further eliminate the fluctuation of density by Boussinesq’s relation, and eventually 42 42 arrive at 43 43 44 γ ¯ 44 ρ(∂¯ t θε + uε ·∇xθε) +¯ρuε ·∇xφε κ θ xθε. (2.22) 45 γ − 1 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 16

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1 Collecting both equations (2.21) and (2.22), we arrive at the coupled system in the small 1 2 ε limit 2 3 3 θ 4 + ·∇ + ∇ +∇ = = 4 ∂t u u xu ¯ xφ xπ ν xu, divx u 0, 5 θ 5 (2.23) 6 γ − 1 6 ∂ θ + u ·∇ θ + u ·∇ φ =¯κ θ, 7 t x γ x x 7 8 8 9 where 9 10 10 γ − 1 κ(θ)¯ 11 κ¯ = . 11 12 γ ρ¯ 12 13 13 14 It is interesting to compare the temperature equation in (2.23) with (2.18); notice that the 14 15 heat conductivity in (2.23) is 1/γ that in (2.18). Besides there is no viscous heating term in 15 16 the temperature equation in (2.23), at variance with (2.18). This, however, is a consequence 16 17 of the scaling considered in the discussion above: indeed, the fluctuations of velocity and 17 18 temperature fields are of the same order of magnitude, so that viscous heating is a lower- 18 19 order effect. On the contrary, if one sets the temperature fluctuations to be of the order of 19 20 the squared fluctuations of velocity field, one recovers a viscous heating term in (2.23). 20 21 The material in this section is fairly classical and can be found in most textbooks on 21 22 fluid mechanics; more information can be gathered from the excellent introductory section 22 23 of [86]; see also the classical treatise [75]. 23 24 24 25 25 3. The Boltzmann equation and its formal properties 26 26 27 27 The Boltzmann equation is the model that governs the evolution of perfect gases in kinetic 28 28 theory. While fluid dynamics describes the state of a fluid with a few scalar or vector fields 29 29 defined on the domain filled by the fluid, such as the temperature or velocity fields, kinetic 30 30 theory describes the state of a gas with the number density (also called the distribution 31 31 function) F ≡ F(t,x,v) 0 that is the density of gas molecules which, at time t 0, 32 32 are located at the position x ∈ R3 and have velocity v ∈ R3. Put in other words, in any 33 33 infinitesimal volume dx dv centered at the point (x, v) ∈ R3 × R3 of the single particle 34 34 phase space, one can find approximately F(t,x,v)dx dv like particles at time t.Inthe 35 35 classical kinetic theory of gases, the molecular radius is neglected, except in the collision 36 36 cross-section: this has important consequences, as will be seen later. 37 37 The Boltzmann equation takes the form 38 38 39 39 ∂t F + v ·∇xF = B(F, F ), (3.1) 40 40 41 where B(F, F ) is the collision integral. This collision integral B(F, F ) is a quadratic inte- 41 42 gral operator acting only on the v-argument of the number density F , and takes the form 42 43 43 44 44 B = − − 45 (F,F)(t,x,v) F F∗ FF∗ b(v v∗,ω)dω dv∗, (3.2) 45 R3×S2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 17

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1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 Fig. 1. The pre- and post-collision velocities in the reference frame of the center of mass of the particle pair. 17 18 18 19 19 where the notations F , F∗, F and F∗ designate respectively the values F(t,x,v), 20 20 F(t,x,v∗), F(t,x,v ) and F(t,x,v ), with v ≡ v (v, v∗,ω) and v ≡ v (v, v∗,ω) given 21 ∗ ∗ ∗ 21 in terms of v, v∗ and ω by the formulas 22 22 23 23 v = v − (v − v∗) · ωω, v = v∗ + (v − v∗) · ωω, 24 ∗ (3.3) 24 25 25 ∈ S2 26 where ω is an arbitrary unit vector. These formulas represent all the solutions 26 ∈ R3 × R3 27 (v ,v∗) of the system of equations 27 28 28 2 2 2 2 29 v + v∗ = v + v∗, v + v∗ =|v| +|v∗| , (3.4) 29 30 30 31 where (v, v∗) ∈ R3 × R3 is given. If (v, v∗) are the velocities of a pair of like particles 31 32 before collision, and (v ,v∗) are the velocities of the same pair of particles after collision – 32 33 or vice versa, equalities (3.4) express the conservation of momentum and kinetic energy 33 34 during the collision. 34 35 Only the binary collisions are accounted for in Boltzmann’s equation. Indeed, since the 35 36 molecular radius is neglected in the kinetic theory of gases, one can show that collisions 36 37 involving more than two particles are events that occur with probability zero, and therefore 37 38 can be neglected for all practical purposes. 38 39 Moreover, kinetic energy is the only form of energy conserved during collisions. In 39 40 fact, the Boltzmann collision integral (3.2) applies only to monatomic gases. Polyatomic 40 41 gases can also be treated by the methods of kinetic theory; however this require using 41 42 complicated variants of Boltzmann’s original collision integral that involve vibrational and 42 43 rotational energies in addition to the kinetic energy of the center of mass of each molecule; 43 44 besides, these additional energy variables are quantized in certain applications. While such 44 45 considerations are important for understanding some real gas effects, they lead to heavy 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 18

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1 technicalities which do not belong to an expository article such as the present one. For these 1 2 reasons, we shall implicitly restrict our attention to monatomic gases and to the collision 2 3 integral (3.2) in the sequel. 3 4 The function b ≡ b(V,ω) is the collision kernel, an a.e. positive function that is of the 4 5 form 5 6 6 7 b(V,ω) =|V |Σ |V |, cos V,ω , (3.5) 7 8 8 9 where Σ is the scattering cross-section (see Section 3.5 for a precise deﬁnition of this 9 10 notion). 10 11 We shall discuss later the physical meaning of the function Σ, together with the usual 11 12 mathematical assumptions on the collision kernel b. For the moment, assume that b is 12 13 locally integrable on R3 × S2, and consider a number density F ≡ F(t,x,v) which, at 13 14 any arbitrary instant of time t and location x, is continuous with compact support in the 14 15 velocity variable v. Then, the collision integral B(F,F)(t,x,v)can be split as 15 16 16 17 17 B(F,F)(t,x,v)= B+(F,F)(t,x,v)− B−(F, F )(t, x, v), (3.6) 18 18 19 19 where 20 20 21 21 B = − 22 +(F,F)(t,x,v) F F∗b(v v∗,ω)dω dv∗, 22 R3×S2 23 (3.7) 23 24 B−(F,F)(t,x,v)= FF∗b(v − v∗,ω)dω dv∗ 24 25 R3×S2 25 26 26 27 are called respectively the gain term and the loss term in the collision integral B(F, F ). 27 28 The physical meaning of both the gain and loss terms – and that of the collision integral 28 29 itself – can be explained in the following manner: 29 30 • B−(F,F)(t,x,v)dv is the number of particles located at x at time t that exit the vol- 30 31 ume element dv centered at v in the velocity space by colliding with another particle 31 32 with an arbitrary velocity v∗ located at the same position x at the same time t, and 32 33 • B+(F,F)(t,x,v)dv is the number of particles located at x at time t that enter the 33 34 volume element dv centered at v in the velocity space as the result of a collision 34 35 involving two particles with pre-collisional velocities v and v∗ at the same time t and 35 36 the same position x. 36 37 Notice that, in this model, collisions are purely local and instantaneous, which is another 37 38 consequence of having neglected the molecular radius. Moreover, it is assumed that the 38 39 joint distribution of any pair of particles located at the same position x at the same time t 39 40 with velocities v and v∗ and that are about to collide is the product 40 41 41 42 F(t,x,v)F(t,x,v∗). 42 43 43 44 In other words, such particles are assumed to be statistically uncorrelated; however, this as- 44 45 sumption, which is crucial in the physical derivation of the Boltzmann equation, is needed 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 19

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1 only for particle pairs about to collide – and is obviously false for a pair of particles having 1 2 just collided. 2 3 Going back to the Boltzmann equation (3.1) in the form 3 4 4

5 ∂t F =−v ·∇xF + B(F, F ), 5 6 6 7 it follows from the above discussion that the first term on the right-hand side represents 7 8 the net number of particles entering the infinitesimal phase-space volume dx dv centered 8 9 at (x, v) as the result of inertial motion of particles between collisions, while the second 9 10 term represents the net number of particles entering that same volume as the result of 10 11 instantaneous and purely local collisions. 11 12 12 13 13 14 3.1. Conservation laws 14 15 15 16 Throughout this subsection, it is assumed that the collision kernel b is locally integrable, 16 17 17 18 ∈ 1 R3 × S2 18 b Lloc . (3.8) 19 19 20 20 The first major result about the Boltzmann collision integral is the following proposition. 21 21

22 3 3 22 PROPOSITION 3.1. Let F ≡ F(v)∈ Cc(R ) and φ ∈ C(R ). Then 23 23 24 24 25 B(F, F )(v)φ(v) dv 25 R3 26 26 27 1 27 = F F∗ − FF∗ φ + φ∗ − φ − φ∗ b(v − v∗,ω)dv dv∗ dω. 28 4 R3×R3×S2 28 29 29 30 This result is essential to understanding the Boltzmann equation and especially its rela- 30 31 tions to hydrodynamics. For this reason, we shall give a complete proof of it. 31 32 32 33 PROOF OF PROPOSITION 3.1. The second relation in (3.4) and the fact that F is com- 33 34 pactly supported shows that the support of (v, v∗,ω) → F F∗ − FF∗ is compact in 34 3 3 2 35 R × R × S . Hence both integrals 35 36 36 37 1 37 F F∗ − FF∗ φ + φ∗ − φ − φ∗ b(v − v∗,ω)dv dv∗ dω 38 4 R3×R3×S2 38 39 39 40 and 40 41 41 42 B(F, F )(v)φ(v) dv 42 R3 43 43 44 44 = − − 45 F F∗ FF∗ φb(v v∗,ω)dv dv∗ dω 45 R3×R3×S2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 20

20 F. G o l s e

1 are well defined since F and φ are continuous and b satisfies (3.8). In the latter inte- 1 2 gral, apply the change of variables (v, v∗) → (v∗,v), while keeping ω fixed: (3.3) show 2 3 that (v ,v∗) is changed into (v∗,v ), so that the expression (F F∗ − FF∗) is invariant, 3 4 while (3.5) shows that the collision kernel satisfies b(v∗ − v,ω) = b(v − v∗,ω). Hence 4 5 5 6 6 − − 7 F F∗ FF∗ φb(v v∗,ω)dv dv∗ dω 7 R3×R3×S2 8 8 9 = F F∗ − FF∗ φ∗b(v − v∗,ω)dv dv∗ dω 9 R3×R3×S2 10 10 11 11 = 1 − + − 12 F F∗ FF∗ (φ φ∗)b(v v∗,ω)dv dv∗ dω. 12 2 R3×R3×S2 13 13 14 14 Now in the latter integral, for a.e. fixed ω ∈ S2, apply the change of variables (v, v∗) → 15 15 (v ,v∗) defined by (3.3). It is easily seen that this transformation is an involution of 16 3 3 16 R × R , so that this change of variables maps (v ,v∗) onto (v, v∗): hence F F∗ − FF∗ is 17 17 transformed into its opposite FF∗ − F F . Formulas (3.3) also show that 18 ∗ 18 19 19 20 v − v∗ = v − v∗ and v − v∗ · ω =−(v − v∗) · ω 20 21 21 22 so that, by (3.5), one has b(v −v∗,ω)= b(v−v∗,ω). Finally, this change of variables is an 22 23 isometry of R3 × R3 by the second relation of (3.4), and therefore preserves the Lebesgue 23 24 measure. Eventually, we have proved that 24 25 25 26 26 F F − FF∗ (φ + φ∗)b(v − v∗,ω) v v∗ ω 27 ∗ d d d 27 R3×R3×S2 28 28 29 =− F F∗ − FF∗ φ + φ∗ b(v − v∗,ω)dv dv∗ dω 29 R3×R3×S2 30 30 31 31 = 1 − + − − − 32 F F∗ FF∗ φ φ∗ φ φ∗ b(v v∗,ω)dv dv∗ dω 32 2 R3×R3×S2 33 33 34 and this entails the announced formula. 34 35 35 36 36 Notice that it may not be necessary to assume that F has compact support in Proposi- 37 37 tion 3.1. For instance, the same result holds for all F and φ ∈ C(R3) if there exists m>0 38 38 such that 39 39 40 40 m −n 41 φ(v) + b(v,ω)dω = O |v| while F(v)= O |v| 41 S2 42 42 43 as |v|→+∞, with n>2m + 3. (3.9) 43 44 44 45 An important consequence of this proposition is the following corollary. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 21

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1 COROLLARY 3.2. Under the same assumptions as in Proposition 3.1 or (3.9), one has 1 2 2 3 3 B = 4 (F, F )(v) dv 0(conservation of mass), 4 R3 5 5

6 vkB(F, F )(v) dv = 0(conservation of momentum), 6 R3 7 7 8 8 1| |2B = 9 v (F, F )(v) dv 0(conservation of energy), 9 R3 2 10 10 11 for k = 1, 2, 3. 11 12 12 13 13 When applied to a solution of the Boltzmann equation F ≡ F(t,x,v), these five re- 14 14 lations are the net conservation of mass – equivalently, of the total number of particles – 15 15 momentum and energy in each phase-space cylinder dx ×R3, where dx is any infinitesimal 16 v 16 phase-space element in the space of positions R3 . 17 x 17 18 18 19 PROOF OF COROLLARY 3.2. Assuming that φ(v) is one of the functions 1, vk for 19 = 1 | |2 20 k 1, 2, 3, and 2 v , one has 20 21 21 22 φ(v)+ φ(v∗) − φ v − φ v∗ = 0 22 23 23 24 24 for each (v, v∗,ω)∈ R3 × R3 × S2, because of (3.4). Applying Proposition 3.1 shows the 25 25 five relations stated in Corollary 3.2. 26 26 27 27 28 Let F ≡ F(t,x,v) be a solution of the Boltzmann equation; assume that F(t,x,·) is 28 R3 ∈ R × R3 29 continuous with compact support on v a.e. in (t, x) + , or satisfies (3.9). Then 29 30 Corollary 3.2 implies that 30 31 31 32 32 ∂t F dv + divx vF dv = 0, 33 R3 R3 33 34 34 35 ∂t vF dv + divx v ⊗ vF dv = 0, (3.10) 35 R3 R3 36 36 37 1 2 1 2 37 ∂t |v| F dv + divx v |v| F dv = 0. 38 R3 2 R3 2 38 39 39 40 These equalities are the local conservation laws of mass, momentum and energy in space– 40 41 time divergence form. 41 42 Assume further, for simplicity, that for a.e. (t, x) ∈ R+ × R3, 42 43 43 44 44 ; 45 F(t,x,v)dv>0 45 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 22

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1 define then 1 2 2 3 ρ(t,x)= F(t,x,v)dv (macroscopic density), 3 4 R3 4 5 5 = 1 6 u(t, x) vF(t,x,v)dv (bulk velocity), (3.11) 6 ρ(t,x) R3 7 7 1 1 8 θ(t,x)= v − u(t, x)2F(t,x,v)dv (temperature). 8 9 ρ(t,x) R3 3 9 10 10 11 With these definitions, the local conservation laws (3.10) take the form 11 12 12 ∂ ρ + div (ρu) = 0, 13 t x 13 14 14 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) =−divx A(v − u)F dv, 15 R3 15 16 (3.12) 16 1 2 3 1 2 5 17 ∂t ρ |u| + θ + divx ρu |u| + θ 17 2 2 2 2 18 18 19 19 =−divx B(v − u)F dv − divx A(v − u) · uF dv, 20 R3 R3 20 21 21 22 where 22 23 23 1 1 24 A(z) = z ⊗ z − |z|2,B(z)= |z|2 − 5 z. 24 25 3 2 25 26 The left-hand side of the equalities above coincides with that of the compressible Euler 26 27 system (2.7) with γ = 5/3 (the adiabatic exponent for point particles, i.e., for particles 27 28 with 3 degrees of freedom). The right-hand side, on the contrary, depends on the solution of 28 29 the Boltzmann equation F and is in general not determined by the macroscopic variables ρ, 29 30 u and θ. 30 31 However, in some limit, it may be possible to approximate the right-hand side of (3.12) 31 32 by appropriate functions of ρ, u and θ, thereby arriving at a system in closed form with 32 33 unknown (ρ,u,θ). 33 34 For instance, deriving the compressible Euler system (2.7) as some asymptotic limit of 34 35 the Boltzmann equation would consist in proving that the right-hand side of the second and 35 36 third equations in (3.12) vanishes in that limit. Deriving the Navier–Stokes system (2.8) 36 37 from the Boltzmann equation would consist in finding some (other) asymptotic limit such 37 38 that 38 39 39 40 40 − − − =− ∇ 41 A(v u)F dv λD(u) and B(v u)F dv κ xθ, 41 R3 R3 42 42 43 and so on. 43 44 The problem of finding such closure relations is the key to all the derivations of hydro- 44 45 dynamic models from the Boltzmann equation. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 23

The Boltzmann equation and its hydrodynamic limits 23

1 3.2. Boltzmann’s H -theorem 1 2 2 3 We have seen in the last subsection how the symmetries of the Boltzmann collision integral 3 4 entail the local conservation of mass, momentum and energy. 4 5 Another important feature of these symmetries is that they also entail a variant of the 5 6 second principle of thermodynamics, as we shall now explain. 6 7 7 8 8 PROPOSITION 3.3 (Boltzmann’s H -theorem). Assume that the collision kernel b satis- 9 9 ﬁes (3.8), that F ∈ C(R3) is positive and rapidly decaying at inﬁnity, and that, for some 10 10 m>0, one has 11 11 12 12 13 b(v,ω)dω + ln F(v) = O |v|m as |v|→+∞. 13 14 S2 14 15 15 16 Then 16 17 17 18 B(F, F ) ln F dv 18 R3 19 19 20 20 =−1 − F F∗ − 21 F F∗ FF∗ ln b(v v∗,ω)dv dv∗ dω 0. 21 4 R3×R3×S2 FF∗ 22 22 23 Moreover, the following conditions are equivalent 23 24 24 B(F, F ) F v = 25 (i) R3 ln d 0, 25 3 26 (ii) B(F, F )(v) = 0 for all v ∈ R , 26 27 (iii) F is a Maxwellian distribution, i.e., there exists ρ,θ > 0 and u ∈ R3 such that 27 28 F = M(ρ,u,θ), where 28 29 29

30 ρ −|v−u|2/(2θ) 3 30 M(ρ,u,θ)(v) := e for each v ∈ R . (3.13) 31 (2πθ)3/2 31 32 32 33 As was already the case of Proposition 3.1, Boltzmann’s H -theorem is so essential in 33 34 deriving hydrodynamic equations from the Boltzmann equation that we give a complete 34 35 proof of it. 35 36 36 37 37 PROOF OF PROPOSITION 3.3. The assumptions on F and b are such that F , b and 38 φ = ln F satisfy the assumption (3.9). Applying Proposition 3.1 implies that 38 39 39 40 40 41 B(F, F ) ln F dv 41 R3 42 42 43 1 F F∗ 43 =− F F∗ − FF∗ ln b(v − v∗,ω)dv dv∗ dω. 44 4 R3×R3×S2 FF∗ 44 45 (3.14) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 24

24 F. G o l s e

1 Since the logarithm is an increasing function, one has 1 2 2 3 f 3 (f − g)ln = (f − g)(ln f − ln g) 0 for each f,g > 0, 4 g 4 5 5 6 so that the expression on the right-hand side of (3.14) is nonpositive. 6 7 If that expression is equal to zero, the integrand must vanish a.e., meaning that 7 8 8 3 3 2 9 F F∗ = FF∗ for a.e. (v, v∗,ω)∈ R × R × S , 9 10 10 11 since the collision kernel b is a.e. positive. This accounts for the equivalence between 11 12 conditions (i) and (ii). That (iii) implies (i) is proved by inspection; for instance, one can 12 13 observe that, if F is a Maxwellian distribution, ln F is a linear combination of 1, v1, v2, 13 2 14 v3 and |v| , so that 14 15 15 3 3 2 16 ln F + ln F∗ − ln F − ln F∗ = 0 for all (v, v∗,ω)∈ R × R × S , 16 17 17 18 because of the microscopic conservation laws (3.4). Finally, (i) implies (iii), as shown by 18 19 the next lemma, and this concludes the proof of Boltzmann’s H -theorem. 19 20 20 21 LEMMA 3.4. Let φ 0 a.e. be such that (1 +|v|2)φ ∈ L1(R3). If 21 22 22 23 3 3 2 23 φ φ∗ = φφ∗ for a.e. (v, v∗,ω)∈ R × R × S , 24 24 25 then φ is either a.e.0or a Maxwellian (i.e., is of the form (3.13)). 25 26 26 27 The following proof is due to Perthame [105]; Boltzmann’s original argument can be 27 28 found in Section 18 of [16]. 28 29 29 30 30 PROOF OF LEMMA 3.4. After translation and multiplication by a constant, one can always 31 assume that 31 32 32 33 33 φ(v) v = , vφ(v) v = 34 d 1 d 0 (3.15) 34 R3 R3 35 35 36 unless φ = 0 a.e. Denoting by φˆ the Fourier transform of φ, our assumptions implies that, 36 37 for a.e. ω ∈ S2, one has 37 38 38 39 39 −iξ·v−iξ∗·v∗ ˆ ˆ ∗ = ∗ 40 φ(ξ)φ(ξ ) φ v φ v∗ e dv dv 40 R3×R3 41 41 −iξ·v−iξ·v 42 = φ(v)φ(v∗)e ∗ dv dv∗ 42 R3×R3 43 43 44 44 = −iξ·v−iξ∗·v∗ i((ξ−ξ∗)·ω)((v−v∗)·ω) 45 φ(v)φ(v∗)e e dv dv∗. 45 R3×R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 25

The Boltzmann equation and its hydrodynamic limits 25

1 (Notice that the second equality follows from the same change of variables (v, v∗) → 1 2 2 (v ,v∗) as in the proof of Proposition 3.1.) In fact, this relation holds for all ω ∈ S since 2 3 both sides of the equality above are continuous in ω. 3 4 Since the left-hand side of the equality above is independent of ω, one can differentiate 4 5 in ω to obtain that 5 6 6 −iξ·v−iξ∗·v∗ 7 0 = φ(v)φ(v∗)e (v − v∗) · ω0 dv dv∗ 7 8 R3×R3 8 9 9 for any ξ = ξ∗ ∈ R3 and ω ∈ S2 such that ω ⊥ (ξ − ξ∗). In other words, 10 0 0 10 11 ˆ ˆ 11 ω0 ⊥ (ξ − ξ∗) ⇒ (∇ξ −∇ξ∗ )φ(ξ)φ(ξ∗) ⊥ ω0. 12 12 13 13 This implies that, for all ξ = ξ∗ ∈ R3, one has 14 14 15 ˆ ˆ 15 (∇ξ −∇ξ∗ )φ(ξ)φ(ξ∗) (ξ − ξ∗). (3.16) 16 16 17 Applying this with ξ∗ = 0 leads to 17 18 18 ˆ 19 ∇ξ φ(ξ) ξ, 19 20 20 ˆ 21 on account of the normalization condition (3.15). Hence φ is of the form 21 22 22 ˆ = | |2 ∈ R3 23 φ(ξ) ψ ξ ,ξ . 23 24 ˆ 24 25 Writing (3.16) with this form of φ, one ﬁnds that 25 26 26 2 2 2 2 | | | ∗| − ∗ | | | ∗| − ∗ 27 ξψ ξ ψ ξ ξ ψ ξ ψ ξ (ξ ξ ). 27 28 28 ξ ξ∗ ξ,ξ∗ ∈ R3 29 Whenever and are not colinear, i.e., for a dense subset of all , this implies 29 that 30 30 31 2 2 2 2 31 ψ |ξ| ψ |ξ∗| = ψ |ξ| ψ |ξ∗| . 32 32 33 33 Since φ ∈ L1((1 +|v|2) dv), φˆ ∈ C2(R) and the normalization conditions (3.15) imply 34 34 that φ(ˆ 0) = 1 and φˆ(0) = 0; hence ψ ∈ C1(R2) and the relation above holds for each 35 35 (ξ, ξ∗) ∈ R3 × R3. This relation implies in turn that ψ is of the form 36 36 37 − 37 ψ(r)= e θr/2. 38 38 39 Hence φˆ is of the form 39 40 40 − | |2 41 φ(ξ)ˆ = e θ ξ /2, 41 42 42

43 so that φ is of the form φ = M(1,0,θ). 43 44 44 45 At this point, it is natural to introduce the notion of collision invariant. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 26

26 F. G o l s e

1 DEFINITION 3.5. A collision invariant is a measurable function φ defined a.e. on R3 that 1 2 satisfies 2 3 3 4 φ(v)+ φ(v∗) − φ v − φ v∗ = 0a.e.in(v, v∗,ω), 4 5 5 6 where v ≡ v (v, v∗,ω)and v∗ ≡ v∗(v, v∗,ω)are defined by (3.3). 6 7 7 8 For instance, in the proof of Boltzmann’s H -theorem, Maxwellian densities are charac- 8 9 terized as the densities whose logarithms are collision invariants. 9 10 A variant of Lemma 3.4 characterizes collision invariants. 10 11 11 12 PROPOSITION 3.6. A function φ is a collision invariant if and only if there exists five 12 13 13 constants a0,a1,a2,a3,a4 ∈ R such that 14 14 15 2 3 15 φ(v)= a0 + a1v1 + a2v2 + a3v3 + a4|v| a.e. in R . 16 16 17 See Section 3.1 in [28] for a proof of the proposition above. 17 18 18 19 19 20 3.3. H -theorem and a priori estimates 20 21 21 22 We conclude with the main application of Boltzmann’s H -theorem, i.e., getting a priori 22 23 estimates on the solution of the Boltzmann equation. We shall discuss four different cases. 23 24 24 25 25 Case 1: The periodic box. Consider the Cauchy problem 26 26 27 ∗ 3 3 27 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × T × R , 28 28 in 29 F |t=0 = F . 29 30 30 31 Let F be a solution of the Boltzmann equation such that, for a.e. (t, x) ∈ R+ ×T3, F(t,x,·) 31 32 satisfies the assumptions of Proposition 3.3. Then the number density F satisfies the lo- 32 33 cal entropy inequality (3.26). Integrating this differential inequality on [0,t]×T3, one 33 34 arrives at 34 35 35 36 F ln F(t,x,v)dx dv 36 37 T3×R3 37 38 t 38 1 F F∗ 39 + F F∗ − FF∗ ln b dv dv∗ dω dx ds 39 4 T3 R3×R3×S2 FF∗ 40 0 40 41 41 = F in ln F in(x, v) dx dv (3.17) 42 T3×R3 42 43 43 44 for each t 0. 44 45 The following definition explains the name “H -theorem”. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 27

The Boltzmann equation and its hydrodynamic limits 27

1 DEFINITION 3.7. Let F 0 a.e. be an element of L1(T3 × R3) such that 1 2 2 3 3 F ln F(x,v) dx dv<+∞. 4 T3×R3 4 5 5 6 One denotes by H(F) the quantity 6 7 7 8 8 H(F)= F ln F(x,v)dx dv. 9 T3×R3 9 10 10 11 Whenever there is no risk of ambiguity, we use the notation H(t) to designate 11 12 H(F(t,·, ·)), when F is a solution of the Boltzmann equation. Equality (3.17) implies 12 13 that H(F) is a nonincreasing function of time; it was this property that Boltzmann called 13 14 “the H -theorem”. Moreover, H(F) is stationary only if F is a Maxwellian (see Sec- 14 15 tion 3.4.2). Hence, from the physical viewpoint, it is natural to think of H(F(t,·, ·)) as 15 16 minus the entropy of the system of particles distributed under F(t,·, ·). 16 17 In order to obtain a bound on the entropy production, it is convenient to introduce another 17 18 (closely related) concept of entropy. 18 19 19

20 DEFINITION 3.8. Let F 0 a.e. and G>0 be two measurable functions on T3 × R3;the 20 21 relative entropy of F with respect to G is 21 22 22 23 F 23 24 H(F|G) = F ln − F + G dx dv. 24 T3×R3 G 25 25 26 Notice that the integrand in the definition of H(F|G) is an a.e. nonnegative measurable 26 27 function, so that the relative entropy H(F|G) is well defined as an element of [0, +∞]. 27 28 Let ρ,θ >0 and u ∈ R3, then 28 29 29 30 30 |v − u|2 31 H(F|M(ρ,u,θ)) = H(t)+ F dx dv 31 T3×R3 2θ 32 32 33 33 − + ρ + 34 1 ln F dx dv ρ. (3.18) 34 (2πθ)3/2 T3×R3 35 35 36 Hence, if F ∈ L1(T3 × R3; (1 +|v|2) dv dx) and if H(0) is finite, then H(t) is finite for 36 37 each t 0, and 37 38 38 39 | |2 + 39 ρ u 1 2 40 − ln + 1 +|v| F dv dx − ρ 40 (2πθ)3/2 θ T3×R3 41 41 42 H(t) H(0). 42 43 43 44 On the other hand, F also satisfies the local conservation of mass, momentum, and en- 44 45 ergy (3.10), so that, integrating these local conservation laws on [0,t]×T3, one arrives at 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 28

28 F. G o l s e

1 the global variant of these conservation laws 1 2 2 3 3 = 4 F(t,x,v)dv dx F(0,x,v)dv dx, 4 T3×R3 T3×R3 5 5 6 vF(t,x,v)dv dx = vF(0,x,v)dv dx, (3.19) 6 T3×R3 T3×R3 7 7 8 8 1| |2 = 1| |2 9 v F(t,x,v)dv dx v F(0,x,v)dv dx. 9 T3×R3 2 T3×R3 2 10 10 11 Since 11 12 12 13 |v − u|2 13 14 F dx dv 14 T3×R3 2θ 15 15 1 1 16 = |v|2 +|u|2 F dx dv − u · vF dx dv, 16 17 2θ T3×R3 θ T3×R3 17 18 18 19 one has 19 20 20 21 21 H F(t) M(ρ,u,θ) = H(t)+ globally conserved quantities 22 22 23 so that 23 24 24 25 25 H F(t)M − H F(0)M = H(t)− H(0). 26 (ρ,u,θ) (ρ,u,θ) 26 27 27 28 Hence, the global entropy relation (3.17) is recast in terms of the relative entropy as 28 29 29 t 30 1 F F∗ 30 F F∗ − FF∗ ln b dv dv∗ dω dx ds 31 4 0 T3 R3×R3×S2 FF∗ 31 32 32 = H F(0)M − H F(t)M (3.20) 33 (ρ,u,θ) (ρ,u,θ) 33 34 34 3 35 for each ρ,θ >0 and each u ∈ R . This implies, in particular, 35 36 • the relative entropy bound 36 37 37 38 0 H F(t) M(ρ,u,θ) H F(0) M(ρ,u,θ) ,t 0; 38 39 39 40 • the following entropy control 40 41 41 42 | |2 + 42 ρ u 1 2 in 43 − ln + 1 +|v| F dv dx − ρ 43 (2πθ)3/2 θ T3×R3 44 44 45 H (F )(t) H(F)(0); 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 29

The Boltzmann equation and its hydrodynamic limits 29

1 • the entropy production estimate 1 2 2 +∞ 3 1 F F∗ 3 F F∗ − FF∗ ln b dv dv∗ dω dx ds 4 4 0 T3 R3×R3×S2 FF∗ 4 5 5 H F(0) M(ρ,u,θ) . 6 6 7 Case 2: A bounded domain with specular reflection on the boundary. The periodic box 7 8 is a somewhat academic choice of a spatial domain for studying the Boltzmann equation. 8 9 The next case that we consider now is very similar but more realistic. Let Ω be a smooth, 9 10 bounded domain of R3. Starting from a given number density F in at time t = 0, we con- 10 11 sider the initial boundary value problem 11 12 12 3 13 ∂t F + v ·∇xF = B(F, F ), (x, v) ∈ Ω × R , 13 14 14 F(t,x,v)= F(t,x,R v), (x,v) ∈ ∂Ω × R3, 15 x 15 in 16 F |t=0 = F , 16 17 17 18 where Rx designates the specular reflection defined by the outward unit normal nx 18 19 at x∈ ∂Ω 19 20 20 R = − · 21 xv v 2(v nx)nx. 21 22 22 Assume that the initial boundary value problem above has a solution F satisfying the as- 23 23 sumptions of Proposition 3.3. One multiplies the Boltzmann equation above by ln F + 1 24 24 and integrates first in v to obtain the identity (3.26), and then integrates in x, which leads to 25 25 26 26 d + · 27 F ln F dx dv F ln Fv nx dσ(x)dv 27 dt Ω×R3 ∂Ω×R3 28 28 29 1 F F∗ 29 =− F F∗ − FF∗ ln b(v − v∗,ω)dv dv∗ dω. 30 4 R3×R3×S2 FF∗ 30 31 31 → = R 32 Changing the v variable in the boundary term by v w xv, one sees that 32 · =− · 33 w nx v nx , while the specular reflection condition satisfied by F on ∂Ω implies 33 F(t,x,v)= F(t,x,w) 34 that ; besides this change of variables preserves the Lebesgue mea- 34 sure dv since R is an isometry. Hence 35 x 35 36 36 37 F ln Fv· nx dσ(x)dv =− F ln Fw· nx dσ(x)dw = 0 37 ∂Ω×R3 ∂Ω×R3 38 38 39 and therefore 39 40 40 d 41 H (F )(t) 41 dt 42 42 43 1 F F∗ 43 =− F F − FF∗ b(v − v∗,ω) v v∗ ω. 44 ∗ ln d d d 44 4 R3×R3×S2 FF∗ 45 (3.21) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 30

30 F. G o l s e

1 Proceeding similarly with the local conservation laws (3.10) shows that 1 2 2 d d 1 3 F dx dv = |v|2F dx dv = 0. (3.22) 3 4 dt Ω×R3 dt Ω×R3 2 4 5 5 = 6 At this point, we apply the formula (3.18) in the case where u 0 and deduce from (3.22) 6 7 that 7 8 8 H(F|M ) = H(F)+ globally conserved quantities. 9 (ρ,0,θ) 9 10 10 Notice that, at variance with the case of the periodic box, the total momentum is not con- 11 11 served, so that the formula above only holds with centered Maxwellians (i.e., Maxwellians 12 with zero bulk velocity). 12 13 Therefore, as in the case of the periodic box, one has, for each ρ,θ >0, 13 14 14 15 t 15 1 F F∗ 16 F F∗ − FF∗ ln b dv dv∗ dω dx ds 16 4 0 Ω R3×R3×S2 FF∗ 17 17 18 = H F(0) M(ρ,0,θ) − H F(t) M(ρ,0,θ) (3.23) 18 19 19 20 for each t 0. Again we obtain 20 • 21 the relative entropy bound 21 22 22 M M ; 23 0 H F(t) (ρ,u,θ) H F(0) (ρ,u,θ) ,t 0 23 24 24 • the following entropy control 25 25 26 26 ρ 1 2 in 27 − ln + 1 +|v| F dv dx − ρ 27 (2πθ)3/2 θ T3×R3 28 28 29 H (F )(t) H(F)(0); 29 30 30 • 31 the entropy production estimate 31 32 +∞ 32 1 F F∗ 33 F F∗ − FF∗ ln b dv dv∗ dω dx ds 33 4 0 Ω R3×R3×S2 FF∗ 34 34 35 35 H F(0) M(ρ,u,θ) . 36 36 37 Case 3: The Euclidean space with Maxwellian equilibrium at inﬁnity. Next we study 37 38 cases where the spatial domain is unbounded. The simplest of such cases is that of a cloud 38 39 of gas which is in Maxwellian equilibrium at inﬁnity. Therefore, we consider the Cauchy 39 40 problem 40 41 41 ∗ 3 3 42 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × R × R , 42 43 43 F(t,x,v)→ M(ρ,u,θ), |x|→+∞, 44 44 in 45 F |t=0 = F . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 31

The Boltzmann equation and its hydrodynamic limits 31

1 We shall assume that F converges to the Maxwellian state M(ρ,u,θ) rapidly enough so that 1 2 the relative entropy 2 3 3 4 H F(t) M(ρ,u,θ) 4 5 5 F 6 = F − F + M x v<+∞ 6 ln M (ρ,u,θ) d d 7 R3×R3 (ρ,u,θ) 7 8 8 for each t 0. We claim that the same entropy relation as in the case of the three-torus 9 9 also holds in the present situation 10 10 11 t 11 1 F F∗ 12 F F∗ − FF∗ ln b dv dv∗ dω dx ds 12 4 0 R3 R3×R3×S2 FF∗ 13 13 14 = H F(0) M(ρ,u,θ) − H F(t) M(ρ,u,θ) (3.24) 14 15 15 16 for each t 0. However, this equality is not obtained in the same way, since neither the 16 17 globally conserved quantities nor the H -function itself are well-deﬁned objects in this 17 18 case (these quantities involve divergent integrals because of the Maxwellian condition at 18 19 inﬁnity). 19 20 Observe instead that, by the same argument as in the case of the three-torus, one has 20 21 21 22 F 22 F ln − F + M(ρ,u,θ) dv 23 R3 M(ρ,u,θ) 23 24 24 1 25 = F ln F dv + |v|2 +|u|2 F dv 25 R3 2θ R3 26 26 27 1 ρ 27 − u · vF dv + 1 + ln F dv + ρ 28 θ R3 (2πθ)3/2 R3 28 29 29 30 while 30 31 31 32 F 32 v F ln − F + M(ρ,u,θ) dv 33 R3 M(ρ,u,θ) 33 34 34 1 2 2 35 = vF ln F dv + v |v| +|u| F dv 35 R3 2θ R3 36 36 1 ρ 37 − vu · vF dv + 1 + ln vF dv. 37 38 θ R3 (2πθ)3/2 R3 38 39 39 40 In other words, 40 41 41 F 42 42 F ln − F + M(ρ,u,θ) dv R3 M(ρ,u,θ) 43 43 44 44 = + 45 F ln F dv locally conserved quantity 45 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 32

32 F. G o l s e

1 while 1 2 2 3 3 F − + M 4 v F ln F (ρ,u,θ) dv 4 R3 M(ρ,u,θ) 5 5 6 = vF ln F dv + ﬂux of that locally conserved quantity 6 7 R3 7 8 8 9 so that 9 10 10 11 F 11 ∂t F ln − F + M(ρ,u,θ) dv 12 R3 M(ρ,u,θ) 12 13 13 F 14 + divx v F ln − F + M(ρ,u,θ) dv 14 R3 M(ρ,u,θ) 15 15 16 16 = + 17 ∂t F ln F dv divx vF ln F dv 17 R3 R3 18 18 19 so that 19 20 20 21 F 21 22 ∂t F ln − F + M(ρ,u,θ) dv 22 R3 M(ρ,u,θ) 23 23 24 F 24 + divx v F ln − F + M(ρ,u,θ) dv 25 R3 M(ρ,u,θ) 25 26 26 F F 27 1 ∗ 27 =− F F∗ − FF∗ ln b dv dv∗ dω. 28 4 R3×R3×S2 FF∗ 28 29 29 30 Integrating further on [0,t]×R3, one arrives at (3.24). 30 31 To summarize, we deduce from (3.24) that 31 32 • the relative entropy bound 32 33 33 34 in 34 0 H F(t) M(ρ,u,θ) H F M(ρ,u,θ) for each t 0; 35 35 36 36 • 37 the entropy production estimate 37 38 38 +∞ 39 1 F F∗ 39 F F − FF∗ ln b dv dv∗ dω dx ds 40 ∗ 40 4 0 R3 R3×R3×S2 FF∗ 41 41 inM 42 H F (ρ,u,θ) . 42 43 43 44 Case 4: The Euclidean space with vacuum at inﬁnity. Finally, we consider the case of a 44 45 cloud of gas expanding in the vacuum. As we shall see, this case is slightly different from 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 33

The Boltzmann equation and its hydrodynamic limits 33

1 the previous ones. Consider the Cauchy problem 1 2 2 ∗ 3 3 3 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × R × R , 3 4 4 F(t,x,v)→ 0, |x|, |v|→+∞, 5 5 in 6 F |t=0 = F . 6 7 7 8 We shall assume that F vanishes rapidly enough at inﬁnity so that the relative entropy 8 9 9 G +∞ 10 H F(t) < for each t 0, 10 11 11 G 12 where is the centered reduced Gaussian 12 13 13 1 −(|x|2+|v|2)/2 14 G(x, v) = e . 14 (2π)3 15 15 16 Assume that 16 17 17 18 F in ln F in +|x|2 +|v|2 + 1 dx dv<+∞. 18 19 R3×R3 19 20 20 21 We claim that, for each 0, 21 22 22 2 2 23 |x − tv| F(t,x,v)dx dv = |x| F(0,x,v)dx dv. (3.25) 23 R3×R3 R3×R3 24 24 25 Indeed 25 26 26 27 d 27 |x − tv|2F(t,x,v)dx dv 28 dt R3×R3 28 29 29 2 30 = ∂t |x − tv| F(t,x,v) dx dv 30 R3×R3 31 31 32 2 32 = (∂t + v ·∇x) |x − tv| F(t,x,v) dx dv 33 R3×R3 33 34 34 2 35 = |x − tv| (∂t + v ·∇x)F (t, x, v) dx dv 35 R3×R3 36 36 37 37 = | − |2B = 38 x tv (F,F)(t,x,v)dv dx 0. 38 R3 R3 39 39 40 Observe that 40 41 41 1 42 H(F|G) = H(F)+ |x|2 +|v|2 F dx dv 42 2 R3×R3 43 43 44 44 + π − + 45 3ln(2 ) 1 F dx dv 1. 45 R3×R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 34

34 F. G o l s e

1 Because of (3.25), one has 1 2 2 3 3 |x|2F(t)dx dv 4 R3×R3 4 5 5 6 2 |x − tv|2F(t)dx dv + 2t2 |v|2F(t)dx dv 6 R3×R3 R3×R3 7 7 8 8 = | |2 + 2 | |2 9 2 x F(0) dx dv 2t v F(0) dx dv 9 R3×R3 R3×R3 10 10 11 so that 11 12 12 13 13 − | |2 in − 1 + 2 | |2 in 14 x F dx dv t v F dx dv 14 R3×R3 2 R3×R3 15 15 16 − 3ln(2π) − 1 F in dx dv − 1 H(t) H(0). 16 17 R3×R3 17 18 18 19 Integrating on [0,t]×R3 the local entropy equality (3.26), one arrives at the equality 19 20 20 21 H(0) − H(t) 21 22 22 t 1 F F∗ 23 = F F∗ − FF∗ ln b dv dv∗ dω dx ds 23 4 R3 R3×R3×S2 FF∗ 24 0 24 25 25 + 2 +| |2 +| |2 + in in 26 C 1 t 1 x v ln F F dv dx. 26 R3×R3 27 27 28 As we shall see below, Cases 1–3 are the most useful in the context of hydrodynamic 28 29 limits. Case 4 is also interesting, although not for hydrodynamic limits: it provides one 29 30 of the important estimates in the construction of global weak solutions to the Boltzmann 30 31 equation by R. DiPerna and P.-L. Lions (see further). 31 32 Another case, which we did not discuss in spite of its obvious interest for applications, 32 33 is that of a spatial domain that is the complement in R3 of a regular compact set, assuming 33 34 specular reflection of the particles at the boundary of the domain. This case is handled by 34 35 35 a straightforward adaptation of the arguments in Cases 2 and 3. 36 36 Let us now briefly discuss some of the main consequences of Boltzmann’s H -theorem. 37 37 38 38 39 39 40 3.4. Further remarks on the H -theorem 40 41 41 42 3.4.1. H -theorem and the second principle of thermodynamics. To begin with, let F be a 42 43 solution of the Boltzmann equation such that, for a.e. (t, x) ∈ R+ × R3, F(t,x,·) satisfies 43 44 the assumptions of Proposition 3.3. Multiplying both sides of the Boltzmann equation by 44 45 ln F + 1 and applying Proposition 3.3 and Corollary 3.2 with φ ≡ 1, one arrives at the 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 35

The Boltzmann equation and its hydrodynamic limits 35

1 identity 1 2 2 3 3 ∂t F ln F dv + divx vF ln F dv 4 R3 R3 4 5 5 1 F F∗ 6 =− F F∗ − FF∗ ln b(v − v∗,ω)dv dv∗ dω 6 R3×R3×S2 FF∗ 7 4 7 8 0. (3.26) 8 9 9 10 It is interesting to compare the equality (3.26) with the second principle of thermodynamics 10 11 applied to any portion of a fluid in a smooth domain Ω. Denoting by nx the outward unit 11 12 normal field on ∂Ω, and by s the entropy per unit of mass in the fluid, one has 12 13 13 14 d q(t,x) 14 ρs(t,x)dt − ρsu(t,x) · nx dσ(x)− · nx dσ(x), 15 dt Ω ∂Ω ∂Ω θ(t,x) 15 16 16 17 where ρ is the density of the fluid, u the velocity field, θ the temperature, q the heat flux 17 18 and dσ(x)the surface element on ∂Ω. The infinitesimal version of this inequality is 18 19 19 20 q 20 ∂t (ρs) + divx ρsu + 0, (3.27) 21 θ 21 22 22 23 which is obviously analogous to (3.26). In particular, 23 24 24 25 the quantity − F ln F dv is analogous to ρs 25 26 R3 26 27 27 and 28 28 29 29 − 30 the quantity vF ln F dv is analogous to ρsu, 30 R3 31 31 32 while the quantity 32 33 33 34 1 F F∗ 34 F F − FF∗ b(v − v∗,ω) v v∗ ω 35 ∗ ln d d d 35 4 R3×R3×S2 FF∗ 36 36 37 is the local entropy production. Notice that fluid dynamics does not in general provide any 37 38 expression of the entropy production in terms of ρ, u, θ, s and q. On the contrary, in the 38 39 kinetic theory of gases, the entropy production is given in terms of the number density by 39 40 the integral above. 40 41 41 42 3.4.2. Relaxation towards equilibrium. One application of the second principle of ther- 42 43 modynamics is the relaxation towards equilibrium for closed systems. Assume that a gas 43 44 described by the Boltzmann equation is enclosed in some container Ω that is a smooth, 44 45 bounded domain of R3. At the microscopic level, we assume that the gas molecules are 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 36

36 F. G o l s e

1 reflected on the surface of the container without exchanging heat. One model for this is 1 2 the ideal situation where each molecule impinging on the boundary of the container is 2 3 specularly reflected, this being Case 2 of Section 3.3. 3 4 Starting from a given number density F in at time t = 0, we consider the initial boundary 4 5 value problem 5 6 6 3 7 ∂t F + v ·∇xF = B(F, F ), (x, v) ∈ Ω × R , 7 8 3 8 F(t,x,v)= F(t,x,Rxv), (x,v) ∈ ∂Ω × R , 9 9 in 10 F |t=0 = F , 10 11 11 12 where Rx designates the specular reflection defined by the outward unit normal nx at 12 13 x ∈ ∂Ω 13 14 14 15 Rxv = v − 2v · nxnx. 15 16 16 17 Now pick any sequence tn →+∞such that 17 18 18 19 Fn(t,x,v):= F(t + tn,x,v)→ E(t,x,v) (3.28) 19 20 20 21 in a weak topology that is compatible with the conservation laws (3.22). Then, by weak 21 22 convergence and convexity, the bound on entropy production obtained in Case 2 of 22 23 Section 3.3 implies that E(t,x,v) is a local Maxwellian – meaning that the function 23 24 v → E(t,x,v) is a.e. a Maxwellian with parameters ρ,u,θ that are functions of t,x – 24 25 which satisfies 25 26 26 + ·∇ = ∈ × R3 27 (∂t E v xE)(t,x,v) 0,(x,v)Ω , 27 (3.29) 28 3 28 E(t,x,v) = E(t,x,Rxv), (x,v) ∈ ∂Ω × R . 29 29 30 Whenever Ω is not rotationally invariant with respect to some axis of symmetry, the only 30 31 local Maxwellians that solve the system of equations (3.29) are the global Maxwellians of 31 32 the form 32 33 33 34 34 E(t,x,v) = M(ρ,0,θ)(v) for some constant ρ,θ >0. (3.30) 35 35 36 Since we assumed that the conservation laws (3.22) are compatible with the topology in 36 37 which the long time limit holds, the constants ρ and θ are given by 37 38 38 39 39 in = | | 1 in = 3 | | 40 F dx dv ρ Ω , F dx dv ρθ Ω . 40 Ω×R3 Ω×R3 2 2 41 41 42 However, if Ω is rotationally invariant around some axis of symmetry, the system (3.29) 42 43 has other solutions than the global Maxwellians (3.30), namely all functions of the form 43 44 44 λ(k×(x−x0))·v 45 E(t,x,v) = M(ρ,0,θ)(v)e , 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 37

The Boltzmann equation and its hydrodynamic limits 37

1 where λ = 0 is a constant, while the axis of rotational symmetry for Ω is the line of 1 2 2 direction k ∈ S passing through x0. 2 3 In fact, Lions proved in Section V of [84] that the convergence (3.28) is locally uniform 3 4 in t with values in the strong topology of L1(Ω × R3). 4 5 Let us mention that, in spite of his apparent simplicity, the problem of relaxation to- 5 6 ward equilibrium for the Boltzmann equation is still open, in spite of recent progress by 6 7 Desvillettes and Villani [35]. The main issue is the lack of a tightness estimate in the 7 8 v variable for |v|2F(t,x,v) as t →+∞that would apply to all initial data of finite mass, 8 9 energy and entropy. Short of such an estimate, the part of the argument above identify- 9 10 ing the temperature in terms of the initial data fails. The only cases where such estimates 10 11 have been obtained correspond to initial data that are already close enough to some global 11 12 Maxwellian, or that are independent of the space variable (i.e., the space homogeneous 12 13 case). 13 14 However incomplete, this discussion shows the importance of Boltzmann’s H -theorem 14 15 whenever one seeks to estimate how close to the class of local Maxwellians a given solution 15 16 of the Boltzmann equation may be. This particular point is of paramount importance for 16 17 hydrodynamic limits. 17 18 18 19 19 20 3.5. The collision kernel 20 21 21 22 So far, our discussion of the Boltzmann equation – in fact, of the Boltzmann collision 22 23 integral – did not use much of the properties of the collision kernel b. Indeed, we only 23 24 took advantage of the symmetries of b in (3.5) and some additional bounds such as (3.8) 24 25 or (3.9). 25 26 However, the derivation of hydrodynamic limits requires further properties of the colli- 26 27 sion integral, for which a more extensive discussion of the collision kernel becomes neces- 27 28 sary. 28 29 First we recall some elementary facts concerning the two-body problem. Consider two 29 30 points of unit mass subject to a repulsive interaction potential U ≡ U(r), where r is the 30 31 distance between these two points. In other words, assume that U satisfies the properties 31 32 32 ∞ ∗ + 33 U ∈ C R+ is decreasing, lim U(r)=+∞, lim U(r)= 0 . 33 r→0+ r→+∞ 34 34 35 It is well known that both points stay in the same plane for all times. Pick a Galilean 35 36 frame where one of the points is at rest; then, the trajectory of the moving point is easily 36 37 expressed in polar coordinates (r, θ) with the fixed point as origin. Choosing the origin of 37 38 polar angles to be the line asymptotic to the trajectory of the moving particle in the past, 38 39 the trajectory is determined as follows. Let v be the speed of the moving particle at infinity, 39 40 and let h be the impact parameter defined by 40 41 41 42 vh = r2(t)θ(t),˙ t ∈ R, 42 43 43 44 this quantity being a well-known first integral of the motion (see Figure 2). In other words, 44 45 h is the distance between the line asymptotic to the trajectory of the moving particle in the 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 38

38 F. G o l s e

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 Fig. 2. Deflection of a particle subject to a radial repulsion potential from a particle at rest. 16 17 17 18 18 past and the parallel line going through the particle at rest (see Figure 2). Let z∗ > 0bethe 19 unique solution to 19 20 20 21 21 − 2 − 4 h = 22 1 z∗ U 0, 22 v2 z∗ 23 23 24 and set 24 25 25 z∗ h ∗ dz 26 r∗ = and θ = . (3.31) 26 27 z∗ 0 1 − z2 − 4/v2U(h/z) 27 28 28 29 The point of polar coordinates (r∗,θ∗) is the apse of the trajectory, i.e., the closest to the 29 30 particle at rest. Then the trajectory of the moving particles is given in polar coordinates by 30 31 the equation 31 32  32  h/r √ dz ∈ ] 33  for θ (0,θ∗ and r>r∗, 33 0 1−z2−4/v2U(h/z) 34 θ = 34  h/r dz  2θ∗ − √ for θ ∈[θ∗, 2θ∗) and r>r∗. 35 0 1−z2−4/v2U(h/z) 35 36 36 37 Notice in particular that the moving particle is deflected of an angle 2θ∗. 37 38 Next we recall the notion of scattering cross-section. Pick an arbitrary relative speed v 38 39 at infinity, and consider the deflection angle χ∗ = π − 2θ∗ as a function of the impact 39 40 parameter h. It is easily seen that 40 41 41 − + 42 χ∗ is decreasing, lim χ∗(h) = π and lim χ∗(h) = 0 . 42 h→0+ h→+∞ 43 43 44 Because the two-body problem is invariant by any rotation around the line D passing 44 45 through the particle at rest that is parallel to the asymptote in the past to the trajectory 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 39

The Boltzmann equation and its hydrodynamic limits 39

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 Fig. 3. The scattering cross-section corresponding to the relative speed v andinthedirectionχ corresponding to 14 15 the impact parameter h. 15 16 16 17 17 18 of the moving particle, consider the map 18 19 19 R2 { } R∗ ×[ π → π ×[ π S2 { } 20 0 + 0, 2 ) (0, ) 0, 2 ) N,S , 20 (3.32) 21 21 (h, φ) → χ∗(h), φ , 22 22 23 23 where the first identification is through polar coordinates in the plane orthogonal to D 24 24 with origin h = 0 (the intersection of D with that plane), while the second identification is 25 25 through spherical coordinates, with D as the polar axis and S2 ∩ D ={N,S}, see Figure 3. 26 26 The image of the Lebesgue measure under this map is a surface measure on S2 of the form 27 27 28 28 S(v,χ)sin χ dχ dφ, 29 29 30 30 31 and S(v,χ) is the scattering cross-section in the direction χ corresponding to the relative 31 32 speed v. Because S is the density with respect to the Euclidean surface element on the unit 32 33 sphere (which is dimensionless) of the image of the two-dimensional Lebesgue measure 33 34 (which has the dimension of a surface) under the map (3.32), it has the dimension of a 34 35 surface, which justifies the name “cross-section”. 35 36 The scattering cross-section clearly depends upon the computation of the deflection an- 36 37 gle in (3.31) 37 38 38 h 39 S(v,χ) = . 39 | | 40 sin χ χ∗(h) χ∗(h)=χ 40 41 41 42 Here are the scattering cross-sections for a few typical interactions: 42 43 • in the case of a hard sphere interaction, 43 44 44

45 U(r)= 0ifr d0 and U(r)=+∞ if 0

40 F. G o l s e

1 In this example, U is not decreasing but only nonincreasing and has finite range. 1 2 Therefore, the definition of the scattering cross-section must be modified as follows. 2 3 The map (3.32) is replaced with 3 4 4 2 5 B(0,d0) (0,d0) ×[0, 2π) →[0, π) ×[0, 2π) S {N,S}, 5 6 (3.33) 6 → ∗ 7 (h, φ) χ (h), φ , 7 8 8 and S(v,χ)sin χ dχ dφ is the image under the above map of the restriction to the disk 9 9 B(0,d ) ⊂ R2 of the Lebesgue measure. With this slightly modified definition, it is 10 0 10 found that 11 11 12 1 12 S(v,χ) = d2; 13 4 0 13 14 14 − 15 • if U(r)= kr s with s>0, set 15 16 16 ζ(l) 17 dz 17 ϑ(l)= , 18 18 0 1 − z2 − 2(z/l)s 19 19 20 where l = (v2/2k)1/sh, and where ζ(l) is the only positive root of the denominator of 20 21 the integrand above; set ϑ → λ(ϑ) to be the inverse of l → ϑ(l) so defined. Then 21 22 22

23 − β((π − χ)/2) 23 S(v,χ) = (2k)2/sv 4/s , 24 sin χ 24 25 25 26 where β(ϑ) = λ(ϑ)λ (ϑ). One finds that β is singular near θ = π/2, which corre- 26 27 sponds to χ = 0 and l →+∞, i.e., to collisions with small deflection angles, or 27 28 equivalently to the case of grazing collisions 28 29 29 − − 30 π 1 2/s π − 30 β(θ) C − θ as θ → , 31 2 2 31 32 32 33 while β(θ) = O(θ) for θ → 0. 33 34 Although the usual definition of the scattering cross-section involves the deflection an- 34 35 gle χ, one might find it easier to use instead the angle θ = (π − χ)/2 (see Figure 2). It 35 36 is easily seen that the scattering cross-section S as above can be expressed in terms of a 36 37 function Σ ≡ Σ(v,µ) defined on R+ ×[0, 1) by the formula 37 38 38 1 39 Σ v,| cos θ| = S(v,π − 2θ)| cos θ|. 39 40 2 40 41 41 42 This function Σ has the following geometric interpretation: the mapping 42 43 43 S2 {N,S} (0, π) ×[0, 2π) → S2 {N}, 44 44 45 ω (θ, φ) → − cos(2θ),sin(2θ)cos φ,sin(2θ)sin φ 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 41

The Boltzmann equation and its hydrodynamic limits 41

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 Fig. 4. The double cover θ → χ. 13 13 14 14 15 is a double cover (see Figure 4) and the image of the surface measure Σ(v,| cos θ|) × 15 16 16 sin θ dθ dφ under the above mapping is 2S(v,χ)sin χ dχ dφ. 17 17 Then, the collision kernel b is given by 18 18 19 19 b(V,ω) =|V |Σ |V |, cos V,ω . 20 20 21 21 22 With the formulas for the scattering cross-section given above, one sees that 22 • 23 in the case of hard-spheres with radius r0, 23 24 24 1 25 = 2| · |; 25 b(V,ω) r0 V ω (3.34) 26 2 26 27 27 • = −s 28 in the case of an interaction potential U(r) kr for s>0, 28 29 29 1 − β(θ) 30 b(V,ω) = (2k)2/s|V |1 4/s with θ = V,ω (3.35) 30 31 4 sin θ 31 32 32 = → + = π − −1−2/s → π− 33 with β(θ) O(θ) as θ 0 while β(θ) O(( /2 θ) ) as θ /2; 33 • = | | 34 whenever s 4, the collision kernel b is independent of V ; such potentials are usu- 34 35 ally referred to as Maxwellian potentials, and considerably facilitate the analysis of 35 36 the collision integral; 36 37 • for s = 1, which is the case of a repulsive Coulomb potential, one has 37 38 38 −3 39 − β(θ) π π 39 b(V,ω) = k2|V | 3 with β(θ) = O − θ as θ → . (3.36) 40 sin θ 2 2 40 41 41 42 Observe that, for any inverse power-law potential U(r)= kr−s , 42 43 43 44 44 =+∞ 45 b(V,ω)dω (3.37) 45 S2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 42

42 F. G o l s e

1 because of the singularity at θ = π/2, so that our earlier assumption (3.8) is violated 1 2 by such potentials. In particular, whenever the particle interaction is given by an inverse 2 3 power-law potential, one cannot split the collision integral as 3 4 4 5 B(F, F ) = B+(F, F ) − B−(F, F ) 5 6 6 7 as was done earlier in this section. One way around this is to define B(F, F ) in the sense 7 8 of distributions, as follows 8 9 9 10 B(F, F ), φ 10 11 11 1 12 = F F∗ − FF∗ φ + φ∗ − φ − φ∗ b(v − v∗,ω)dv dv∗ dω 12 R3×R3×S2 13 4 13 14 14 ∈ 1 R3 ∈ 1 R3 15 for each F Cc ( ) and φ Cc ( ). Then 15 16 16 2 17 F F∗ − FF∗ φ + φ∗ − φ − φ∗ = O (v − v∗) · ω 17 18 18 1−2/s 19 so that the integrand in the right-hand side above is O((π/2 − θ) ). This procedure 19 −s 20 can handle all inverse power-law potentials U(r)= kr for s>1; however, the Coulomb 20 21 case s = 1 remains excluded. 21 22 Observe that, in addition to the singularity in the deflection angle at θ = π/2, the col- 22 −3 23 lision kernel of the Coulomb potential also has a |v − v∗| singularity in the velocity 23 24 variable. Physicists deal with the latter singularity by introducing a further truncation near 24 25 V = 0; the dependence upon this truncation parameter of the collision integral is only loga- 25 26 rithmic, so that the result does not depend “too much” on the truncation parameter. In other 26 27 words, the collision integral is computed modulo a scaling factor known as the Coulomb 27 28 logarithm (see [76], Section 41). 28 29 Another way of avoiding the singularity of the collision kernel at θ = π/2 consists in 29 30 assuming that the interaction potential is truncated at large distance, in other words, that 30 31 31 32 U(r)= U(rC) whenever r rC. (3.38) 32 33 33 34 In this case (as in the hard sphere case), U is not decreasing but only nonincreasing, and the 34 35 definition of the scattering cross-section must be modified as in (3.33) with rC in the place 35 36 of d0.CallbC the collision kernel corresponding to the potential truncated as in (3.38), 36 37 and SC, ΣC the associated scattering cross-section. Then 37 38 38 2π π 39 39 bC(V, ω) dω =|V | ΣC |V |, | cos θ| sin θ dθ dφ 40 S2 0 0 40 41 2π π 41 42 =|V | SC |V |,χ sin χ dχ dφ 42 0 0 43 43 44 rc 2π 44 =| | =| |π 2 45 V h dh dφ V rC 45 0 0 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 43

The Boltzmann equation and its hydrodynamic limits 43

1 since SC(v, χ) sin χ dχ dφ is the image under the map (3.33) of the Lebesgue measure on 1 2 the two-dimensional disk of radius rC. Hence the truncation (3.38) leads to a collision ker- 2 3 nel whose integral over the angle variables is finite, thereby avoiding the divergence (3.37) 3 4 that occurs for any infinite range, inverse power law potential. 4 5 Yet another way of avoiding the singularity of the collision kernel at θ = π/2 was pro- 5 6 posed by Grad [64]. He considered molecular force laws for which the collision kernel 6 7 satisfies the condition 7 8 8 9 b(V,ω) − 9 C |V |+|V |1 ε , (3.39) 10 | cos(V, ω)| 10 11 11 −s 12 where C>0 and ε ∈ (0, 1). Comparing the case of a power law potential U(r) = kr 12 13 with Grad’s assumption above, we see that the collision kernel in (3.35) satisfies (3.39) 13 14 provided that one modifies the function β near θ = π/2 so that β(θ) = O(π/2 − θ).One 14 15 possibility is to replace β with 15 16 16 17 ˜ = 17 β(θ) β(θ)1θθ0 , (3.40) 18 18 19 ∈ π 19 where θ0 (0, 2 ) is some arbitrary value. Then, the associated truncated collision kernel 20 b˜ satisfies the bounds 20 21 21 22 22 23 23 ˜ +| | 1−4/s 1 +| | 1−4/s 24 b(V,ω) Cb 1 V and b(V,ω)dω 1 V 24 S2 Cb 25 (3.41) 25 26 26

27 for some positive constant Cb. The potential U is called a hard cut-off potential if s 4, 27 28 and a soft cut-off potential if s<4. 28 29 Grad defined more general classes of hard and soft cut-off potentials; specifically, a gen- 29 30 eral hard cut-off potential corresponds to the condition 30 31 31 | | 32 c V 32 33 b(V,ω)dω 33 S2 1 +|V | 34 34 35 while a soft cut-off potential is defined by the condition 35 36 36 37 − 37 b(V,ω)dω c 1 +|V |ε 1 38 S2 38 39 39 40 for some c>0 and ε ∈ (0, 1), in addition to the bound (3.39). In the sequel, we shall mostly 40 41 restrict our attention to those hard cut-off potentials that satisfy the same bound (3.41) as 41 42 in the inverse power law case. 42 43 The terminology “cut-off potential” attached to Grad’s cut-off prescription, is somewhat 43 44 unfelicitous. Indeed, it is not equivalent to truncating the potential at large intermolecu- 44 45 lar distances as in (3.38). Indeed, the angular truncation (3.40) prohibits grazing collisions 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 44

44 F. G o l s e

1 with deflection angle less than a threshold π − 2θ0 that is independent of the relative veloc- 1 2 ity V . On the contrary, with a potential truncated as in (3.38), there exist grazing collisions 2 3 with deflection angle arbitrarily small for large enough relative velocity |V |. Hence, it 3 4 would be more appropriate to refer to Grad’s procedure as leading to a “cut-off scattering 4 5 cross-section” rather than a “cut-off potential”. Yet the latter terminology is commonly 5 6 used in the literature, so that changing it would only cause confusion. 6 7 Let us conclude this subsection with a few words on the physical relevance of Grad’s 7 8 cut-off assumption. Grad observed that, in gases of neutral particles with short range inter- 8 9 actions, grazing collisions are not statistically dominant as in the case of plasmas, where 9 10 the long range effect of the Coulomb interaction must be accounted for. The latter case 10 11 requires using a mean-field description of the long-range interaction potential, in addition 11 12 to the Landau collision integral, an approximation of Boltzmann’s collision integral in the 12 13 regime of essentially grazing collisions. 13 14 That the Coulomb potential, probably the best known interaction in physics, is a sin- 14 15 gular point in the theory of the Boltzmann collision integral may seem highly regrettable. 15 16 However, in view of the remark above, the reader should bear in mind that the Boltzmann 16 17 equation is essentially meant to model collisional processes in neutral gases with short- 17 18 range molecular interactions, and not in plasmas, so that the Coulomb potential is not 18 19 really relevant in this context. 19 20 20 21 21 22 3.6. The linearized collision integral 22 23 23 3 24 Let ρ and θ>0, and pick u ∈ R ; the linearization at M(ρ,u,θ) of Boltzmann’s collision 24 25 integral is defined as follows 25 26 26 − 27 L =− M 1 B M M 27 M(ρ,u,θ) f 2 (ρ,u,θ) ( (ρ,u,θ), (ρ,u,θ)f), (3.42) 28 28 29 where B is the bilinear operator obtained by polarization from the Boltzmann collision 29 30 integral. In other words, 30 31 31 32 L = + − − − M 32 M(ρ,u,θ) f f f∗ f f∗ b(v v∗,ω) (ρ,u,θ)(v∗) dv∗ dω, 33 R3×S2 33 34 (3.43) 34 35 35 36 where f∗, f and f∗ are the values of f at v∗, v and v∗, respectively, and where v and v∗ 36 37 are determined in terms of v, v∗ and ω by the usual collision relations (3.3). 37 38 The dependence on the parameters ρ, u and θ of the linearized collision integral is 38 39 handled most easily by using the translation and scaling invariance of L. 39 40 40 41 L 41 3.6.1. Translation and scaling invariance of M(ρ,u,θ) . We introduce the following nota- 42 tion for the actions of translation and scaling transformations on functions defined on R3: 42 43 43 44 − v 44 τ f(v)= f(v− u), m f(v)= λ 3f . (3.44) 45 u λ λ 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 45

The Boltzmann equation and its hydrodynamic limits 45

1 For instance, with these notations 1 2 2 3 M = √ M 3 (ρ,u,θ) ρm θ τu (1,0,1). (3.45) 4 4 5 Given a collision kernel b ≡ b(z,ω), we denote by Bb Boltzmann’s collision integral de- 5 6 fined by this collision kernel. With the notation so defined, a straightforward change of 6 7 variables in the collision integral leads to the following relation 7 8 8 9 b b 9 τuB (Φ, Φ) = B (τuΦ,τuΦ), 10 (3.46) 10 11 b 3 mλb 11 mλB (Φ, Φ) = λ B (mλΦ,mλΦ) 12 12 13 13 for each continuous, rapidly decaying Φ ≡ Φ(v), where, in the expression mλb, it is un- 14 derstood that the scaling transformation acts on the first argument of b, i.e., on the relative 14 15 velocity. The analogous formula for the linearized collision operator is 15 16 16 17 √ 17 √ b 3/2 m θ b √ m τuL φ = ρθ L (m τuφ). (3.47) 18 θ M(1,0,1) M(ρ,u,θ) θ 18 19 19 20 This relation shows that it is enough to study the linearization of the collision integral at 20 21 the centered reduced Gaussian 21 22 22 23 M = M(1,0,1) 23 24 24 25 with an arbitrary collision kernel b. 25 26 26 27 L R 27 3.6.2. Rotational invariance of M(1,0,1) . The orthogonal group O3( ) acts on functions 28 on R3 by the formula 28 29 29 30 T 3 30 fR(v) = f R v ,R∈ O3(R), v ∈ R ; (3.48) 31 31 32 32 likewise its action on vector fields is defined by 33 33 34 34 = T ∈ R ∈ R3 35 UR(v) RU R v ,RO3( ), v , (3.49) 35 36 36 37 while its action on symmetric matrix fields is given by 37 38 38 T T 3 39 SR(v) = RS R v R ,R∈ O3(R), v ∈ R . (3.50) 39 40 40 41 The Boltzmann collision integral is obviously invariant under the action of O3(R) – indeed, 41 42 the microscopic collision process is isotropic. In fact, an elementary change of variables in 42 43 the collision integral shows that 43 44 44

45 B(ΦR,ΦR) = B(Φ, Φ)R (3.51) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 46

46 F. G o l s e

1 for each continuous, rapidly decaying Φ. Since the centered unit Gaussian M = M(1,0,1) 1 2 is a radial function, this rotation invariance property goes over to LM 2 3 3 4 4 LM (φR) = (LM φ)R. (3.52) 5 5 6 3 6 Extending LM to act componentwise on vector or matrix fields on R , one finds that 7 7 8 8 L = L 9 M (UR) ( M U)R (3.53) 9 10 10 11 for continuous, rapidly decaying vector fields U and 11 12 12 13 LM (SR) = (LM S)R (3.54) 13 14 14 15 15 for continuous, rapidly decaying symmetric matrix fields S, where the notations UR and SR 16 are as in (3.48)–(3.50). 16 17 17 As we shall see below, this O3(R)-invariance of LM has important consequences: it 18 implies in particular that the viscosity and heat conductivity are scalar quantities (and not 18 19 matrices). 19 20 20 21 3.6.3. The Fredholm property. Henceforth, we assume that the collision kernel b satisfies 21 22 22 a hard cut-off assumption in the sense of Grad [64], i.e., there exists α ∈[0, 1] and Cb > 0 23 23 such that, for a.e. z ∈ R3 and ω ∈ S2, one has 24 24 25 25 +| | α 26 0

45 Kφ = K1φ − K2φ, (3.58) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 47

The Boltzmann equation and its hydrodynamic limits 47

1 where 1 2 2 3 3 K1φ = φ∗b(v − v∗,ω)M∗ dv∗ dω 4 R3×S2 4 5 5 6 K2φ = φ + φ∗ b(v − v∗,ω)M∗ dv∗ dω (3.59) 6 R3×S2 7 7 8 8 = 2 φ b(v − v∗,ω)M∗ dv∗ dω. 9 R3×S2 9 10 10 11 (This last formula is not entirely obvious; it rests on a change of variables that will be 11 12 2 12 explained later.) It is clear that K1 is a compact operator on L (M dv); that K2 shares the 13 same property is much less obvious, and was proved by Hilbert in the hard sphere case. 13 14 Fifty years later, Grad introduced the cut-off assumption which now bears his name and 14 15 used it in particular to extend Hilbert’s result to all cut-off potentials. 15 16 16 17 17 LEMMA 3.9 (Hilbert [69], Grad [64]). Assume that b is a collision kernel that satisﬁes the 18 18 hard cut-off assumption (3.55). Then the operator K is compact on L2(M dv). 19 2 19 20 20 21 We shall not give the proof of the Hilbert–Grad lemma here, since it is rather long and 21 22 technical; the interested reader is referred to the lucid exposition by Glassey [50], or to the 22 23 treatise by Cercignani, Illner and Pulvirenti [28] for the hard sphere case. 23 24 Instead, we shall digress from our discussion of the linearized collision integral and 24 25 mention a new result by Lions which can be viewed as a nonlinear analogue of the Hilbert– 25 26 Grad result. 26 27 27 28 LEMMA 3.10 (Lions [84]). Assume that b is the collision kernel of a hard sphere gas: 28 29 b(z,ω) =|z · ω|, and consider the gain term in the Boltzmann collision integral 29 30 30 31 31 32 B+(F, F ) = F F∗ (v − v∗) · ω dv∗ dω. 32 R3×S2 33 33 34 34 B+ 2 R3 1 R3 35 Then maps Lcomp( ) continuously into Hloc( ). 35 36 36 37 Actually, this is not exactly Lions’ result; in the reference [84], only compactly supported 37 2 38 collision kernels are considered. The proof in [84] is based upon the L -continuity of 38 39 Fourier integral operators of order 0. The statement above was later proved by Bouchut 39 40 and Desvillettes [17] by a very simple and elegant argument. 40 41 Deriving the Hilbert–Grad lemma from the Lions lemma would require additional es- 41 42 timates in order to handle contributions from large |v|. Since obtaining these estimates 42 43 requires essentially as much work as does the proof of the Hilbert–Grad lemma, we do not 43 44 claim that the Lions lemma leads to a simpler proof of the compactness of K; it is however 44 45 of great independent interest. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 48

48 F. G o l s e

1 The main properties of L are now summarized in the following theorem. 1 2 2 3 THEOREM 3.11. Assume that the collision kernel b satisfies the hard cut-off assump- 3 4 tion (3.55). Then L is an unbounded self-adjoint nonnegative Fredholm operator, with do- 4 5 main D(L) = L2(a2M dv) (a being the collision frequency defined in (3.57)). Its nullspace 5 6 is the space of collision invariants 6 7 7 8 2 8 Ker L = span 1,v1,v2,v3, |v| . 9 9 10 10 L L ⊥ 11 Finally, satisfies the following relative coercivity property on (Ker ) : there exists 11 ∈ 2 L 12 C0 > 0 such that, for each φ L (aM dv) – the form domain of – one has 12 13 13 14 2 14 φLφM dv C0 (φ − Πφ) aM dv, (3.60) 15 R3 R3 15 16 16 17 where Π is the L2(M dv)-orthogonal projection on Ker L. 17 18 18 19 SKETCH OF THE PROOF. Let us briefly explain how these various facts are established. It 19 20 follows from the Hilbert–Grad lemma that K is a compact operator on L2(M dv). Hence 20 21 L = a − K is an unbounded Fredholm operator with domain D(L) = L2(a2M dv). That 21 22 L is self-adjoint comes from the symmetries of Boltzmann’s collision integral, especially 22 23 from the computation of 23 24 24 25 25 26 B(F, F )φ dv 26 R3 27 27 28 28 as explained in Proposition 3.1. By the same argument, one sees that 29 29 30 30 1 2 31 φLφM dv = φ + φ∗ − φ − φ∗ bM dvM∗ dv∗ dω 0 31 R3 R3×R3×S2 32 4 32 (3.61) 33 33 34 34 L L = 35 so that is a nonnegative operator; moreover, if φ 0, the integral on the right-hand 35 36 side of the above equality vanishes, so that 36 37 37 38 φ + φ − φ − φ∗ = 0a.e.inv,v∗,ω. 38 39 39 40 Hence φ is a collision invariant (see Definition 3.5) and this characterizes Ker L, see Propo- 40 41 sition 3.6. As for the last inequality, the spectral theory of compact operators implies that 41 42 L has a spectral gap in L2(M dv) 42 43 43 44 44 L ∗ − 2 45 φ φM dv C (φ Πφ) M dv. (3.62) 45 R3 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 49

The Boltzmann equation and its hydrodynamic limits 49

1 On the other hand, by continuity of K on L2(M dv), one has 1 2 2 3 φLφM dv = (φ − Πφ)L(φ − Πφ)Mdv 3 4 R3 R3 4 5 5 2 2 2 6 (φ − Πφ) aM dv −K (φ − Πφ) M dv. 6 R3 R3 7 7 8 Combining both inequalities leads to the announced estimate. 8 9 9 10 The improved spectral gap estimate above is due to Bardos, Caflisch and Nicolaenko [6]. 10 11 Also, an elegant explicit estimate for the L2(M dv) spectral gap can be found in [5]. 11 12 In fact, it is interesting to notice that the weighted spectral gap estimate (3.60) is in some 12 13 sense more intrinsic than the unweighted one. Indeed, in the case of soft cut-off potentials, 13 14 (3.62) is false, but (3.60) (which is of course a weaker statement since the collision fre- 14 15 quency vanishes for large velocities in the soft potential case) holds true: this was proved 15 16 in [58], based on decay estimates on the gain term in the linearized collision integral due to 16 17 Grad and Caflisch [23]. In other words, while the spectral gap estimate (??) holds for hard 17 18 potentials only, the weighted spectral gap estimate holds for hard as well as soft potentials, 18 19 in the cut-off case. 19 20 An important consequence of Theorem 3.11 is that the integral equation 20 21 21 2 22 Lφ = ψ, ψ ∈ L (M dv), (3.63) 22 23 23 24 satisfies the Fredholm alternative: 24 25 • either ψ ⊥ Ker L, in which case (3.63) has a unique solution 25 26 26 ∈ 2 2 ∩ L ⊥; 27 φ0 L a M dv (Ker ) 27 28 28 29 then any solution of (3.63) is of the form 29 30 30 φ = φ + φ , φ L; 31 0 1 where 1 is an arbitrary element of Ker 31 32 32 • or ψ/∈ (Ker L), in which case (3.63) has no solution. 33 33 34 34 EXAMPLE. Consider the vector field 35 35 36 1 36 B(v) = |v|2 − 5 v 37 2 37 38 38 39 and the matrix field 39 40 40 1 2 41 A(v) = v ⊗ v − |v| I. 41 3 42 42 43 Clearly 43 44 44

45 Ajk ⊥ Ker L,Bl ⊥ Ker L,Ajk ⊥ Bl, j,k,l= 1, 2, 3. (3.64) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 50

50 F. G o l s e

1 In fact, more is true 1 2 2 3 3 A(v)f |v|2 M dv = 0, 4 R3 4 5 5 2 6 A(v)vf |v| M dv = 0, 6 R3 7 7 8 8 B(v)f |v|2 M dv = 0, 9 R3 9 10 10 11 B(v) · vM dv = 0. 11 R3 12 12 13 The second and third formulas are obvious since A is even and B odd. As for the first 13 14 formula, observe that A is an isotropic matrix, in the sense that A(Rv) = RA(v)RT for 14 15 15 each R ∈ O3(R) – with the notation (3.50) for the action of O3(R) on symmetric matrices, 16 16 AR = A for each R ∈ O3(R). Hence the matrix 17 17 18 18 2 19 A(v)f |v| M dv 19 R3 20 20 21 21 commutes with any R ∈ O3(R) – as can be seen by changing v into Rv in the above 22 integral – and is therefore a scalar multiple of the identity matrix. But 22 23 23 24 24 2 2 25 trace A(v)f |v| M dv = trace A(v)f |v| M dv = 0 25 R3 R3 26 26 27 and hence this scalar multiple of the identity matrix is null. The fourth and last formula is 27 28 based on the following elementary recursion formula for Gaussian integrals 28 29 29 30 30 n n−2 31 |v| M dv = (n + 1) |v| M dv, n 2 (3.65) 31 R3 R3 32 32 33 (to see this, use spherical coordinates and integrate by parts). 33 34 In particular, the Fredholm alternative implies the existence of a matrix field A and of a 34 35 35 vector field B such that 36 36 37 37 LA= A and A⊥ Ker L, 38 (3.66) 38 39 LB = B and B ⊥ Ker L. 39 40 40 41 Observe that 41 42 42 43 43 L AR = AR = A, AR ⊥ Ker L for all R ∈ O3(R), 44 44 45 L BR = BR = B, BR ⊥ Ker L for all R ∈ O3(R), 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 51

The Boltzmann equation and its hydrodynamic limits 51

1 so that, by the uniqueness part in the Fredholm alternative 1 2 2 3 AR = A and BR = B for all R ∈ O3(R). 3 4 4 5 An elementary geometric argument (see [34]) shows the existence of two scalar functions 5 6 6

7 a : R+ → R, b : R+ → R 7 8 8 9 such that 9 10 10 11 11 A(v) = a |v|2 A(v) and B(v) = b |v|2 B(v). (3.67) 12 12 13 13 As we shall see further, the viscosity and heat conductivity of a gas are expressed as 14 14 Gaussian integrals of the scalar functions a and b, and therefore are scalar quantities them- 15 15 selves. 16 16 17 17 18 18 19 4. Hydrodynamic scalings for the Boltzmann equation 19 20 20 21 This short section introduces in particular the Knudsen number, Kn, a dimensionless pa- 21 22 rameter of considerable importance for the derivation of hydrodynamic models from the 22 23 kinetic theory of gases. 23 24 24 25 25 26 4.1. Notion of a rarefied gas 26 27 27 28 Think of a monatomic gas as a cloud of N hard spheres of radius r confined in a container 28 29 of volume V ; we shall call excluded volume the volume Ve that the N gas molecules would 29 30 occupy if tightly packed somewhere in the container (as oranges in a grocery store). Clearly 30 31 31 32 4 3 3 32 N πr Ve N (2r) , 33 3 33 34 34 35 and we shall call a perfect gas one for which Ve V . It is well known that the equation of 35 36 state for a perfect gas is given by the Boyle–Mariotte law 36 37 37 38 p = kρθ, 38 39 39 40 where p, ρ and θ are the pressure, the density and the temperature in the gas, while k des- 40 −23 −1 41 ignates the Boltzmann constant (k = 1.38 · 10 JK ). 41 42 42 43 EXAMPLE. For a monatomic gas at room temperature and atmospheric pressure, about 43 − 44 N = 1020 molecules with radius 10 8 cm are to be found in a volume V of 1 cm3. Hence, 44 20 −8 3 −4 3 45 the excluded volume satisfies Ve 10 · (2 · 10 ) = 8 · 10 cm , so that Ve V . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 52

52 F. G o l s e

1 Another important notion is that of mean free path: it is the average distance between 1 2 two successive collisions for one gas molecule picked at random. There is more than one 2 3 precise definition of that notion, in particular, because there are several choices of proba- 3 4 bility measures for computing the mean. One choice could be to use the empirical notion 4 5 of mean: Pick a typical particle, wait until after this particle has collided n 1 times with 5 6 other particles; the ratio of the distance traveled by the particle between the initial time and 6 7 the last collision time to the number n of collisions should converge, in the large n limit, 7 8 to one notion of mean free path. This definition, however, does not provide us with an easy 8 9 way of estimating the mean free path, because it is fairly difficult to compute the trajectory 9 10 of one particle in the cloud of the N − 1 other particles. 10 11 Intuitively, one expects that the larger the number of particles in the container, the 11 12 smaller the mean free path; likewise, the bigger the particles, the smaller the mean free 12 13 path. This suggests using the following formula to estimate the order of magnitude of the 13 14 mean free path, 14 15 15 1 16 mean free path , 16 17 (N /(V − Ve)) × A 17 18 18 where A is the surface area of the section of the particles. A mathematical justification of 19 19 the above formula for the mean free path can be found in [111]; see also [39]. 20 20

21 20 21 EXAMPLE. For the same monatomic gas as above, N = 10 molecules, V − Ve 22 22 V = 1cm3, while A = π · (10−8)2 cm2. This gives a mean free path of the order 23 23 of 3 · 10−5 cm 1 cm (the size of the container). Hence, a gas molecule will bump 24 24 into roughly 104 other particles when traveling a distance comparable to the size of the 25 25 container. 26 26 27 The example above shows that even in the case of a perfect gas (i.e., for an excluded 27 28 volume negligible when compared to the size of the container) any given particle can col- 28 29 lide with a large number of other particles. See, for instance, [2] for more information on 29 30 the importance of the excluded volume in this context. 30 31 31 32 32 EXAMPLE. For the same monatomic gas as above, lower the pressure from p = 1 atm to 33 p = 10−4 atm. Then, only N = 1016 molecules are to be found in the container, which 33 34 gives a mean free path of the order of 0.3 cm, comparable to the size of the container. 34 35 35 36 These examples suggest that the natural way of measuring the degree of rarefaction in 36 37 the gas is by using the Knudsen number, Kn, defined as the ratio of the mean free path to 37 38 the size of the container, or more generally, as 38 39 39 40 mean free path 40 Kn = . 41 macroscopic length scale 41 42 42 43 In other words, 43 44 • a rarefied gas is a gas for which Kn 1, while 44 45 • a gas in hydrodynamic regime satisfies Kn 1. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 53

The Boltzmann equation and its hydrodynamic limits 53

1 4.2. The dimensionless Boltzmann equation 1 2 2 3 In the discussion above, the introduction of the Knudsen number was based on physical 3 4 arguments. We shall see below that it also appears naturally when writing the Boltzmann 4 5 equation in dimensionless variables. 5 6 Choose a macroscopic length scale L and time scale T , and a reference temperature Θ. 6 7 This defines two velocity scales: 7 8 • one is the speed at which some macroscopic portion of the gas is transported over a 8 9 distance L in time T , i.e., 9 10 10 11 L 11 U = ; 12 T 12 13 13 14 14 • the other one is the thermal speed of the molecules with energy 3 kΘ;infact,itis 15 2 15 more natural to define this velocity scale as 16 16 17 17 5 kΘ 18 c = , 18 19 3 m 19 20 20 21 m being the molecular mass, which is the speed of sound in a monatomic gas at the 21 22 temperature Θ. 22 23 Define next the dimensionless variables involved in the Boltzmann equation, i.e., the di- 23 24 mensionless time, space and velocity variables as 24 25 25 t x v 26 tˆ = , xˆ = and vˆ = . 26 27 T L c 27 28 28 29 Define also the dimensionless number density 29 30 30 L3c3 31 F t,ˆ x,ˆ vˆ = F(t,x,v), 31 32 N 3 32 33 33 34 where N is the total number of gas molecules in a volume L3. Finally, we must rescale the 34 35 collision kernel b. As mentioned earlier, b(z,ω) is the relative velocity multiplied by the 35 36 scattering cross-section of the gas molecules; define 36 37 37 38 1 z 38 b(ˆ z,ˆ ω) = b(z,ω) with zˆ = , 39 c × πr2 c 39 40 40 41 where r is the molecular radius. 41 42 If f satisfies the Boltzmann equation 42 43 43 44 44 + ·∇ = − − 45 ∂t F v xF F F∗ FF∗ b(v v∗,ω)dv∗ dω, 45 R3×S2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 54

54 F. G o l s e

1 then 1 2 2 N π 2 3 L +ˆ·∇ = r − ˆ ˆ −ˆ ˆ 3 ∂tˆF v xˆ F F F∗ F F∗ b(v v∗,ω)dv∗ dω. 4 cT L2 R3×S2 4 5 5 6 The factor multiplying the collision integral is 6 7 7 N × πr2 L 1 8 L × = = , 8 9 L3 mean free path Kn 9 10 10 11 where Kn is the Knudsen number defined above. The factor multiplying the time derivative 11 12 12 (1/T)× L 13 =: St 13 14 c 14 15 15 is called the kinetic Strouhal number (by analogy with the notion of Strouhal number used 16 16 in the dynamics of vortices). Hence the dimensionless form of the Boltzmann equation (see 17 17 Section 2.9 in [115]) is 18 18 19 19 +ˆ·∇ = 1 − ˆ ˆ −ˆ ˆ 20 St ∂tˆF v xˆ F F F∗ F F∗ b(v v∗,ω)dv∗ dω. (4.1) 20 Kn R3×S2 21 21 22 There is some arbitrariness in the way the length, time and temperature scales L, T , Θ are 22 23 chosen. For instance, if F in ≡ F in(x, v) is the initial number density (at time t = 0), one 23 24 can choose 24 25 25 in 26 1 R3×R3 |∇xF | dx dv 26 = , 27 in 27 L R3×R3 F dx dv 28 28 in 29 3 | 3 vF dv| dx 29 U = R R , 30 in 30 R3×R3 F dx dv 31 31 2 in 32 1/(3k) R3×R3 m|v| F dx dv 32 Θ = , 33 in 33 R3×R3 F dx dv 34 34 35 and define T = L/U. In addition to Sone’s book [115], we also refer to the Introduction 35 36 of [10] for a presentation of the Boltzmann equation in dimensionless variables. 36 37 All hydrodynamic limits of the Boltzmann equation correspond to situations where the 37 38 Knudsen number, Kn, satisfies 38 39 39 40 Kn 1. 40 41 41 42 In other words, the Knudsen number governs the transition from the kinetic theory of 42 43 gases to hydrodynamic models, just as the Reynolds number in fluid mechanics governs 43 44 the transition from laminar to turbulent flows, except that the hydrodynamic limit is much 44 45 better understood than the latter situation. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 55

The Boltzmann equation and its hydrodynamic limits 55

1 But there is no universal prescription for the Strouhal number in the context of the hy- 1 2 drodynamic limit; as we shall see below, various hydrodynamic regimes can be derived 2 3 from the Boltzmann equation by appropriately tuning the Strouhal number. 3 4 4 5 5 6 5. Compressible limits of the Boltzmann equation: Formal results 6 7 7 8 In this section, we study the dimensionless Boltzmann equation (4.1) in the case where 8 9 9 Kn = ε 1, St = 1. 10 10 11 The dimensionless Boltzmann equation takes the form 11 12 12 13 1 13 ∂ F + v ·∇ F = B(F ,F ), (5.1) 14 t ε x ε ε ε ε 14 15 15 16 where 16 17 17 18 B(F, F ) = F F∗ − FF∗ b(v − v∗,ω)dv∗ dω. 18 R3×S2 19 19 20 The collision kernel b is assumed to satisfy Grad’s hard cut-off assumption 20 21 21 22 α 1 α 22 0 0. 25 26 26 27 27 5.1. The compressible Euler limit: The Hilbert expansion 28 28 29 29 In [69], Hilbert proposed to seek the solution of (5.1) as a formal power series in ε, 30 30 31 k 31 Fε(t,x,v)= ε Fk(t,x,v) (5.3) 32 32 k0 33 33 34 ∞ 3 3 34 with Fk ∈ C (R+ × R ; S(R )) for each k 0. 35 35 The equations governing the coefficients Fk are obtained by inserting the right-hand side 36 of (5.3) in (5.1) and equating the coefficients multiplying the successive powers of ε. 36 37 37 38 Order ε−1. One finds 38 39 39

40 0 = B(F0,F0), 40 41 41 42 hence F0 is a local Maxwellian, meaning that there exists ρ ≡ ρ(t,x)>0, θ ≡ θ(t,x)>0 42 3 43 and u ≡ u(t, x) ∈ R such that 43 44 44 45 F0(t,x,v)= M(ρ(t,x),u(t,x),θ(t,x))(v) a.e. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 56

56 F. G o l s e

1 Order ε0. One ﬁnds 1 2 2 3 (∂t + v ·∇x)F0 = B(F1,F0) + B(F0,F1) 3 4 4 5 which can be recast in terms of the linearization at the local Maxwellian F0 of the collision 5 6 integral (see (3.42)) as 6 7 7 F 8 L 1 =− + ·∇ 8 F0 (∂t v x) ln F0. (5.4) 9 F0 9 10 10 This is precisely an integral equation for F /F of the form (3.63) studied above. 11 1 0 11 Let us compute the right-hand side of the above equality, i.e., the expression 12 12 13 13 (∂t + v ·∇x) ln F0. 14 14 15 For convenience, we shall denote by V the vector 15 16 16 17 1 17 V = √ (v − u). 18 θ 18 19 19 20 Then 20 21 21 22 (∂t + v ·∇x) ln M(ρ,u,θ) 22 23 1 3 1 23 24 = (∂t + v ·∇x)ρ − (∂t + v ·∇x)θ 24 ρ 2 θ 25 25 2 26 v − u |v − u| 26 + (∂t + v ·∇x)u + (∂t + v ·∇x)θ. 27 θ 2θ 2 27 28 28 29 We shall rearrange the right-hand side of the above equality and express it as a linear 29 combination of the functions 30 30 31 31 1 2 1,Vj , |V | − 3 ,A(V)kl and B(V )j , j,k,l= 1, 2, 3. 32 2 32 33 33 34 One ﬁnds that 34 35 35 + ·∇ M 36 (∂t v x) ln (ρ,u,θ) 36 37 1 37 = (∂ ρ + u ·∇ ρ + ρ div u) 38 ρ t x x 38 39 39 40 1 θ 40 + √ V · ∂t u + u ·∇xu +∇xθ + ∇xρ 41 θ ρ 41 42 42 1 2 1 2 43 + |V | − 3 ∂t θ + u ·∇xθ + θ divx u 43 2 θ 3 44 √ 44 45 + A(V ) :∇xu + 2B(V ) ·∇x θ. (5.5) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 57

The Boltzmann equation and its hydrodynamic limits 57

1 Because of the orthogonality relations (3.64), the last two terms on the right-hand side 1 ⊥ 2 L L 2 of (5.5) belong to (Ker F0 ) , while the ﬁrst three terms there are in Ker F0 . Therefore, 3 the solvability condition for the Fredholm integral equation (5.4) consists in setting to 0 3 4 1 | |2 − 4 the coefﬁcients of the functions 1, Vj and 2 ( V 3), i.e., 5 5

6 ∂t ρ + u ·∇xρ + ρ divx u = 0, 6 7 θ 7 8 ∂t u + u ·∇xu +∇xθ + ∇xρ = 0, 8 9 ρ 9 10 2 10 ∂t θ + u ·∇xθ + θ divx u = 0, 11 3 11 12 12 13 or in other words, 13 14 14 15 ∂t ρ + divx(ρu) = 0, 15 16 1 16 ∂ u + u ·∇ u + ∇ (ρθ) = 0, (5.6) 17 t x ρ x 17 18 18 19 2 19 ∂t θ + u ·∇xθ + θ divx u = 0. 20 3 20 21 21 We recognize the system of Euler equations for a compressible fluid (2.7), in the case of a 22 22 perfect monatomic gas; i.e., for γ = 0, so that the pressure law and the internal energy are 23 23 given respectively by 24 24 25 25 3 26 pressure = ρθ and internal energy = θ. 26 27 2 27 28 28 Assuming that ρ, u and θ satisfy these Euler equations, we solve for F1 the Fredholm 29 integral equation (5.4) to find 29 30 30 31 1 √ 31 F =− F a θ,|V | A(V ) ·∇ u + 2b θ,|V | B(V ) ·∇ θ 32 1 ρ 0 x x 32 33 33 34 ρ1 1 1 2 θ1 34 + F0 + √ V · u1 + |V | − 3 . 35 ρ θ 2 θ 35 36 36 37 The second term on the right-hand side of the formula giving F1 represents the arbitrary 37 L 38 element of Ker F0 that appears in the general solution of the integral equation (3.63), 38 39 while the scalar quantities a(θ, V ) and b(θ, V ) satisfy (see (3.67)) 39 40 40 41 L | | = L | | = 41 F0 a θ, V A(V ) A(V ) and F0 b θ, V B(V ) B(V ). 42 42 43 More precisely, let b be the collision kernel satisfying (5.2) considered in this chapter. 43 44 Applying the notation in Section 3.6.1, and especially (3.44), we define scalar functions 44 45 · · √ 45 a(θ, ) and b(θ, ) as in (3.67) with a collision kernel m1/ θ b. dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 58

58 F. G o l s e

1 Order ε1. One ﬁnds that 1 2 2

3 ∂t F1 + v ·∇xF1 − B(F1,F1) = 2B(F0,F2) 3 4 4 5 which can be put in the form 5 6 6 7 F F F 1 7 L 2 = Q 1 , 1 − (∂ + v ·∇ )F 8 F0 F0 t x 1 8 F0 F0 F0 F0 9 9 10 Q 10 and this is therefore a Fredholm integral equation of the type (3.63). Here F0 is the Boltz- 11 mann collision integral intertwined with the multiplication by the local Maxwellian F0, 11 12 i.e., 12 13 13 − 14 Q = 1B 14 F0 (φ, φ) F0 (F0φ,F0φ). 15 15 16 For this equation to have a solution, one must verify the compatibility conditions 16 17     17 18 1 1 18     19 ∂t v F1 dv + divx v ⊗ v F1 dv = 0. 19 20 R3 1 | |2 R3 1 | |2 20 2 v 2 v 21 21

22 These five compatibility conditions are five PDEs for the five unknown functions ρ1, u1 22 23 and θ1. 23 24 24 25 Order εn. One finds 25 26 26 27 ∂t Fn + v ·∇xFn − B(Fk,Fl) = 2B(F0,Fn+1) 27 28 k+l=n,1k,l,n 28 29 29 30 which is a Fredholm equation of the same type as above. 30 31 Here again, the compatibility condition reduces to 31 32     32 33 1 1 33     34 ∂t v Fn dv + divx v ⊗ v Fn dv = 0. 34 R3 R3 35 1 | |2 1 | |2 35 2 v 2 v 36 36 37 More generally, the compatibility condition at order n + 1 (to guarantee the existence 37 38 of Fn+1) provides the system of five equations satisfied by that part of Fn which belongs 38 39 L 39 to the nullspace of F0 . See [115] or Chapter V, Section 2 in [27] for more on Hilbert’s 40 expansion. 40 41 41 42 Conclusion. The Hilbert expansion method shows that the leading order in ε of Hilbert’s 42 43 formal solution (5.3) of the scaled Boltzmann equation (5.1) with Strouhal number St = 1 43 44 and Knudsen number Kn = ε 1 is a local Maxwellian state whose parameters are gov- 44 45 erned by the Euler equations of gas dynamics (for a perfect monatomic gas). 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 59

The Boltzmann equation and its hydrodynamic limits 59

1 In general, one cannot hope that Hilbert’s formal power series (5.3) has a positive radius 1 2 of convergence. Yet, Hilbert’s expansion method can be used to obtain a rigorous derivation 2 3 of the Euler equations of gas dynamics from the Boltzmann equation, as we shall see in 3 4 Section 9. 4 5 5 6 6 7 7 8 5.2. The compressible Navier–Stokes limit: The Chapman–Enskog expansion 8 9 9 10 In this subsection, we shall seek higher-order (in ε) corrections to the compressible Euler 10 11 system. The Hilbert expansion presented above is not well suited for this purpose, because 11 12 linear combinations of collision invariants (i.e., hydrodynamic modes) appear at each order 12 13 in ε instead of being all concentrated in the leading order term. 13 14 For that reason, we shall use a slightly different expansion method, the Chapman– 14 15 Enskog expansion. Thus, we seek a solution of the scaled Boltzmann equation (5.1) as 15 16 a Chapman–Enskog formal power series, 16 17 17 18 n (n) 18 Fε(t,x,v)= ε F P(t,x) (v), (5.7) 19 19 n0 20 20 21 21 22 parametrized by the vector P of conserved densities of Fε. 22 23 23 24 NOTATION. F n[P(t,x)](v) designates a quantity that depends smoothly on P and any 24 25 ﬁnite number of its derivatives with respect to the x-variable at the same point (t, x), and 25 26 on the v-variable. 26 27 In particular, F n[P(t,x) ](v) does not contain time-derivatives of P: the Chapman– 27 28 28 Enskog method is based on eliminating ∂t P in favor of x-derivatives via conservation 29 laws satisﬁed by P. 29 30 30 31 31 That P is the vector of conserved densities of F means that 32 ε 32 33   33 34 1 34 (0)   = 35 F P (v) v dv P, 35 R3 1 | |2 36 2 v 36 37   (5.8) 37 1 38 38 (n)  v  = 39 F P (v) dv 0,n1. 39 R3 1 | |2 40 2 v 40 41 41 42 These conserved densities satisfy a formal system of conservation laws of the form 42 43 43 44 n (n) 44 ∂t P = ε divx Φ P , (5.9) 45 n0 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 60

60 F. G o l s e

1 where the formal ﬂuxes Φ(n) are obtained from the local conservation laws associated to 1 2 the Boltzmann equation by the formulas 2 3   3 4 1 4 (n)   (n) 5 Φ P =− v ⊗ v F P (v) dv (5.10) 5 R3 1 | |2 6 2 v 6 7 7 8 for n 0. 8 9 Let us analyze the ﬁrst orders in ε of the Chapman–Enskog expansion. 9 10 10 Order 0. One has 11 11 12 (0) (0) (0) 12 B F P ,F P = 0 and thus F P = M(ρ,u,θ), 13 13 14 here 14 15     15 16 ρ ρu 16   (0)  ⊗  17 P = ρu ,ΦP =− ρu 2 + ρθI . 17 18 ρ(1 |u|2 + 3 θ) 1 | |2 + 5 18 2 2 ρu(2 u 2 θ) 19 19 20 Hence the formal conservation law at order 0 is 20 21 21 = (0) 22 ∂t P divx Φ P mod O(ε) 22 23 23 whose explicit form is 24 24 25 25 ∂t ρ + u ·∇xρ + ρ divx u = 0, 26 26 27 1 27 ∂t u + (u ·∇x)u + ∇x(ρθ) = 0modO(ε), (5.11) 28 ρ 28 29 2 29 ∂t θ + u ·∇xθ + θ divx u = 0. 30 3 30 31 31 32 In other words, the 0th order of the Chapman–Enskog expansion gives the compressible 32 33 Euler system, as does the Hilbert expansion. 33 34 The Hilbert and Chapman–Enskog methods differ at order 1 in ε, as we shall see below. 34 35 35 36 Order 1. One has 36 37 37 (∂ + v ·∇ )F (0) P = B F (0) P ,F(1) P . 38 t x 2 (5.12) 38 39 39 ∂ F (0)[P] 40 Using the formal conservation law at order 0, we eliminate t and replace it with 40 x-derivatives of F (0)[P], by using (5.5) and the conservation laws at order 0 (5.11) (i.e., 41 41 Euler’s system) as follows 42 42 43 43 (∂t + v ·∇x)M(ρ,u,θ) 44 √ 44 45 = M(ρ,u,θ) A(V ) : D(u) + 2B(V ) ·∇x θ + O(ε) (5.13) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 61

The Boltzmann equation and its hydrodynamic limits 61

1 with the notation 1 2 2 3 v − u 1 1 3 V = √ ,A(V)= V ⊗ V − |v|2I, B(V) = V |V |2 − 5 , 4 θ 3 2 4 5 5 6 and where D(u) is the traceless part of the deformation tensor of u 6 7 7 8 8 9 1 T 2 9 D(u) = ∇xu + (∇xu) − divx uI . 10 2 3 10 11 11 12 In view of (5.12) and (5.13), F (1)[P] is determined by the conditions 12 13 13 14 14 √ F (1)[P] 15 A(V ) : D(u) + 2B(V ) ·∇ θ =−LM , 15 x (ρ,u,θ) M 16 (ρ,u,θ) 16   17 17 1 18 F (1) P (v)  v  dv = 0. 18 19 1 | |2 19 2 v 20 20 21 21 L 2 22 By Hilbert’s lemma, M(1,u,θ) is a Fredholm operator on L (M dv); thus 22 23 23 √ 24 (1) 24 F P (v) =−M(1,u,θ) a θ,|V | A(V ) : D(u) + 2b θ,|V | B(V ) ·∇x θ , 25 25 26 26 27 where we recall that the scalar functions a and b are defined by 27 28 28 29 L | | = | | ⊥ L 29 M(1,u,θ) a θ, V A(V ) A(V ) and a θ, V A(V ) Ker M(1,u,θ) 30 30 31 while 31 32 32 33 33 L | | = | | ⊥ L 34 M(1,u,θ) b θ, V B(V ) B(V ) and b θ, V B(V ) Ker M(1,u,θ) . 34 35 35 36 Hence the first-order correction to the fluxes in the formal conservation law is 36 37 37 38 0 38 39 Φ(1) P = µ(θ)D(u) . 39 40 µ(θ)D(u) · u + κ(θ)∇xθ 40 41 41 42 Therefore, the formal conservation law at first order is 42 43 43 44 44 ∂ P = div Φ(0) P + ε div Φ(1) P mod O ε2 , 45 t x x 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 62

62 F. G o l s e

1 i.e., the compressible Navier–Stokes system with O(ε) dissipation terms 1 2 2

3 ∂t ρ + divx(ρu) = 0, 3 4 2 4 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) = ε divx µD(u) mod O ε , 5 5 6 6 1 2 3 1 2 5 (5.14) ∂t ρ |u| + θ + divx ρu |u| + θ 7 2 2 2 2 7 8 8 9 = ε divx(κ∇xθ)+ ε divx µD(u) · u . 9 10 10 11 The viscosity µ and heat conduction κ are computed as follows 11 12 12 13 2 13 θ a θ,|V | Aij (V )Akl(V )M(1,u,θ) dv = µ(θ) δikδjl + δilδjk − δij δkl , 14 R3 3 14 15 15 16 θ b θ,|V | Bi(V )Bj (V )M(1,u,θ) dv = κ(θ)δij , 16 R3 17 17 18 or in other words, 18 19 19 20 +∞ 20 2 6 −r2/2 dr 21 µ(θ) = θ a(θ, r)r e √ , 21 15 0 2π 22 (5.15) 22 +∞ 23 1 − 2 dr 23 κ(θ)= θ b(θ, r)r4 r2 − 5 2e r /2 √ . 24 6 0 2π 24 25 25 26 In the hard sphere case, one finds that the viscosity and heat conduction are of the form 26 27 √ √ 27 28 µ(θ) = µ0 θ, κ(θ)= κ0 θ, 28 29 29 30 where µ0 and ν0 are positive constants. 30 31 31 32 Conclusion. The Chapman–Enskog expansion method shows that the first-order (in ε) 32 33 correction to the compressible Euler system in the limit of (5.1) – with St = 1 and 33 34 Kn = ε → 0 – is the compressible Navier–Stokes system (5.14). 34 35 Notice that the class of Navier–Stokes systems obtained in this way is by no means the 35 36 most general: only the equation of state of a perfect monatomic gas can be obtained in this 36 37 way, as in the case of the compressible Euler limit of the Boltzmann equation. In addition, 37 38 the viscous dissipation tensor obtained in this way involves only one viscosity coefficient 38 39 instead of two. 39 40 Finally, the Chapman–Enskog expansion can be pushed further to obtain higher-order 40 41 corrections to the compressible Euler equations. For instance, the second-order correction 41 42 to the compressible Euler equations is a system known as the Burnett equations; further 42 43 corrections have also been computed and are known as the super-Burnett equations. See 43 44 Chapter 5, Section 3 in [27] and Section 25 in [63] for more material on the Chapman– 44 45 Enskog expansion, as well as for a comparison with Hilbert’s expansion. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 63

The Boltzmann equation and its hydrodynamic limits 63

1 It should be noted however that these further corrections to the Euler system, beyond 1 2 the Navier–Stokes system, i.e., the Burnett and super-Burnett equations, are in general 2 3 not well posed, so that their practical interest in fluid mechanics is unclear. Therefore, we 3 4 shall not pursue this line of investigation. However, Levermore recently proposed a subtle 4 5 modification of the Chapman–Enskog expansion which leads to well-posed variants of the 5 6 Burnett systems [82]. 6 7 7 8 8 9 5.3. The compressible Euler limit: The moment method 9 10 10 11 11 We shall now present a method for deriving hydrodynamic equations from the Boltzmann 12 12 equation that differs from either the Hilbert or Chapman–Enskog expansions. It consists of 13 13 passing to the limit as the Knudsen number vanishes in the local conservation laws of mass, 14 14 momentum and energy that are satisfied by “well-behaved” solutions of the Boltzmann 15 15 equation. We describe this method on the derivation of the compressible Euler equations 16 16 from the scaled Boltzmann equation (5.1). 17 17 Start from the Cauchy problem for the Boltzmann equation in the periodic box 18 18 19 19 1 ∗ 3 3 20 ∂ F + v ·∇ F = B(F ,F ), (t, x, v) ∈ R+ × T × R , 20 t ε x ε ε ε ε 21 (5.16) 21 | = M 22 Fε t=0 (ρin,uin,θin). 22 23 23 24 As before, the collision kernel b in the Boltzmann collision integral satisfies the cut-off 24 25 assumption (5.2). 25 26 26 27 THEOREM 5.1. Let ρin 0 a.e., uin and θ in > 0 a.e. be such that 27 28 28 29 29 in + in in2 + in + in + in +∞ 30 ρ 1 u u θ ln ρ ln θ dx< . 30 T3 31 31 32 32 For each ε>0, let Fε beasolutionof (5.16) that satisfies the local conservation laws of 33 mass, momentum, and energy, as well as the local entropy relation. Assume that 33 34 34 35 35 Fε → F a.e. 36 36 37 37 as well as 38 38 39 39 3 40 Fε dv → F dv in C R+; D T , 40 R3 R3 41 41 42 3 42 vFε dv → vF dv in C R+; D T , R3 R3 43 43 44 44 | |2 → | |2 R ; D T3 45 v Fε dv v F dv in C + , 45 R3 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 64

64 F. G o l s e

1 while 1 2 2 3 ∗ 3 3 v ⊗ vFε dv → v ⊗ vF dv in D R+ × T , R3 R3 4 4 5 5 | |2 → | |2 D R∗ × T3 6 v v Fε dv v v F dv in + , 6 R3 R3 7 7 8 ∗ 3 8 Fε ln Fε dv → F ln F dv in D R+ × T , R3 R3 9 9 10 10 → D R∗ × T3 11 vFε ln Fε dv vF ln F dv in + , 11 R3 R3 12 12 13 as ε → 0. Then 13 14 14 15 F = M(ρ,u,θ), 15 16 16 17 where (ρ,u,θ)is an entropic solution of the compressible Euler system 17 18 18 19 ∂t ρ + divx(ρu) = 0, 19 20 20 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) = 0, (5.17) 21 21 22 1 2 3 1 2 5 22 ∂t ρ |u| + θ + divx ρu |u| + θ = 0, 23 2 2 2 2 23 24 24 25 that satisﬁes the initial condition 25 26 26 | = in in in 27 (ρ,u,θ) t=0 ρ ,u ,θ . (5.18) 27 28 28 29 PROOF. The moment method involves three steps. 29 30 30 31 Step 1 (Entropy production bound implies convergence to local equilibrium). The entropy 31 32 relation in the 3-torus implies the entropy production bound 32 33 33 1 t F F 34 − ε ε∗ 34 FεFε∗ FεFε∗ ln b dv dv∗ dω 35 4 0 T3 R3×R3×S2 FεFε∗ 35 36 M M 36 εH (ρin,uin,θin) , (5.19) 37 37 38 38 where M is any global Maxwellian state, for instance, one could choose 39 39 40 M = M 40 in in in , 41 (ρ ,u ,θ ) 41 42 42 where 43 43 44 44 in = in 45 ρ ρ (x) dx, 45 T3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 65

The Boltzmann equation and its hydrodynamic limits 65 1 1 uin = ρinuin(x) dx, 1 2 ρin T3 2 3 3 1 1 2 4 θ in = ρin uin − uin + θ in (x) dx. 4 5 ρin T3 3 5 6 6 3 3 7 Next we apply Fatou’s lemma: assuming that Fε → F a.e. on R+ × T × R , one has 7 8 8 9 t 9 F F∗ 10 0 F F∗ − FF∗ ln b dv dv∗ dω dx ds 10 0 T3 R3×R3×S2 FF∗ 11 11 t 12 − FεFε∗ 12 lim FεFε∗ FεFε∗ ln b dv dv∗ dω dx ds 13 ε→0 0 T3 R3×R3×S2 FεFε∗ 13 14 14 = 0. 15 15 16 16 17 Hence F is a local Maxwellian, i.e., 17 18 18 19 F(t,x,v)= M(ρ(t,x),u(t,x),θ(t,x))(v) 19 20 20 21 for some ρ(t,x) 0, θ(t,x)>0 and u(t, x) ∈ R3. 21 22 22 23 Step 2 (Passing to the limit in the local conservation laws). For each positive ε, the num- 23 24 24 ber density Fε satisﬁes the local conservation laws recalled below 25 25 26 26 27 ∂t Fε dv + divx vFε dv = 0, 27 R3 R3 28 28 29 29 ∂t vFε dv + divx v ⊗ vFε dv = 0, (5.20) 30 R3 R3 30 31 31 1 2 1 2 32 ∂t |v| Fε dv + divx v |v| Fε dv = 0. 32 R3 2 R3 2 33 33 34 34 It follows from our assumptions and Step 1 that 35 35 36 36 37 37 Fε dv → M(ρ,u,θ) dv = ρ, R3 R3 38 38 39 39 → M = 40 vFε dv v (ρ,u,θ) dv ρu, 40 R3 R3 41 41

42 v ⊗ vFε dv → v ⊗ vM(ρ,u,θ) dv = ρ(u⊗ u + θI), 42 R3 R3 43 43 44 44 | |2 → | |2M = 1| |2 + 5 45 v v Fε dv v v (ρ,u,θ) dv ρu u θ , 45 R3 R3 2 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 66

66 F. G o l s e

1 in the limit as ε → 0. Hence the functions ρ and θ, and the vector ﬁeld u satisfy the system 1 2 of PDEs (5.17). It also satisﬁes the initial condition (5.18) because the convergence of the 2 3 conserved densities is locally uniform in t. 3 4 4 5 Step 3 (Passing to the limit in the local entropy relation). Finally, we recall the local 5 6 entropy relation 6 7 7 8 8 9 ∂t Fε ln Fε dv + divx vFε ln Fε dv =−local entropy production rate 0. 9 R3 R3 10 10 11 11 It follows from our assumptions that 12 12 13 13 14 Fε ln Fε dv → M(ρ,u,θ) ln M(ρ,u,θ) dv 14 R3 R3 15 15 16 ρ 3 16 = ρ ln − 1 + ln(2π) ρ, 17 θ 3/2 2 17 18 18

19 vFε ln Fε dv → vM(ρ,u,θ) ln M(ρ,u,θ) dv 19 R3 R3 20 20 21 ρ 3 21 = ρuln − 1 + ln(2π) ρu, 22 θ 3/2 2 22 23 23 24 so that, by passing to the limit in the local entropy relation, on account of the continuity 24 25 equation in (5.17), one arrives at the differential inequality 25 26 26 27 ρ ρ 27 28 ∂ ρ ln + div ρuln 0. (5.21) 28 t θ 3/2 x θ 3/2 29 29 30 30 In other words, (ρ,u,θ)is a solution of the compressible Euler equations that satisﬁes the 31 31 Lax–Friedrichs entropy condition. 32 32 33 33 34 Theorem 5.1 and its proof are taken from [7]. 34 35 35 36 36 37 5.4. The acoustic limit 37 38 38 39 Start from the Boltzmann equation with the same scaling as before, but with initial data 39 40 that are small perturbations of a uniform Maxwellian equilibrium. By Galilean invariance, 40 41 one can assume without loss of generality that this uniform equilibrium is the centered 41 42 reduced Gaussian 42 43 43 44 44 1 −|v|2/2 M(v)= M( , , )(v) = e . 45 1 0 1 (2π)3/2 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 67

The Boltzmann equation and its hydrodynamic limits 67

1 Thus, the problem (5.16) reduces to 1 2 2 3 3 1 ∗ 3 3 ∂ F + v ·∇ F = B(F ,F ), (t, x, v) ∈ R+ × T × R , 4 t ε x ε ε ε ε 4 5 (5.22) 5 F | = = M in in in , 6 ε t 0 (1+ηερ ,ηεu ,1+ηεθ ) 6 7 7 8 where 0 <ε 1 and 0 <ηε 1. The same moment method as above shows that the 8 9 limiting behavior of the solution to (5.22) under these assumptions is governed by the 9 10 acoustic system. 10 11 11 in in in 2 3 12 THEOREM 5.2. Assume that ρ , u and θ belong to L (T ) and that ηε → 0 as ε → 0. 12 13 For each ε>0, let Fε beasolutionof (5.22) that satisﬁes the local conservation laws of 13 14 mass momentum and energy, as well as the local entropy relation. 14 15 Assume that 15 16 16 17 Fε − M 17 gε = → g in the sense of distributions, 18 ηεM 18 19 19 20 while 20 21 21 22 22 ηεB(Fε − M,Fε − M)→ 0 in the sense of distributions, 23 23 24 24 as well as 25 25 26 26 3 27 gεM dv → gM dv in C R+; D T , 27 R3 R3 28 28 29 3 29 vgεM dv → vgM dv in C R+; D T , 30 R3 R3 30 31 31 2 2 3 32 |v| gεM dv → |v| gM dv in C R+; D T , 32 R3 R3 33 33 34 and 34 35 35 36 36 ∗ 3 37 v ⊗ vgεM dv → v ⊗ vgM dv in D R+ × T , 37 R3 R3 38 38 39 2 2 ∗ 3 39 v|v| gεM dv → v|v| gM dv in D R+ × T 40 R3 R3 40 41 41 42 as ε → 0. Then 42 43 43 1 44 g(t,x,v) = ρ(t,x)+ u(t, x) · v + θ(t,x) |v|2 − 3 , 44 45 2 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 68

68 F. G o l s e

1 where (ρ,u,θ)is the solution of the acoustic system 1 2 2 3 ∂t ρ + divx u = 0, 3 4 4 ∂ u +∇ (ρ + θ)= 0, (5.23) 5 t x 5 6 3 6 ∂t θ + divx u = 0, 7 2 7 8 8 9 that satisfies the initial condition 9 10 in in in 10 | = = 11 (ρ,u,θ) t 0 ρ ,u ,θ . (5.24) 11 12 12 The proof of this theorem is an easy variant of the formal compressible Euler limit, and 13 13 is left to the reader. See [53] for the missing details of this formal proof. 14 14 15 15 16 16 6. Incompressible limits of the Boltzmann equation: Formal results 17 17 18 18 So far, we have discussed various limits of the kinetic theory of gases leading to hydrody- 19 19 namic models for compressible fluids that satisfy the equation of state of perfect gases. As 20 20 we shall see in this section, incompressible hydrodynamic models describing incompress- 21 21 ible flows of perfect gases can also be derived from the Boltzmann equation. 22 22 23 23 24 24 6.1. The incompressible Navier–Stokes limit 25 25 26 26 The scaling on the Boltzmann equation that leads to the incompressible Navier–Stokes 27 27 equations in the hydrodynamic limit is defined by 28 28 29 29 Kn = St = ε 1. 30 30 31 However, this scaling is not sufficient by itself: as in all long time scalings, one should 31 32 assume that the length and time scales L and T that enter the definition of St capture the 32 33 speed of the fluid motion. In other words, situations where 33 34 34 35 L 35 u(t, x) must be excluded, 36 T 36 37 37 38 where u is the bulk velocity of the gas, i.e., 38 39 39 40 3 vF dv 40 u(t, x) = R . 41 41 R3 F dv 42 42 43 In other words, since St = (L/T )/speed of sound, one must take 43 44 44 45 Ma = O(St). 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 69

The Boltzmann equation and its hydrodynamic limits 69

1 The case Ma ∼ St corresponds to the largest possible velocity field compatible with 1 2 the above condition, and therefore leads to the Navier–Stokes equation, while the case 2 3 Ma = o(ε) leads to the linearized version of the Navier–Stokes equations, i.e., the Stokes 3 4 equations. 4 5 Hence the complete Navier–Stokes scaling is 5 6 6 7 Kn = St = Ma = ε 1. (6.1) 7 8 8 9 We consider therefore the scaled Boltzmann equation posed on the spatial domain R3 with 9 10 uniform Maxwellian equilibrium at infinity – without loss of generality, this Maxwellian 10 11 equilibrium is assumed to be the centered reduced Gaussian distribution 11 12 12 13 1 −|v|2/2 13 M(v)= M(1,0,1)(v) = e . 14 (2π)3/2 14 15 15 16 The problem to be studied is therefore 16 17 17 18 1 ∗ 3 3 18 ε∂t Fε + v ·∇xFε = B(Fε,Fε), (t, x, v) ∈ R+ × R × R , 19 ε 19 (6.2) 20 20 Fε(t,x,v)→ M as |x|→+∞. 21 21 22 22 That Ma = O(ε) is seen on the number density Fε, and not on the Boltzmann equation 23 itself. Here is an example of number density with Ma = O(ε). 23 24 24 25 25 EXAMPLE 1. Take Fε of the form 26 26 27 27 Fε(t,x,v)= M(1,εu(t,x),1)(v). 28 28 29 Indeed, 29 30 30 31 vF dv 31 R3 ε := 32 εu. 32 R3 Fε dv 33 33 34 34 5 35 The speed of sound for the state of the gas described by Fε is 3 θ where 35 36 36 37 (1/3)|v − εu|2F dv 37 :≡ R3 ε = 38 θ(t,x) 1. 38 R3 Fε dv 39 39 40 40 Hence the Mach number for the state of the gas associated to Fε is 41 41 42 ε|u(t, x)| 42 √ = O(ε). 43 (5/3)θ(t, x) 43 44 44 45 Here is another example that also involves fluctuations of temperature. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 70

70 F. G o l s e

1 EXAMPLE 2. Take Fε of the form 1 2 2 3 3 Fε(t,x,v)= M εu(t,x) 1 (v) (1−εθ(t,x), − , − ) 4 1 εθ(t,x) 1 εθ(t,x) 4

5 ∞ 5 ∈ R+ × R3 ∞ 6 with θ L ( ) and ε θ L < 1. One easily checks that 6 7 7 R3 vFε dv 8 = εu + O ε2 , 8 9 R3 Fε dv 9 10 10 11 while 11 12 12 2 R3 (1/3)|v − εu(t,x)/(1 − εθ(t, x))| Fε dv 13 = 1 + εθ + O ε2 . 13 14 R3 Fε dv 14 15 15 16 Hence the Mach number is 16 17 17 εu + O(ε2) 18 = O(ε). 18 19 (5/3)(1 + εθ + O(ε2)) 19 20 20 21 More generally, if Fε is a number density of the form 21 22 22 23 1 23 Fε = M(1 + εgε) such that gε − a.e., (6.3) 24 ε 24 25 25 ∞ 26 one can check that, provided that gεL = O(1), the Mach number for the state of the 26 27 gas defined by Fε is O(ε). 27 28 Hence, we shall supplement the scaled Boltzmann equation (6.2) with the initial condi- 28 29 tion 29 30 30 31 F | = ≡ M in (v), 31 ε t 0 − in εu (x) 1 (6.4) (1 εθ (x), − in , − in ) 32 1 εθ (x) 1 εθ (x) 32 33 33 where 34 34 35 35 div uin = 0. 36 x 36 37 37 38 For each ε>0, define the number density fluctuation 38 39 39 Fε − M 40 gε = . 40 41 εM 41 42 42 In terms of the number density fluctuation gε, the Boltzmann equation (6.2) takes the form 43 43 44 44 1 3 45 ε∂ g + v ·∇ g + L g = Q (g ,g ), t > 0,x,v∈ R , (6.5) 45 t ε x ε ε M ε M ε ε dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 71

The Boltzmann equation and its hydrodynamic limits 71

1 where QM is the collision integral intertwined with the multiplication by M 1 2 2 −1 3 QM (φ, φ) = M B(Mφ, Mφ). (6.6) 3 4 4 5 We shall also need notation for moments 5 6 6 7 7 φ= φ(v)M(v)dv. 8 R3 8 9 9 10 Hence the local conservation laws of mass momentum and energy satisﬁed by Fε take the 10 11 form 11 12 12 13 ε∂t gε+divxvgε=0 (mass), 13 14 14 ε∂t vgε+divxv ⊗ vgε=0 (momentum), (6.7) 15 15 16 1 2 1 2 16 ε∂t |v| gε + divx v |v| gε = 0 (energy). 17 2 2 17 18 18 19 THEOREM 6.1 (Bardos, Golse and Levermore [8,9]). For each ε>0, let Fε be a solution 19 20 of (6.2)–(6.4). Assume that 20 21 21 − 22 Fε M ∗ 3 22 → g in the sense of distributions on R+ × R , 23 εM 23 24 24 25 and that Fε satisﬁes the local conservation laws of mass, momentum and energy, and that 25 26 26 2 2 3 27 vgε→vg and |v| − 5 gε → |v| − 5 g in C R+, D R 27 28 28 29 while 29 30 30 ∗ 3 31 LM gε → LM g in D R+ × R 31 32 32 33 and all formally small terms vanish in the sense of distributions as ε → 0. Assume further 33 34 that 34 35 35

36 v ⊗ vgε→v ⊗ vg, Bgε→Bg, 36 37 37 AQM (gε,gε) → AQM (gε,gε) and A ⊗ vgε → A ⊗ vg , 38 38 39 39 BQM (gε,gε) → BQM (gε,gε) and B ⊗ vgε → B ⊗ vg 40 40 41 ∗ 3 41 in the sense of distributions on R+ × R . 42 Then g is of the form 42 43 43 44 44 1 2 45 g(t,x,v) = u(t, x) · v + θ(t,x) |v| − 5 , 45 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 72

72 F. G o l s e

1 where (u, θ) satisfy the incompressible Navier–Stokes–Fourier system 1 2 2 3 ∂t u + divx(u ⊗ u) +∇xp = ν xu, divx u = 0, 3 4 (6.8) 4 + = 5 ∂t θ divx(uθ) κ xθ, 5 6 6 7 where 7 8 +∞ 8 1 2 6 −r2/2 dr 9 ν = A : A = a(1,r)r e √ , 9 10 15 0 2π 10 (6.9) 10 +∞ 11 2 1 − 2 dr 11 κ = B· B = b(1,r)r2 r2 − 5 2e r /2 √ . 12 12 15 15 0 2π 13 13 14 We recall that 14 15 15 1 1 16 A(v) = v ⊗ v − |v|2I, B(v) = |v|2 − 5 v, (6.10) 16 17 3 2 17 18 18 ∈ 2 R3 19 and that there exists A and B L ( ,Mdv) uniquely determined by 19 20 20 L = ⊥ L 21 M A A, A Ker M , 21 (6.11) 22 22 LM B = B, B ⊥ Ker LM . 23 23 24 Furthermore, there exists two scalar functions a and b such that 24 25 25 26 A(v) = a 1, |v| A(v) and B(v) = b 1, |v| B(v). (6.12) 26 27 27 28 PROOF OF THEOREM 6.1. We shall use the moment method, although either the Hilbert 28 29 or the Chapman–Enskog expansions would also lead to the incompressible Navier–Stokes 29 30 limit. However, the moment method is the closest to a complete (instead of formal) con- 30 31 vergence proof for global solutions without restriction on the size of the initial data. 31 32 32 33 Step 1 (Asymptotic form of the ﬂuctuations). Multiply the Boltzmann equation (6.5) by ε, 33 34 so that 34 35 35 2 36 ε ∂t gε + εv ·∇xgε + LM gε = εQM (gε,gε). 36 37 37 38 Passing to the limit as ε → 0 in the above equation, we arrive at 38 39 39

40 LM g = 0. 40 41 41 3 42 Hence, for a.e. (t, x) ∈ R+ × R , g(t,x,·) ∈ Ker LM , which means that g is of the form 42 43 43 44 1 44 g(t,x,v) = ρ(t,x)+ u(t, x) · v + θ(t,x) |v|2 − 3 . (6.13) 45 2 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 73

The Boltzmann equation and its hydrodynamic limits 73

1 Step 2 (Passing to the limit in the local conservation laws). Passing to the limit in (6.7) 1 2 we arrive at 2 3 3

4 divxvg=0, 4 5 5 divxv ⊗ vg=0, 6 6 2 7 divx v|v| g = 0. 7 8 8 9 Together with the relation (6.13), the ﬁrst and the last relation reduce to the incompress- 9 10 ibility condition for u 10 11 11 12 12 divx u = 0. (6.14) 13 13 14 Together with the relation (6.13), the second relation gives 14 15 15 16 16 ∇x(ρ + θ)= 0. 17 17 18 ∞ 18 Since g ∈ L (R+; L2(R3,Mdv)), this implies the Boussinesq relation 19 19 20 20 ρ + θ = 0. (6.15) 21 21 22 22 With this last relation, the asymptotic form of g becomes 23 23 24 24 1 2 25 g(t,x,v) = u(t, x) · v + θ(t,x) |v| − 5 . (6.16) 25 2 26 26 27 It remains to derive the motion and heat equations from the local conservation laws. To 27 28 do this, we recast the local conservation law of momentum as 28 29 29 30 30 1 1 1 2 ∂t vgε+divx Agε+∇x |v| gε = 0, 31 ε ε 3 31 32 32 33 and we combine the local conservation laws of mass and energy into 33 34 34 35 35 1 2 1 ∂t |v| − 5 gε + divx Bgε=0. 36 2 ε 36 37 37 38 One easily checks with (6.16) that 38 39 39 40 40 vgε→vg=u, 41 (6.17) 41 42 1 2 1 2 5 42 |v| − 5 gε → |v| − 5 g = θ 43 2 2 2 43 44 44 45 in C(R+; D (R3)) as ε → 0. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 74

74 F. G o l s e

1 Next we pass to the limit in the flux terms. Since LM is self-adjoint, one has 1 2 2 3 1 1 1 3 Agε= LM A gε = A LM gε . 4 ε ε ε 4 5 5 1 L 6 Then we eliminate the term ε M gε with (6.5) 6 7 7 8 1 8 A LM gε = AQM (gε,gε) − A(ε∂t + v ·∇x)gε 9 ε 9 10 10 11 so that, passing to the limit as ε → 0 leads to 11 12 12 1 ∗ 3 13 Agε→ AQM (g, g) − Av ·∇xg in D R+ × R . (6.18) 13 14 ε 14 15 Likewise 15 16 16 17 1 ∗ 3 17 Bgε→ BQM (g, g) − Bv ·∇xg in D R+ × R , (6.19) 18 ε 18 19 19 as ε → 0. 20 20 With (6.16), one easily finds that 21 21 22 22 T 2 23 Av ·∇xg = ν ∇xu + (∇xu) − (divx u)I , 23 3 24 24 (6.20) 25 5 25 Bv ·∇xg = κ∇xθ, 26 2 26 27 27 where ν and κ are given by (6.9). 28 28 The nonlinear term is slightly more difficult. Its computation involves in particular the 29 29 following classical lemma. 30 30 31 31 LEMMA 6.2. Let φ ∈ Ker LM , then 32 32 33 33 1 2 34 QM (φ, φ) = LM φ . 34 2 35 35 36 PROOF. Differentiate twice the relation 36 37 37 38 B(M(ρ,u,θ), M(ρ,u,θ)) = 0 38 39 39 40 with respect to ρ, u and θ. See, for instance, [25]or[9] for the missing details. 40 41 41 42 Then 42 43 43 1 2 1 2 1 2 1 2 44 AQM (g, g) = ALM g = LM A g = Ag = u ⊗ u − |u| I, 44 2 2 2 3 45 (6.21) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 75

The Boltzmann equation and its hydrodynamic limits 75

1 and likewise 1 2 2 3 1 2 1 2 1 2 5 3 BQM (g, g) = BLM g = LM B g = Bg = uθ. (6.22) 4 2 2 2 2 4 5 5 Observe that 6 6 7 7 1 2 1 2 8 divx u ⊗ u − |u| I = divx(u ⊗ u) − ∇x|u| , 8 3 3 9 9 10 while 10 11 11 12 T 2 2 12 divx ∇xu + (∇xu) − (divx u)I = xu +∇x(divx u) − ∇x(divx u) 13 3 3 13 14 14 = xu, 15 15 16 16 because of the divergence-free condition on u. 17 17 Gathering (6.17)–(6.22), we arrive at 18 18 19 19 ∂t u + divx(u ⊗ u) − ν xu = 0 modulo gradients, 20 20 21 ∂t θ + divx(uθ) − κ xθ = 0. 21 22 22 23 23 24 6.2. The incompressible Stokes and Euler limits 24 25 25 26 By the same moment method, one can derive other incompressible models from the Boltz- 26 27 mann equation. We just state the results below without giving the proofs (which are anyway 27 28 simpler than that of the Navier–Stokes limit). 28 29 29 30 6.2.1. The Stokes limit. The Stokes–Fourier system is the linearization about u = 0 and 30 31 θ = 0 of the Navier–Stokes–Fourier system. Thus, in order to derive the Stokes–Fourier 31 32 system from the Boltzmann equation, one keeps the same scaling as for the incompressible 32 33 Navier–Stokes limit on the Boltzmann equation, i.e., 33 34 34 35 Kn = St = ε 1 35 36 36 37 and one scales the Mach number as 37 38 38 39 Ma = ηε = o(ε). 39 40 40 41 In other words, we start from the following Cauchy problem 41 42 42 43 1 ∗ 3 3 43 ε∂t Fε + v ·∇xFε = B(Fε,Fε), (t, x, v) ∈ R+ × R × R , 44 ε 44 (6.23) 45 Fε(t,x,v)→ M as |x|→+∞, 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 76

76 F. G o l s e

1 with initial condition 1 2 2 3 3 F | = = M in . (6.24) ε t 0 − in ηεu 1 4 (1 ηεθ , , ) 4 1−ηεθin 1−ηεθin 5 5 6 In this subsection, the number density ﬂuctuation gε is deﬁned to be 6 7 7 8 F − M 8 g = ε . 9 ε 9 ηεM 10 10

11 THEOREM 6.3 (Bardos, Golse and Levermore [9]). For each ε>0, let Fε be a solution 11 12 of (6.23)–(6.24). Assume that 12 13 13 14 − 14 Fε M → R∗ × R3 15 g in the sense of distributions on + , 15 ηεM 16 16 17 17 that Fε satisﬁes the local conservation laws of mass, momentum and energy, and that 18 18 19 2 2 3 19 vgε→vg and |v| − 5 gε → |v| − 5 g in C R+, D R 20 20 21 21 while 22 22 23 23 L → L D R∗ × R3 24 M gε M g in + 24 25 25 26 and all formally small terms vanish in the sense of distributions as ε → 0. Assume further 26 27 that 27 28 28

29 v ⊗ vgε→v ⊗ vg, Bgε→Bg, 29 30 30 ⊗ → ⊗ ⊗ → ⊗ 31 A vgε A vg and B vgε B vg 31 32 32 ∗ 3 33 in the sense of distributions on R+ × R . 33 34 Then g is of the form 34 35 35 36 1 36 g(t,x,v) = u(t, x) · v + θ(t,x) |v|2 − 5 , 37 2 37 38 38 39 where (u, θ) satisfy the incompressible Stokes–Fourier system 39 40 40 41 ∂t u +∇xp = ν xu, divx u = 0, 41 42 (6.25) 42 = 43 ∂t θ κ xθ, 43 44 44 45 with ν and κ given by formula (6.9). 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 77

The Boltzmann equation and its hydrodynamic limits 77

1 6.2.2. The incompressible Euler limit. The incompressible Euler equations are formally 1 2 the inviscid limit of the Navier–Stokes equations. Thus, in order to derive the incompress- 2 3 ible Euler equations from the Boltzmann equation, one chooses a scaling that increases the 3 4 strength of the nonlinear term. In other words, one takes 4 5 5 6 6 St = Ma = ε while Kn = ηε = o(ε). 7 7 8 In fact, this is consistent with the von Karman relation, which relates the Mach, Knudsen 8 9 and Reynolds numbers as follows 9 10 10 11 Ma 11 12 Kn = . (6.26) 12 Re 13 13 14 14 Indeed, one gets Re = ε/ηε →+∞as ε → 0: therefore, the limiting equation obtained in 15 this way is incompressible (since Ma → 0) and inviscid (since Re →+∞). 15 16 In other words, we start from the following Cauchy problem 16 17 17 18 18 1 ∗ 3 3 19 ε∂t Fε + v ·∇xFε = B(Fε,Fε), (t, x, v) ∈ R+ × R × R , 19 ηε 20 (6.27) 20 21 Fε(t,x,v)→ M as |x|→+∞, 21 22 22 23 with initial condition 23 24 24

25 F | = = M in . (6.28) 25 ε t 0 (1−εθin, εu , 1 ) 26 1−εθin 1−εθin 26 27 27 28 In this subsection, the number density fluctuation gε is defined to be 28 29 29 − 30 Fε M 30 gε = . 31 εM 31 32 32 33 THEOREM 6.4 (Bardos, Golse and Levermore [9]). For each ε>0, let Fε be a solution 33 34 of (6.27)–(6.28). Assume that 34 35 35 − 36 Fε M ∗ 3 36 → g in the sense of distributions on R+ × R , 37 εM 37 38 38 39 that Fε satisfies the local conservation laws of mass, momentum and energy, and that 39 40 40 → | |2 − → | |2 − R D R3 41 vgε vg and v 5 gε v 5 g in C +, 41 42 42 43 while 43 44 44 ∗ 3 45 LM gε → LM g in D R+ × R 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 78

78 F. G o l s e

1 and all formally small terms vanish in the sense of distributions as ε → 0. Assume further 1 2 that 2 3 3

4 v ⊗ vgε→v ⊗ vg, Bgε→Bg, 4 5 5 AQ (g ,g ) → AQ (g ,g ) and BQ (g ,g ) → BQ (g ,g ) 6 M ε ε M ε ε M ε ε M ε ε 6 7 7 R∗ × R3 8 in the sense of distributions on + . 8 9 Then g is of the form 9 10 10 1 11 g(t,x,v) = u(t, x) · v + θ(t,x) |v|2 − 5 , 11 12 2 12 13 13 where (u, θ) satisfy the system 14 14 15 15 ∂t u + divx(u ⊗ u) +∇xp = 0, divx u = 0, 16 (6.29) 16 17 17 ∂t θ + divx(uθ) = 0. 18 18 19 19 20 6.3. Other incompressible models 20 21 21 22 There are many possible variants of the incompressible Navier–Stokes–Fourier limit de- 22 23 scribed above. To begin with, it is possible to include a conservative force in the Boltzmann 23 24 equation. The scaling is as follows. Start from equation 24 25 25 26 1 26 ε∂ F + v ·∇ F − ε∇ φ(x)·∇ F = B(F ,F ), 27 t ε x ε x v ε ε ε ε 27 28 28 29 where φ ≡ φ(x) is a given (smooth) potential. Writing the (x, v)-derivative in the 29 30 Boltzmann equation above as a Poisson bracket, i.e., 30 31 31 32 1 2 32 v ·∇xFε − ε∇xφ(x)·∇vFε = |v| + εφ(x); Fε 33 2 33 34 34 35 suggests to seek the solution Fε in the form 35 36 36 −εφ(x) 37 Fε(t,x,v)= e M(v) 1 + εgε(t,x,v) . 37 38 38 39 Indeed, 39 40 40 41 − 1 −| |2 − 41 e εφ(x)M(v)= e v /2 εφ(x) 42 (2π)3/2 42 43 43 44 { 1 | |2 + 44 is both a Maxwellian and an element of the nullspace of the Poisson bracket 2 v 45 εφ(x);·}. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 79

The Boltzmann equation and its hydrodynamic limits 79

1 The same formal argument as above shows that 1 2 2 3 1 2 3 gε(t,x,v)→ u(t, x) · v + θ(t,x) |v| − 5 , 4 2 4 5 5 6 where u and θ satisfy 6 7 7 8 ∂t u + divx(u ⊗ u) +∇xp − θ∇xφ = ν xu, 8 9 2 9 10 ∂t θ + divx(uθ) + u ·∇xφ = κ xθ, 10 5 11 11 12 with 12 13 13 14 1 2 14 ν = A: A and κ = B· B 15 10 15 15 16 16 17 as in the incompressible Navier–Stokes–Fourier limit theorem above. Of course, this is in 17 18 agreement with the discussion in Section 2.7. We refer the interested reader to [8]formore 18 19 details on the formal derivation. 19 20 In still another variant of the Navier–Stokes–Fourier limit presented above, it is possible 20 21 to recover viscous heating terms as in Section 2.6. As explained in that subsection, viscous 21 22 heating terms should appear when the fluctuations of velocity field are of the order of the 22 23 square root of temperature fluctuations. In [13], Levermore and Bardos used the following 23 24 very elegant approach: start from the Boltzmann equation in the Navier–Stokes scaling 24 25 (6.2) and seek the number density Fε as 25 26 26 27 = + − + 2 + 27 Fε(t,x,v) M(v) 1 εgε (t,x,v) ε gε (t,x,v) , 28 28 29 − + 29 where gε is odd in v while gε is even in v. Because the Boltzmann collision integral is 30 rotation invariant (see Section 3.6.2 and especially (3.51)), 30 31 31 32 + + − − 32 B(Φ ,Φ ) and B(Φ ,Φ ) are even in v, while 33 33 B(Φ−,Φ+) B(Φ+,Φ−) v. 34 and are odd in 34 35 35 36 Levermore and Bardos gave a formal argument showing that 36 37 37 − → · 38 gε (t,x,v) u(t, x) v, 38 39 39 40 while 40 41 41 42 + → + 1 2 + 1| |2 − 42 gε (t,x,v) ρ(t,x) u(t, x) 3θ(t,x) v 1 43 2 3 43 44 44 1 45 + A : u(t, x) ⊗ u(t, x) − A :∇xu(t, x), 45 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 80

80 F. G o l s e

1 where ρ, u and θ satisfy the relations 1 2 2 3 divx u = 0,p= ρ + θ, 3 4 4 ∂t u + divx(u ⊗ u) +∇xp = ν xu, 5 5 6 5 1 2 5 1 2 6 ∂t θ + |u| − p + divx u θ + |u| 7 2 2 2 2 7 8 8 5 T 9 = κ xθ + µ divx ∇xu + (∇xu) · u . 9 2 10 10 11 In the system above, ν and κ are given by the same formulas as in the Navier–Stokes– 11 12 Fourier limit theorem, i.e., (6.9). 12 13 13 14 14 7. Mathematical theory of the Cauchy problem for hydrodynamic models 15 15 16 16 In this section we have gathered a few mathematical results bearing on the various hy- 17 17 drodynamic models that appear as limits of the Boltzmann equation. We shall leave aside 18 the compressible Navier–Stokes system, since its derivation from the Boltzmann equation 18 19 leads to dissipation terms that are of the order of the Knudsen number, and therefore van- 19 20 ish in the hydrodynamic limit. Put in other words, the compressible Navier–Stokes system 20 21 is an asymptotic expansion of the Boltzmann equation in the Knudsen number, and not a 21 22 limit thereof. Hence, a mathematical treatment of this limit would involve existence results 22 23 on the compressible Navier–Stokes system that are uniform in the Reynolds and Péclet 23 24 numbers, which is beyond current knowledge on this model at the time of this writing. 24 25 25 26 26 27 7.1. The Stokes and acoustic systems 27 28 28 29 We begin with the simplest hydrodynamic models 29 30 • the Stokes–Fourier system, and 30 31 • the acoustic system, 31 32 which are variants of the heat and the wave equations. 32 33 33 34 7.1.1. The Stokes–Fourier system. Consider the Stokes–Fourier system 34 35 ∗ 35 ∂ u +∇ p = ν u, div u = 0,(t,x)∈ R × R3, 36 t x x x + 36 37 ∂t θ = κ xθ, 37 38 in in 38 (u, θ)| = = u ,θ , 39 t 0 39 40 where ν and κ>0. 40 41 41 42 in in 2 3 in 42 THEOREM 7.1. For each (u ,θ ) ∈ L (R ) such that divx u = 0, there exists a unique 43 solution (u, θ) of the Stokes–Fourier system such that 43 44 44 2 3 ∗ 2 3 45 (u, θ) ∈ C R+; L R and p ∈ C R+; L R R . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 81

The Boltzmann equation and its hydrodynamic limits 81

1 In addition, the pressure p is constant and 1 2 2 ∞ ∗ 3 3 (u, θ) ∈ C R+ × R . 3 4 4 5 PROOF. Applying the divergence to both sides of the motion equation gives 5 6 6 7 7 xp = 0; 8 8 9 since for each t>0 the function p(t,·) ∈ L2(R3)/R is harmonic, it is a constant in 9 10 variable x. Since the divergence operator commutes with the heat operator on R3,the 10 11 Stokes–Fourier system above reduces to a system of uncoupled heat equations, whence the 11 12 12 announced result. 13 13 14 14 7.1.2. The acoustic system. Consider the acoustic system 15 15 16 16 + = 17 ∂t ρ divx u 0, 17 ∗ 3 18 ∂t u +∇x(ρ + θ)= 0,(t,x)∈ R+ × R , 18 19 19 3 20 ∂t θ + divx u = 0, 20 2 21 21 in in in 22 (ρ,u,θ)|t=0 = ρ ,u ,θ . 22 23 23

24 THEOREM 7.2. For each (ρin,uin,θin) ∈ L2(R3), there exists a unique solution (ρ,u,θ) 24 25 of the acoustic system such that 25 26 26 27 2 3 27 (ρ,u,θ)∈ C R+; L R . 28 28 29 29 PROOF. Applying the Helmholtz decomposition to u(t, ·) gives 30 30 31 31 u(t, ·) = us(t, ·) −∇ φ(t,·), div u = 0. 32 x x s 32 33 33 34 Hence the acoustic system becomes 34 35 35 3 36 ∂t ρ − xφ = 0, ∂t θ − xφ = 0, 36 37 2 37 38 ∂t φ − ρ − θ = 0,∂t us = 0. 38 39 39 40 Hence 40 41 41 42 5 42 ∂t φ − ρ − θ = 0,∂t (ρ + θ)− xφ = 0, 43 3 43 44 3 44 ∂t ρ − θ = 0. 45 2 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 82

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1 Therefore, letting ψ = ρ + θ, we arrive at 1 2 2 3 5 3 ∂ttφ − xφ = 0, 4 3 4 5 5 5 6 ∂ttψ − xψ = 0. 6 3 7 7 8 This is a system of two uncoupled wave equations; then, (ρ,u,θ) is reconstructed from 8 9 ψ and µ ≡ µ(x) = ρ(t,x)− 3 θ(t,x) by the formulas 9 10 2 10 11 11 3 2 12 ρ(t,x)= ψ(t,x)+ µ(x), 12 13 5 5 13 2 14 θ(t,x)= ψ(t,x)− µ(x) , 14 15 5 15 16 16 17 while 17 18 18 s 19 u(t, x) = u (0,x)−∇xφ(t,x). 19 20 20 21 Applying the classical theory of the Cauchy problem for the wave equations satisﬁed by 21 22 φ and ψ leads to the announced result. 22 23 23 24 24 25 25 7.2. The incompressible Navier–Stokes equations 26 26 27 27 28 Next, we consider the incompressible Navier–Stokes equations. The mathematical theory 28 29 of the Cauchy problem for these equations was developed by J. Leray in the early 1930s. 29 30 Here is a quick summary of his results, see for instance [30,45,86] for more information 30 31 on this subject. 31 32 Consider the Navier–Stokes equations 32 33 33 ∗ D 34 ∂t u + divx(u ⊗ u) +∇xp = ν xu, divx u = 0,(t,x)∈ R+ × R , 34 35 in 35 u|t=0 = u , 36 36 37 37 38 where ν>0. 38 39 We begin with the three-dimensional case. 39 40 40 in 2 3 in 41 THEOREM 7.3 (Leray [80]). For each u ∈ L (R ) such that divx u = 0, there exists 41 42 a solution in the sense of distributions to the Cauchy problem for the Navier–Stokes equa- 42 43 tions such that 43 44 44 2 3 2 1 3 45 u ∈ C R+; w − L R ∩ L R+; H R . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 83

The Boltzmann equation and its hydrodynamic limits 83

1 → · R 1 Moreover, the function t u(t, ) L2 is nonincreasing on + and satisfies, for each 2 t>0, the Leray energy inequality 2 3 3 t 4 1 2 + ∇ 2 1 in2 4 u(t) L2 ν xu(s, x) dx ds u L2 . 5 2 0 RD 2 5 6 6 7 A solution of the Navier–Stokes equations in the sense of distributions that belongs to 7 2 3 2 1 3 8 C(R+; w − L (R )) ∩ L (R+; H (R )) and satisfies the Leray energy inequality is called 8 9 a “Leray solution”. 9 10 In fact, a modification of Leray’s original argument allows constructing weak solutions 10 11 that satisfy the local energy inequality 11 12 12 13 1 2 1 2 2 1 2 13 ∂t |u| + divx u |u| + p + ν|∇xu| ν x |u| (7.1) 14 2 2 2 14 15 15 R∗ × R3 16 in the sense of distributions on + . 16 17 It is not known whether Leray solutions are uniquely determined by their initial data; 17 18 however, Leray was able to prove that regular solutions of the Navier–Stokes equations are 18 19 unique within the class of Leray solutions. 19 20 20 in ∈ 2 R3 in = 21 THEOREM 7.4 (Leray [80]). Let u L ( ) such that divx u 0. Assume that there 21 22 exists a classical solution, 22 23 23 ∈ 1 R ; 1 R3 ∩ R ; 2 R3 ∇ ∈ ∞ R × R3 24 v C + H C + H such that xv L + , 24 25 25 of the Navier–Stokes equations with initial condition 26 26

27 in 27 v|t= = u . 28 0 28 29 29 Then, any Leray solution u of the Navier–Stokes equations with initial data 30 30 31 in 31 u|t=0 = u 32 32 33 coincides with v a.e. 33 34 34 35 Whether the space dimension is D = 2orD = 3 leads to fundamental differences in the 35 36 regularity theory for the Navier–Stokes equations. 36 37 37 38 in 2 2 in 38 THEOREM 7.5 (Leray [79]). Let u ∈ L (R ) such that divx u = 0. Then there exists a 39 unique weak solution u to the Navier–Stokes equations with initial data 39 40 40 41 in 41 u|t=0 = u 42 42 43 such that 43 44 44 2 2 2 1 2 45 u ∈ C R+; L R ∩ L R+; H R . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 84

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1 Furthermore, this solution is smooth for t>0 1 2 2 ∞ ∗ 2 3 u ∈ C R+ × R 3 4 4 5 and satisfies the energy equality 5 6 6 7 t 7 1 2 + ∇ 2 = 1 in2 8 u(t) L2 ν xu(s, x) dx ds u L2 8 2 0 RD 2 9 9 10 for each t 0. 10 11 11 12 In space dimension 3, it is an outstanding open problem to determine whether Leray 12 13 solutions with C∞ initial data remain C∞ for all times. 13 14 What is known to this date is the following partial regularity theorem which improves 14 15 on an earlier result by Scheffer [112]. 15 16 16 17 17 THEOREM 7.6 (Caffarelli, Kohn and Nirenberg [21]). In space dimension 3, let u be a 18 Leray solution of the Navier–Stokes equations that satisfies the local variant of Leray’s 18 19 energy inequality (7.1). Let the singular set of u be 19 20 20 21 3 21 S(u) = (t, x) ∈ R+ × R u is not bounded in a neighborhood of (t, x) . 22 22 23 23 Then, S(u) has parabolic Hausdorff dimension less than 1. 24 24 25 25 This definition of the singular set S(u) comes from a bootstrap argument due to 26 26 Serrin [113] showing that, if a Leray solution u of the Navier–Stokes equations is bounded 27 27 in B((t, x), R), then u is C∞ in B((t, x), R/2). 28 28 The parabolic Hausdorff dimension is defined through coverings with translates of 29 29 (−r2,r2) × B(0,r) in R × R3 (the usual Hausdorff dimension being defined with balls 30 30 for the Euclidean metric of R4). 31 31 This result implies that the singular set S(u) must be smaller than a curve in space– 32 32 time: in other words singularities of solutions to the Navier–Stokes equations in space 33 33 dimension 3 are rare (if they exist at all). 34 34 35 35 36 36 37 7.3. The incompressible Navier–Stokes–Fourier system 37 38 38 39 By mimicking the compactness method in the proof of the Leray existence theorem, we 39 40 can also treat the case of the Navier–Stokes–Fourier system 40 41 41 ∗ 3 42 ∂t u + divx(u ⊗ u) +∇xp = ν xu, divx u = 0,(t,x)∈ R+ × R , 42 43 ∗ 3 43 ∂t θ + divx(uθ) = κ xθ, (t,x)∈ R+ × R , 44 44 in in 45 (u, θ)|t=0 = u ,θ , 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 85

The Boltzmann equation and its hydrodynamic limits 85

1 where κ>0 and ν>0. 1 2 The analogue of Leray’s theorem for the Navier–Stokes–Fourier system is the following 2 3 one. 3 4 4 5 in in 2 3 in 5 THEOREM 7.7. For each (u ,θ ) ∈ L (R ) such that divx u = 0, there exists a solution 6 in the sense of distributions to the Cauchy problem for the Navier–Stokes–Fourier system 6 7 such that 7 8 8 9 2 3 2 1 3 9 (u, θ) ∈ C R+; w − L R ∩ L R+; H R . 10 10 11 11 This solution satisﬁes, for each t>0 12 12 13 13 1 t 1 14 2 + ∇ 2 in2 14 u(t) L2 ν xu(s, x) dx ds u L2 , 15 2 0 RD 2 15 16 t 16 1 2 + ∇ 2 1 in2 17 θ(t) L2 κ xθ(s,x) dx ds θ L2 . 17 2 0 RD 2 18 18 19 19 20 7.4. The compressible Euler system 20 21 21 22 22 The compressible Euler system is a quasilinear hyperbolic system. The existence and 23 23 uniqueness theory for this system is not entirely satisfying in its present state, especially in 24 24 25 space dimension greater than or equal to 2. More information on the theory of hyperbolic 25 26 system of conservation laws can be found for instance in [32,78] and [19]. 26 27 Consider the Cauchy problem for the compressible Euler system (with perfect mon- 27 28 atomic gas equation of state) 28 29 29 30 ∂t ρ + divx(ρu) = 0, 30 31 31 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) = 0, (7.2) 32 32 33 1 2 3 1 2 5 33 ∂t ρ |u| + θ + divx ρ |u| + θ = 0. 34 2 2 2 2 34 35 35 36 We begin with a local existence and uniqueness result which is a particular case of a gen- 36 37 eral theorem on quasilinear symmetrizable systems. The theory of symmetric hyperbolic 37 38 system was developed very early by Friedrichs; the importance of the notion of symmetriz- 38 39 able systems was recognized by Godunov [52], and then by Friedrichs and Lax [43]–see 39 40 also [46] and [81] for more information on the theory of hyperbolic systems. The result 40 41 below comes from [91]; the case of general systems is studied in [78] and [72]. 41 42 42 43 THEOREM 7.8. Let D 1 and (ρin,uin,θin) ∈ H m(RD) with m>D/2 + 1. There exists 43 − 44 T>0 and a unique solution (ρ,u,θ)∈ C([0,T); H m(RD)) ∩ C1([0,T); H m 1(RD)) of 44 45 the compressible Euler system in the sense of distributions on (0,T)× RD. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 86

86 F. G o l s e

[ − ] 1 Such solutions are regular (since H m(RD) ⊂ C m D/2 (RD) by Sobolev embedding). 1 2 In general, one does not expect that regular solutions to (7.2) may exist for all times. Sin- 2 3 gularities – such as shock waves – are expected to appear in ﬁnite time for a large class of 3 4 smooth initial data. Here is an interesting result in this direction, due to Sideris [114]. 4 5 5 ∞ 6 THEOREM 7.9. Let D = 3, and let R>0. Pick the initial data (ρin,uin,θin) ∈ C (R3) 6 7 to be such that ρin,θin > 0 on R3 with 7 8 8 in in 9 supp ρ − 1 ⊂ B(0,R), supp u ⊂ B(0,R), 9 10 10 in in in 3/2 3 11 supp θ − 1 ⊂ B(0,R) and ρ θ on R . 11 12 12 13 Assume further the existence of R0 ∈ (0,R)such that 13 14 14 (|x|−r)2 15 ρin(x) − 1 dx>0 15 16 |x|>r |x| 16 17 17 18 and 18 19 19 (|x|2 − r2) 20 ρin(x)x · uin(x) x 20 3 d 0 21 |x|>r |x| 21 22 22 1 23 for each r ∈ (R0,R). Then the life-span T of the C solution to (7.2) with such initial data 23 24 is ﬁnite. 24 25 25 26 In dimension greater than or equal to 2, there is no satisfying theory of weak solutions 26 27 that could extend classical solutions after blow-up time. 27 28 At variance, in the one-dimensional case, there is a rather complete theory of weak solu- 28 29 tions, that are obtained as superpositions of interacting Riemann problems (i.e., a Cauchy 29 30 problem for (5.6) whose initial data is a step function with only one jump). Liu studied 30 31 the compressible Euler system in space dimension 1 written in Lagrangian coordinates. 31 32 Denoting by a the Lagrangian particle label, this system reads 32 33 33 34 ∂ V − ∂ U = 0, 34 t a 35 Θ 35 ∂ U + ∂ = 0, 36 t a V 36 37 (7.3) 37 38 1 2 3 ΘU 38 ∂t U + Θ + ∂a = 0, 39 2 2 V 39 40 in in in 40 (V,U,Θ)|t=0 = V ,U ,Θ , 41 41 42 42 with the notation 43 43 44 1 44 V(t,a)= , 45 ρ(t,X(t,0,a)) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 87

The Boltzmann equation and its hydrodynamic limits 87 1 U(t,a)= u t,X(t,0,a) , 1 2 2 Θ(t,a)= θ t,X(t,0,a) , 3 3 4 4 t → X(t, ,a) a t = 5 where 0 is the path of the particle which is at the position at time 0. 5 6 The following existence result is based on Glimm’s algorithm [51] for constructing 6 7 BV solutions to hyperbolic systems of conservation laws in the one-dimensional case that 7 8 are global in time for initial data of small total variation. 8 9 9 in in in ∈ R in 10 THEOREM 7.10 (Liu [89]). Assume that (V ,u ,θ ) BV( ) with θ θ∗ > 0 while 10 in ∗ R 11 V V on . There exists η0 > 0 such that the Cauchy problem (7.3) has a global weak 11 in in in 12 solution provided that TV(V ,u ,θ ) η0. Moreover, this solution is “entropic”, i.e., 12 13 13 2/3 14 ∂t ln V Θ 0. 14 15 15 16 (In other words, the entropy density cannot decrease along particle paths.) 16 17 17 18 18 19 7.5. The incompressible Euler equations 19 20 20 21 There are essentially two main directions in the mathematical theory of the incompressible 21 22 Euler equations: 22 23 • the PDE viewpoint, and 23 24 • the geometric viewpoint. 24 25 In the geometric viewpoint, Euler’s equations for incompressible ﬂuids in the periodic 25 D 26 box T are the equations of geodesics on the group of volume preserving diffeomorphisms 26 D 27 of T , endowed with the metric deﬁned by the kinetic energy. We shall say nothing of 27 28 this part of the theory, for which we refer the reader to the beautiful book by Arnold and 28 29 Khesin [4]. 29 30 Instead, we shall just recall a few results on the Euler equations as nonlinear PDEs 30 D 31 on T . The reader is advised to read [92] and [86] for more information on that topic. 31 32 The incompressible Euler equations are 32 33 33 ∗ D 34 ∂t u + divx(u ⊗ u) +∇xp = 0, divx u = 0,(t,x)∈ R+ × T , 34 35 in 35 | = = 36 u t 0 u . 36 37 37 38 We begin with a local existence result for classical solutions in the three-dimensional 38 39 case, due to Kato. 39 40 40 in ∈ m T3 ∈ N in = 41 THEOREM 7.11 (Kato [71]). Let u H ( ), m , m 3, such that divx u 0. 41 42 Then, there exists T>0 and a unique local solution u of the incompressible Euler equa- 42 in 43 tions with initial data u , such that 43 44 44 ∈ [ ; m T3 ∩ [ ; m−1 T3 45 u C 0,T) H ACloc 0,T) H . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 88

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1 Whether the life-span of such solutions is finite is an outstanding open problem in the 1 2 theory of nonlinear PDEs. What is known is that the blow-up time, if finite, does not depend 2 3 on the regularity index m, as shown by the following beautiful result. 3 4 4 5 THEOREM 7.12 (Beale, Kato and Majda [15]). Under the same assumptions as in the 5 6 previous theorem, if T is finite, then 6 7 • either 7 8 8 T 9 9 curl u(t, ·) ∞ dt =+∞, 10 x L 10 0 11 11 12 • or u ∈ C([0,T]; H m(T3)). 12 13 (In the second case, the solution u can be extended to an interval of time [0,T) with 13 14 T >T.) 14 15 15 16 In the two-dimensional case, there is global existence and uniqueness of a classical so- 16 17 lution to the Cauchy problem for the incompressible Euler equations. 17 18 18 19 THEOREM 7.13 (Yudovich [123]). Let uin ∈ H m(T2), m ∈ N, m 3, such that 19 20 in 20 divx u = 0. Then there exists a unique global solution u of the incompressible Euler 21 equations with initial data uin such that 21 22 22 23 m 2 1 m−1 2 23 u ∈ C R+; H T ∩ C R+; H T . 24 24 25 A good reference on the two-dimensional case of the incompressible Euler equations 25 26 is [29]. 26 27 27 In view of the importance of the vorticity field curlx u for the regularity of the solution 28 to the incompressible Euler equations, the main difference between the two- and the thre- 28 29 dimensional cases comes from the following observation. 29 30 If one considers the two-dimensional flow as given by a three-dimensional velocity field 30 31 u of the form 31 32   32 33 u1(t, x1,x2) 33 34   34 u(t, x) = u2(t, x1,x2) , 35 35 0 36 36 37 the vorticity field is 37 38 38 39 0 39 40 40 curlx u(t, x) = 0 , 41 41 ω(t,x1,x2) 42 42 43 where 43 44 44 45 = − 45 ω(t,x1,x2) ∂x1 u2(t, x1,x2) ∂x2 u1(t, x1,x2). dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 89

The Boltzmann equation and its hydrodynamic limits 89

1 The scalar quantity ω satisfies the transport equation 1 2 2 3 3 ∂t ω + u ·∇xω = 0. 4 4 5 5 The maximum principle holds for the transport equation above, so that 6 6 7 7 ∞ = · 8 ω L (R+×T2) ω(0, ) L∞(T2). 8 9 9 10 Hence, in the two-dimensional case, if uin is sufficiently regular, the vorticity is globally 10 11 bounded and therefore, by the Beale–Kato–Majda criterion, the regularity of the initial data 11 12 is propagated for all times. 12 13 13 In the three-dimensional case, the vorticity curlx u is a vector field that satisfies the 14 analogue of the scalar transport equation above for vectors 14 15 15 16 16 ∂t curlx u + (u ·∇x) curlx u =∇xu · curlx u. 17 17 18 18 The length of the vector curl u can be amplified, or damped, by the matrix ∇ u: this mech- 19 x x 19 20 anism is called “vortex stretching” and so far, there is no satisfying method for controlling 20 21 it. Therefore, there is no a priori bound on the vorticity as in the two-dimensional case, and 21 22 this is why the question of global existence or finite-time blow-up for classical solutions to 22 23 the incompressible Euler equations in the three-dimensional case remains very much open. 23 24 Finally, we conclude this section with an important class of global solutions on the 24 25 periodic box to the incompressible Euler equations. Choose 25 26   26 27 u1(t, x1,x2) 27   3 28 U(t,x)= u2(t, x1,x2) ,(t,x)∈ R+ × T , 28 29 29 w(t,x1,x2) 30 30 31 where 31 32 32 33 u (t, x ,x ) 33 = 1 1 2 34 u(t, x1,x2) 34 u2(t, x1,x2) 35 35 36 36 is a C1-solution of the two-dimensional incompressible Euler equations on R+ × T2.If 37 37 w satisfies the transport equation 38 38 39 39 + = ∈ R∗ × T2 40 ∂t w divx(wu) 0,(t,x) + , 40 41 41 42 the vector field U solves the three-dimensional incompressible Euler equation on R+ ×T3. 42 43 Such solutions are referred to as 2D–3C solutions of the incompressible Euler equations – 43 44 see, in particular, Section S4.3, pp. 150–153 of [86] for an interesting application of 2D–3C 44 45 solutions to the problem of a priori estimates on the Euler equations. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 90

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1 8. Mathematical theory of the Cauchy problem for the Boltzmann equation 1 2 2 3 8.1. Global classical solutions for “small” data 3 4 4 5 In this subsection, we briefly review two early existence theories for the Boltzmann equa- 5 6 tion: 6 7 • Ukai’s theory for perturbations of uniform Maxwellian states; and 7 8 • the Illner–Shinbrot theory for small perturbations of the vacuum state. 8 9 Consider the Cauchy problem for the Boltzmann equation 9 10 10 11 ∗ 3 3 11 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × R × R , 12 12 in 13 F |t=0 = F . 13 14 14 15 8.1.1. Small perturbations of the vacuum. The collision kernel b in the collision integral 15 16 is supposed to satisfy 16 17 17 18 0 b(z,ω) C 1 +|z| for a.e. (z, ω) ∈ R3 × S2. 18 19 19 20 20 THEOREM 8.1 (Illner and Shinbrot [70]). Pick c>0; there exists η>0 such that for each 21 C ∈ (0,η)and each initial data F in satisfying 21 22 22 23 23 in −c(|x|2+|v|2) ∈ R3 24 0 F (x, v) Ce ,x,v , 24 25 25 in 26 then the Cauchy problem for the Boltzmann equation with initial data F has a unique 26 27 global (mild ) solution. 27 28 28 29 The key to this result is to dominate the solution F of the Boltzmann equation with a 29 −c|x−tv|2 30 Maxwellian traveling wave e . 30 31 Obviously this result is not useful in the context of incompressible hydrodynamic limits: 31 32 it bears on rarefied clouds of gas that expand in the vacuum, and therefore never approach 32 33 any global Maxwellian equilibrium. 33 34 34 35 8.1.2. Small perturbations of a global Maxwellian state. The following result, due to 35 36 Ukai, is the first global existence and uniqueness result proved on the (space inhomoge- 36 37 neous) Boltzmann equation. It bears on the dynamics of a gas whose state is a perturbation 37 38 of a global Maxwellian equilibrium. It uses a detailed spectral analysis of the linearization 38 39 of the collision integral at the background Maxwellian state, see [40]. 39 40 40

41 THEOREM 8.2 (Ukai [116]). Assume that b(z,ω) =|z·ω| (hard sphere case). Let s>3/2, 41 42 and β>3; deﬁne 42 43 43 44 = +| | β · 44 f s,β sup 1 v f(,v) s 3 45 H (R ) 45 v∈R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 91

The Boltzmann equation and its hydrodynamic limits 91

1 and let 1 2 2 ∞ 3 s = ∈ R3; s +∞ 3 Hβ f Lloc v Hx f s,β < . 4 4 5 5 There exists η>0 such that, for any f in ≡ f in(x, v) satisfying 6 6 7 7 in in −M1/2 8 f s,β <η and f (1,0,1), 8 9 9 10 the Cauchy problem for the Boltzmann equation with initial data 10 11 11 1/2 12 in = M + M in 12 F (1,0,1) (1,0,1)f 13 13 14 has a unique solution F such that 14 15 15 16 ∈ ∞ R s ∩ R s−ε ∩ 1 R s−1−ε 16 F L +;Hβ C +;Hβ−ε C +;Hβ−1−ε 17 17 18 18 for each ε>0. 19 19 20 20 Ukai’s theory describes the evolution of number densities that are close to a uniform 21 21 Maxwellian state, and therefore, one could think of using it in the context of incompressible 22 22 hydrodynamic limits. However, one should bear in mind that the parameter η that monitors 23 23 the size of the initial number density ﬂuctuation is not uniform in the Knudsen number, so 24 24 that applying Ukai’s ideas to derive, say, the incompressible Navier–Stokes equations from 25 25 the Boltzmann equation requires nontrivial modiﬁcations, due to Bardos and Ukai [14]. 26 26 27 Hence, for the purpose of deriving hydrodynamic models from the Boltzmann equation, 27 28 it is desirable to have at one’s disposal a global existence theory based on a priori estimates 28 29 that are uniform in the Knudsen number. The only such existence theory so far is the 29 30 DiPerna–Lions theory of weak solutions of the Boltzmann equation that is described below. 30 31 31 32 32 33 8.2. The DiPerna–Lions theory 33 34 34 35 As already mentioned in our presentation of the Boltzmann equation, the collision integral 35 36 is local in t and x and an integral operator in v. In other words, the collision integral acts as 36 37 a multiplication operator in the variables (t, x), and as kind of convolution in the variable v. 37 38 On the other hand, the natural a priori bound for the number density is 38 39 39 ∞ 40 ∈ 40 F Lt (L ln Lx). 41 41 42 For such an F , expressions like 42 43 43 44 44 2 45 F or F F dv 45 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 92

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1 are deﬁned only as measurable functions, and not as distributions. Hence the collision 1 2 integral does not deﬁne a distribution for all number densities that satisfy the natural a priori 2 3 bounds on solutions of the Boltzmann equation. 3 4 To get around this, DiPerna and Lions proposed to use the following notion of solution. 4 5 5 6 DEFINITION 8.3. A nonnegative function F ∈ C(R+; L1(R3 × R3)) is a renormalized 6 7 solution of the Boltzmann equation if 7 8 8 B(F, F ) 9 √ ∈ 1 9 Lloc( dt dx dv), 10 1 + F 10 11 11 1 12 and if, for each Γ ∈ C (R+) such that 12 13 13 C 14 Γ (Z) √ for all Z 0, 14 15 1 + Z 15 16 16 17 one has 17 18 18 + ·∇ = B 19 (∂t v x)Γ (F ) Γ (F ) (F, F ) 19 20 20 in the sense of distributions on R∗ × R3 × R3. 21 + 21 22 22 In this subsection, we shall consider collision kernels that satisfy the following weak 23 23 cut-off assumption 24 24 25 1 25 26 lim b(v − v∗,ω)dω dv∗ = 0 for each R>0. (8.1) 26 | |→+∞ 2 v 1 +|v| | ∗| S2 27 v

The Boltzmann equation and its hydrodynamic limits 93

1 • the energy inequality: for each t 0, 1 2 2 1 1 3 |v|2F(t)dv dx |v|2F in dv dx; 3 4 R3×R3 2 R3×R3 2 4 5 5 6 • and the H inequality: for each t>0, one has 6 7 7 8 F ln F(t)dv dx 8 9 R3×R3 9 10 t 10 F F∗ 11 + F F∗ − FF∗ ln b dv dv∗ dω dx ds 11 0 R3 R3×R3×R3 FF∗ 12 12 13 F in ln F in dv dx. 13 14 R3×R3 14 15 15 16 A complete description of the proof of the DiPerna–Lions theorem is beyond the scope 16 17 of the present work. We shall just explain the main ideas in it. 17 18 18 ≡ 19 8.2.1. The role of the normalizing nonlinearity. Assume that F F(t,x,v) 0a.e.on 19 R × R3 × R3 20 + satisfies 20 21 21 2 22 1 +|v| F(t,x,v)dv dx C for a.e. t 0, (8.2) 22 R3×R3 23 23 24 and 24 25 25 26 T 26 F F∗ 27 F F∗ − FF∗ ln b dv dv∗ dω dx ds CT (8.3) 27 0 R3 R3×R3×R3 FF∗ 28 28 29 for each T>0. A solution to the Boltzmann equation satisfies the first inequality (by 29 30 the global conservation law of energy) and the second estimate by the entropy production 30 31 bound deduced from Boltzmann’s H -theorem. 31 32 Then 32 33 33 |B(F, F )| 34 √ ∈ 1 R × R3 × R3 34 Lloc + . 35 1 + F 35 36 36 37 Indeed, we first recall the elementary inequality 37 38 √ √ 38 2 1 39 X − Y (X − Y)(ln X − ln Y) for each X, Y > 0. 39 40 4 40 41 Then 41 42 42 43 43 F F∗ − FF∗ = F F∗ + FF∗ F F∗ − FF∗ 44 44 2 45 F F∗ − FF∗ + 2 FF∗ F F∗ − FF∗ . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 94

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1 Hence 1 2 2 T 3 |B(F, F )| 3 √ dv dx dt 4 0 R3 |v|

The Boltzmann equation and its hydrodynamic limits 95

1 convolution operator in the variable v and a multiplication operator in the variables t and x, 1 2 one should seek compactness “with respect to the t and x variables only” – a notion which, 2 3 of course, remains to be defined. 3 4 The appropriate class of compactness theorems was discovered a few years before being 4 5 applied to the Boltzmann equation. They are known as “velocity averaging results”, and 5 6 were first introduced by Golse, Perthame and Sentis in [56] within the context of the diffu- 6 7 sion approximation of the neutron transport equation. (Independently, analogous regularity 7 8 results for the coefficients of the spherical harmonic expansion of the solution to the free 8 9 transport equation were announced in [1].) 9 10 1 10 Obviously, whatever compactness in the strong topology of Lloc is to be found on a 11 sequence of solutions to the Boltzmann equation (regularized or not) has to come from the 11 12 streaming (free transport) operator ∂t + v ·∇x . 12 13 Being hyperbolic, the transport operator v ·∇x propagates singularities along charac- 13 14 teristics. Therefore, at first sight it seems hopeless that one might obtain any regularizing 14 15 effect from the free streaming part of the Boltzmann equation, or of any other similar ki- 15 16 netic model. One can think of the following elementary example. 16 17 17 2 2 18 EXAMPLE.Letf ∈ L (R); for a.e. x,v ∈ R , define F(x,v)= f(v2x1 − v1x2). Clearly, 18 19 ∈ 2 R2 × R2 ·∇ = 2 R 19 F Lloc( ) and v xF 0. However, since f can be any function in L ( ), 20 ∈ s R2 × R2 ·∇ ∈ 2 R2 × R2 20 F/Hloc( ) for any s>0, although v xF Lloc( ). 21 21 22 22 The key to obtaining regularizing effects from the transport operator v ·∇x is to seek 23 those effects not on the number density itself, but on velocity averages thereof, in other 23 24 words, on the macroscopic densities. Here is the prototype of all velocity averaging results. 24 25 25 26 2 D 26 THEOREM 8.5 (Golse, Perthame and Sentis [56]). Let Fε be a bounded family in L (R × 27 D 2 D D 27 R ). Assume that the family v ·∇xFε is also bounded in L (R × R ). Then, for each 28 2 D 28 φ ∈ L (R ), the family of moments ρε defined by 29 29 30 30 ρε(x) = Fε(x, v)φ(v) dv 31 RD 31 32 32 33 2 RD 33 is relatively compact in Lloc( ). 34 34 35 35 PROOF. Set Gε = Fε + v ·∇xFε, and let Fε and Gε denote respectively the Fourier trans- 36 36 forms of Fε and Gε in the x variable only. The assumptions on Fε imply that the family 37 2 D D 37 Gε is bounded in L (R × R ), and that 38 38 39 39 Gε(ξ, v)φ(v) dv 40 ρˆε(ξ) = . 40 RD 1 + iv · ξ 41 41 42 42 We need to study how ρˆε(ξ) decays for |ξ| large. By the Cauchy–Schwarz inequality 43 43 44 44 ˆ 2 1 2 45 ρε(ξ) Gε(ξ, v) dv, 45 m(ξ) RD dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 96

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1 with 1 2 2 1 |φ(v)|2 dv |φ(v)|2 dv 3 := = , 3 4 m(ξ) RD 1 +|v · ξ|2 RD 1 +|v · ω|2|ξ|2 4 5 5 D−1 D 6 where ω = ξ/|ξ|∈S for all ξ ∈ R \{0}. The latter integral is a decreasing fam- 6 7 ily indexed by |ξ| of continuous functions of ω; this family vanishes pointwise in ω 7 8 as |ξ|→+∞ by dominated convergence. By Dini’s theorem, it vanishes uniformly in 8 D−1 9 ω ∈ S , and therefore m(ξ) →+∞as |ξ|→+∞. Since the family 9 10 10 11 2 2 11 ρˆε(ξ) m(ξ) dξ Gε(ξ, v) dξ dv 12 RD RD×RD 12 13 13 2 RD 14 is bounded by Plancherel’s theorem, ρε is relatively compact in Lloc( ) (by a variant of 14 15 Rellich’s compactness theorem). 15 16 16 + ·∇ −1 + ·∇ = 17 Since the operator (I v x) (which maps G on the solution F of F v xF G) 17 1 RD ×RD ∞ RD ×RD 18 is a contraction mapping on both L ( ) and L ( ), the velocity averaging 18 p ∈ +∞ 1 19 result above also holds in L for all p (1, ) by interpolation. However, it fails in L , 19 ∞ 20 as the following example shows. (It also fails in L ,see[55], p. 124.) 20 21 21 1 RD × RD 22 EXAMPLE ([55], pp. 123–124). Consider Gε, a bounded family of L ( ), and for 22 + ·∇ = → ⊗ ∗ 23 each ε,letFε be the solution to Fε v xFε Gε. Assume that Gε δ0 δv weakly, 23 | ∗|= ·∇ 1 RD × RD 24 where v 1. Then both Fε and v xFε are bounded in L ( ) and 24 25 +∞ 25 −t 26 ρε(x) = Fε(x, v) dv = e Gε(x − tv,v)dv dt 26 27 RD 0 RD 27 28 28 ∈ RD 29 so that, for any test function φ Cc( ), 29 30 +∞ 30 −t ∗ 31 ρε(x)φ(x) dx → e φ tv dt 31 D 32 R 0 32 33 ∗ 33 as ε → 0. Hence ρε converges weakly to a density carried by the half-line R+v , so that in 34 1 D 34 particular the family ρε is not relatively compact in L (R ). 35 loc 35 36 36 This example rests on the possible build-up of concentrations in Fε and v ·∇xFε.If 37 37 such concentrations are ruled out, the same interpolation argument as above entails the 38 38 following L1 variant of velocity averaging. 39 39 40 40 PROPOSITION 8.6 (Golse, Lions, Perthame and Sentis [55]). Let Fε be a family of mea- 41 surable functions on RD × RD such that, for each compact subset K of RD, both families 41 42 D 42 Fε and v ·∇xFε are uniformly integrable on K × R . Then the family ρε deﬁned by 43 43 44 44 = 45 ρε(x) Fε(x, v) dv 45 RD dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 97

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1 1 RD 1 is relatively compact in Lloc( ). 2 2 ∞ 3 ≡ RD 3 PROOF.Letχ χ(x) belongs to Cc ( ). Set 4 4 5 Gε(x, v) = χ(x)Fε(x, v) + v ·∇x χ(x)Fε(x, v) 5 6 6 7 and, for each λ>0, decompose χρε as follows 7 8 8 = > + < 9 χρε χρλ,ε χρλ,ε 9 10 10 11 with 11 12 12 > −1 13 = + ·∇ + 13 χρλ,ε (I v x) (Gε1|Gε|>λ) dv χFε dv, |v|λ |v|>λ 14 14 15 < −1 15 χρ = (I + v ·∇ ) (G 1| | ) dv. 16 R,λ,ε x ε Gε λ 16 |v|λ 17 17 18 → 18 The assumptions on Fε imply that Gε is uniformly integrable, so that Gε1|Gε|>λ 0 and 19 → 1 RD × RD →+∞ > → 1 RD 19 χFε1|v|>λ 0inL ( ) uniformly in ε as λ ; thus χρl,ε 0inL ( ) 20 uniformly in ε as λ →+∞. 20 21 2 RD 21 On the other hand, for each λ, the family Gε1|Gε|>λ indexed by ε is bounded in L ( ); 2 < 2 RD 22 thus, by velocity averaging in L , χρλ,ε is relatively compact in L ( ) – and therefore 22 23 in L1(RD), since it has support in supp χ which is compact. 23 24 1 D 24 Hence χρε is relatively compact in L (R ). Since χ is arbitrary, this eventually implies 25 1 RD 25 that ρε is relatively compact in Lloc( ). 26 26 27 In fact, by a further interpolation argument, one can get rid of the assumption of uniform 27 28 integrability on derivatives. 28 29 29

30 THEOREM 8.7 (Golse and Saint-Raymond [60]). Let Fε be a family of measurable func- 30 D D D 31 tions on R × R such that, for each compact subset K of R , the family Fε is uniformly 31 D 1 D 32 integrable on K × R , while v ·∇xFε is bounded in L (K × R ). Then the family ρε 32 33 deﬁned by 33 34 34 35 35 ρε(x) = Fε(x, v) dv 36 RD 36 37 37 1 RD 38 is relatively compact in Lloc( ). 38 39 39 D 40 PROOF. Without loss of generality, assume that all the Fε are supported in K × R . Write 40 41 41 42 −1 42 ρε(x) = λ (λI + v ·∇x) Fε(x, v) dv RD 43 43 44 44 + + ·∇ −1 ·∇ 45 (λI v x) (v xFε)(x, v) dv. 45 RD dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 98

98 F. G o l s e

1 Since 1 2 2 1 3 + ·∇ −1 3 (λI v x) L(L1 ) , 4 x,v λ 4

5 1 5 the second term on the right-hand side of the equality above is O(1/λ) in Lx uniformly 6 1 6 in ε, while the first term is strongly relatively compact in Lx for each λ>0 by the previous 7 1 7 proposition. Hence the family ρε is strongly relatively compact in L . 8 x 8 9 9 There are several extensions and variants of the velocity averaging results recalled here, 10 see for instance [36,38,47–49,106]. Except for the extension of the above results to the 10 11 evolution problem, which is needed in the construction of renormalized solutions to the 11 12 Boltzmann equation, we shall not discuss these extensions in the present notes, but refer 12 13 the interested reader to Chapter 1 of [18] for a survey of that theory as of 2000. 13 14 Here is the analogue of the L1-variant of velocity averaging for evolution problems. 14 15 15 16 16 THEOREM 8.8. Consider Fε ≡ Fε(t,x,v), a family of measurable functions on R+ × 17 D D D 17 R × R such that, for each T>0 and each compact K ⊂ R , Fε is uniformly integrable 18 D 1 D 18 on [0,T]×K × R while (∂t + v ·∇x)Fε is bounded in L ([0,T]×K × R ). Then the 19 19 family ρε defined by 20 20 21 21 ρε(t, x) = Fε(t,x,v)dv 22 RD 22 23 23 24 1 R × RD 24 is relatively compact in Lloc( + ). 25 25 26 The proof is a straightforward variant of the arguments for the steady transport operator 26 27 v ·∇x given above. 27 28 28 29 8.2.3. Conclusion. Let us briefly explain how the renormalization procedure is combined 29 30 with compactness by velocity averaging in the proof of the DiPerna–Lions theorem. 30 31 Choose as normalizing nonlinearity the function 31 32 32 33 1 33 Γδ(Z) = ln(1 + δZ), δ > 0, 34 δ 34 35 and consider the truncated Boltzmann equation (8.5). We leave it to the reader to verify 35 36 that, under the condition (8.4), the map 36 37 37 38 B 38 → (F, F ) 39 F 39 1 + (1/n) R3 F dv 40 40 41 1 R3 × R3 41 is Lipschitz continuous on L ( x v), so that the truncated Boltzmann equation (8.5) 42 has a global solution. Because the truncation factor 42 43 43 44 1 44 45 45 1 + (1/n) R3 F dv dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 99

The Boltzmann equation and its hydrodynamic limits 99

1 is independent of v, the symmetries of the collision integral that imply the local conserva- 1 2 tion laws of mass, momentum and energy, and the H -theorem hold also for the truncated 2 3 collision integral 3 4 4 B 5 (F, F ) 5 6 6 1 + (1/n) R3 F dv 7 7 8 so that, in particular, 8 9 9 10 10 2 2 11 1 +|x − tv| +|v| Fn(t,x,v)dx dv 11 R3×R3 12 12 13 = 1 +|x|2 +|v|2 F in(x, v) dx dv 13 14 R3×R3 14 15 15 16 for all t 0, while 16 17 17 18 in in 18 Fn ln Fn(t,x,v)dx dv F ln F (x, v) dx dv. 19 R3×R3 R3×R3 19 20 20 21 As explained in Case 4 of Section 3.3, this implies the existence of a positive constant C 21 22 such that 22 23 23 24 2 2 2 24 1 +|x| +|v| Fn(t,x,v)dx dv C 1 + t and 25 R3×R3 25 (8.6) 26 26 | | + 2 27 Fn ln Fn (t,x,v)dx dv C 1 t for all t 0. 27 R3×R3 28 28

29 1 3 3 29 Hence Fn is weakly relatively compact in L (R+ × R × R ) by the Dunford–Pettis 30 loc 30 theorem. Since, for each δ>0, one has 31 31 32 32 33 Γδ(Z) Z for each Z 0, 33 34 34 1 R ; 1 R3 × 35 the sequence of functions Γδ(Fn) is also weakly relatively compact in Lloc( + L ( 35 3 36 R )). On the other hand, the discussion in Section 8.2.1 shows that 36 37 37 B 38 (Fn,Fn) 38 (∂t + v ·∇x)Γδ(Fn) = 39 (1 + δFn)(1 + (1/n) R3 Fn dv) 39 40 40 41 1 R × R3 × R3 41 is bounded in Lloc( + ). By velocity averaging (Theorem 8.8), this implies that 42 the sequence 42 43 43 44 44 45 Γδ(Fn) dv 45 R3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 100

100 F. G o l s e

1 1 R ; 1 R3 1 is strongly relatively compact in Lloc( + L ( )). On the other hand, because of the 2 bound (8.6), 2 3 3 4 4 Fn dv − Γδ(Fn) dv → 0 5 R3 R3 5 6 6 7 1 R ; 1 R3 → 7 in Lloc( + L ( )) as δ 0, uniformly in n 1. Hence we conclude that the sequence 8 8 9 9 Fn dv 10 R3 10 11 11 12 1 R ; 1 R3 12 is strongly relatively compact in Lloc( + L ( )). 13 Hence, modulo extraction of a subsequence 13 14 14 15 → 1 R × R3 × R3 15 Fn F weakly in Lloc + 16 16 17 while 17 18 18 19 → 1 R × R3 19 Fn dv F dv strongly in Lloc + . 20 R3 R3 20 21 21 22 Since the collision integral acts as a convolution in the v-variable and a multiplication op- 22 3 23 erator in the t and x variables, this compactness theorem implies that, for each φ ∈ Cc(R ), 23 24 24 25 25 B(Fn,Fn)φ dv → B(F, F )φ dv in measure on [0,T]×K 26 R3 R3 26 27 27 3 28 for each T>0 and each compact K ⊂ R . At ﬁrst sight, this is not enough to pass to the 28 29 limit in the sense of distributions in both sides of the truncated Boltzmann equation (8.5). 29 30 Instead, one integrates the truncated Boltzmann equation along characteristics by treat- 30 31 ing the gain term in the truncated collision integral as a source term. One easily sees that 31 32 the limit F is a supersolution of the limiting Boltzmann equation integrated along charac- 32 33 teristics; notice that this does not make use of the renormalization procedure. That F is a 33 34 subsolution is more involved, we refer to [37] for a complete proof. 34 35 35 36 36 37 8.3. Variants of the DiPerna–Lions theory 37 38 38 39 The original DiPerna–Lions theorem considers a cloud of gas that expands in the vacuum, 39 40 without any restriction on its degree of rarefaction, i.e., for an initial number density of 40 41 arbitrary size that has ﬁnite mass, energy, second moment in x and entropy. As explained 41 42 in Section 8.1.1, this kind of situation is not compatible with incompressible hydrodynamic 42 43 limits. 43 44 For that purpose, we describe below two variants of the DiPerna–Lions theory that are 44 45 particularly relevant in the context of incompressible hydrodynamic limits. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 101

The Boltzmann equation and its hydrodynamic limits 101

1 8.3.1. The periodic box. The first variant of the DiPerna–Lions theory that we discuss 1 2 here is the case of the spatial domain T3 (the periodic box). The collision kernel b satisfies 2 3 the weak cut-off condition (8.1). Let M be the centered reduced Gaussian 3 4 4 5 1 −| |2 5 M(v)= e v /2. 6 (2π)3/2 6 7 7 1 3 3 8 We shall say that F ∈ C(R+; L (R × R )) is a renormalized solution of the Boltzmann 8 9 equation 9 10 10 ∗ 3 3 11 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × T × R , 11 12 in 12 F |t=0 = F , 13 13 14 14 relative to M if and only if, for each normalizing nonlinearity Γ ∈ C1(R+) such that 15 15 16 16 C 17 Γ (Z) √ ,Z 0, 17 1 + Z 18 18 19 one has 19 20 20 21 F F 21 M(∂t + v ·∇x)Γ = Γ B(F, F ) 22 M M 22 23 23 ∗ 3 3 24 in the sense of distributions on R+ × T × R . 24 25 25 26 THEOREM 8.9. Let F in 0 a.e. be a measurable function such that H(Fin|M) < +∞. 26 27 There exists a renormalized solution F relative to M of the Cauchy problem for the Boltz- 27 28 mann equation with initial data F in. This solution satisfies 28 29 • the continuity equation 29 30 30 31 31 ∂t F dv + divx vF dv = 0; 32 R3 R3 32 33 33 34 • the following variant of the local conservation of momentum 34 35 35 36 36 ∂t vF dv + divx v ⊗ vF dv + divx m = 0, 37 R3 R3 37 38 38 ∈ ∞ R ; M T3 R 39 where m L ( + ( ,M3( ))) with values in nonnegative symmetric matrices; 39 • 40 the following energy relation 40 41 41 1 1 42 |v|2F(t,x,v)dx dv + trace m(t) 42 T3×R3 2 2 T3×R3 43 43 44 44 = 1| |2 in 45 v F (x, v) dx dv 45 T3×R3 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 102

102 F. G o l s e

1 for each t>0; 1 2 • and the H inequality: for each t>0, one has 2 3 3 4 t 4 1 − F F∗ 5 F F∗ FF∗ ln b dv dv∗ dω dx ds 5 4 0 T3 R3×R3×R3 FF∗ 6 6 1 7 H F inM − H F(t)M − trace m(t) . 7 8 2 8 9 9 10 The above result – especially the existence of the defect measure m – is due to Lions 10 11 and Masmoudi [88]. 11 12 12 13 8.3.2. The Euclidean space with uniform Maxwellian state at infinity. The next variant 13 14 of the DiPerna–Lions theory that we consider is the case of a spatial domain that is the 14 15 Euclidean space with uniform Maxwellian equilibrium at infinity. Consider the Cauchy 15 16 problem 16 17 17 ∗ 3 3 18 ∂t F + v ·∇xF = B(F, F ), (t, x, v) ∈ R+ × R × R , 18 19 F(t,x,v)→ M as |x|→+∞, 19 20 20 in 21 F |t=0 = F , 21 22 22 23 where M is the centered reduced Gaussian 23 24 24 25 1 −| |2 25 M(v)= e v /2. 26 (2π)3/2 26 27 27 28 Here again, the collision kernel b satisfies the weak cut-off assumption (8.1). The notion 28 29 of “renormalized solution relative to M” of the Cauchy problem above is the same as in 29 30 the case of the periodic box. 30 31 31 32 THEOREM 8.10 (Lions [83]). Let F in 0 a.e. be a measurable function such that 32 33 H(Fin|M) < +∞. There exists a renormalized solution F relative to M of the Cauchy 33 34 problem for the Boltzmann equation with initial data F in. This solution satisfies 34 35 • the continuity equation 35 36 36 37 37 ∂t F dv + divx vF dv = 0; 38 R3 R3 38 39 39 40 • and the H -inequality: for each t>0, one has 40 41 41 t 42 1 F F∗ 42 F F − FF∗ b v v∗ ω x s 43 ∗ ln d d d d d 43 4 0 R3 R3×R3×R3 FF∗ 44 44 in − 45 H F M H F(t) M . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 103

The Boltzmann equation and its hydrodynamic limits 103

1 9. The Hilbert expansion method: Application to the compressible Euler limit 1 2 2 3 In this section we describe a first method for deriving hydrodynamic models from the 3 4 Boltzmann equation. In spite of its numerous shortcomings (which we shall discuss at the 4 5 end of the present section), this method is extremely robust, and can be applied to various 5 6 kinetic models other than the Boltzmann equation. 6 7 Start from the Boltzmann equation in the compressible Euler scaling 7 8 8 9 1 ∗ 3 3 9 ∂t Fε + v ·∇xFε = B(Fε,Fε), (t, x, v) ∈ R+ × T × R . (9.1) 10 ε 10 11 11 12 For simplicity, we only consider in this section the case of a hard sphere gas, so that the 12 B 13 collision kernel b in Boltzmann’s collision integral is given by the expression 13 14 14 =| · | ∈ R3 × S2 15 b(z,ω) z ω ,(z,ω) . 15 16 16 17 Consider next the compressible Euler system for a perfect monatomic gas 17 18 18 + = 19 ∂t ρ divx(ρu) 0, 19 20 20 ∂t (ρu) + divx(ρu ⊗ u) +∇x(ρθ) = 0, 21 (9.2) 21 22 1 2 3 1 2 5 22 ∂t ρ |u| + θ + divx ρu |u| + θ = 0, 23 2 2 2 2 23 24 in in in 24 (ρ,u,θ)|t=0 = ρ ,u ,θ , 25 25 26 where 26 27 27 28 ρin,uin and θ in ∈ H 5 T3 ,ρin 0 and θ in > 0onT3. (9.3) 28 29 29 30 Let (ρ,u,θ)be the solution to (9.2) predicted by Theorem 7.8 under the assumption (9.3), 30 31 and call T>0 its lifespan. Finally, define the local Maxwellian 31 32 32 33 33 E(t,x,v) = M(ρ(t,x),u(t,x),θ(t,x)). 34 34 35 35 THEOREM 9.1 (Caflisch [22]). There exists ε0 such that, for each ε ∈ (0,ε0), there is a 36 3 3 36 unique solution Fε of the Boltzmann equation (9.1) on [0,T)× T × R satisfying the 37 estimate 37 38 38 39 · · − · · = → + 39 sup F(t, , ) E(t, , ) L2(T3×R3) O(ε) as ε 0 40 0tT 40 41 41 42 for each T

104 F. G o l s e

1 First, the solution Fε is sought as a truncated Hilbert expansion, plus a remainder term 1 2 2 3 6 3 k 3 4 Fε(t,x,v)= ε Fk(t,x,v)+ ε Rε(t,x,v), (9.4) 4 5 k=0 5 6 6 such that the last term in the truncated expansion satisﬁes 7 7 8 1 8 9 9 v F6 dv = 0. (9.5) 10 R3 |v|2 10 11 11

12 We recall from our discussion in Section 5.1 that the projection of a term Fk in Hilbert’s 12 13 expansion on the space of collision invariants is determined by postulating the existence of 13 14 the next term in that expansion, i.e., Fk+1. In the truncated expansion above, we obviously 14 15 do not postulate the existence of a F7, so that there is a certain amount of arbitrariness in 15 16 the choice of F6, which is resolved by condition (9.5). The other terms Fk, k = 0,...,5, 16 17 are computed as in Section 5.1. 17 18 Next we write an equation for the remainder Rε: inserting the right-hand side of (9.4) in 18 19 the scaled Boltzmann equation (9.1) we arrive at 19 20 20 21 6 21 2 k−1 2 22 (∂ + v ·∇ )R = B(F ,R ) + 2B ε F ,R + ε B(R ,R ) 22 t x ε ε 0 ε k ε ε ε 23 k=1 23 24 k+l−4 3 24 + ε B(Fk,Fl) − ε (∂t + v ·∇x)F6. (9.6) 25 25 k+l7 26 26 27 27 Indeed, for k = 0,...,5, the terms Fk are chosen as explained in Section 5.1, so that 28 28 29 5 29 k l+m−1 30 (∂t + v ·∇x) ε Fk = ε B(Fl,Fm). 30 31 k=0 l+m6 31 32 32 33 Notice that, in (9.6), the nonlinear term is multiplied by an ε2 factor: hence (9.6) is a 33 34 weakly nonlinear equation. However, the linear term in (9.6) is 34 35 35 36 2 36 B(F0,Rε) 37 ε 37 38 38 2 39 at leading order, which defines a nonpositive operator in a weighted L space in the 39 → B 2 R3; −1 40 v-variable. Specifically, the operator R (F0,R) is self-adjoint in L ( F0 dv). 40 − 41 1 41 A slight difficulty in this setting is that the weight F0 depends on (t, x). 42 To avoid this, we use a slightly different definition of the linearized collision integral 42 43 than in Section 3.6. Define 43 44 44 −1/2 1/2 45 LM φ =−2M B M,M φ , 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 105

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1 where M is a Maxwellian density. As in Section 3.6, the operator LM is split into 1 2 2 3 LM φ(v)= a |v| φ(v)− KM φ(v), 3 4 4 5 where 5 6 6 7 7 aM |v| = (v − v∗) · ω M∗ dv∗ dω 8 R3×S2 8 9 9 10 and 10 11 √ √ 11 12 KM φ(v)= Mφ∗ − M∗φ − M φ∗ (v − v∗) · ω M∗ dv∗ dω. 12 R3×S2 13 13 14 14 The properties of LM are summarized in the following theorem. 15 15 16 2 3 ∗ 16 THEOREM 9.2. The operator KM is compact on L (R ), while aM (|v|) ∼ a |v| 17 17 as |v|→+∞. Hence the operator LM is an unbounded self-adjoint Fredholm operator 18 on L2(R3) with domain L2(R3; (1 +|v|) dv) and nullspace 18 19 19 √ √ √ √ √ 20 2 20 Ker LM = span M, Mv1, Mv2, Mv3, M|v| . 21 21 22 22 The advantage in using LM instead of the operator R → B(F0,R) is that the former 23 operator is self-adjoint on the unweighted space L2(R3). 23 24 24 Formulating (9.6) in terms of the operator LM leads to an equation of the form 25 25 26 1 26 (∂ + v ·∇ ) E−1/2R + L E−1/2R 27 t x ε ε E ε 27 28 28 6 29 − − − − 29 = ε2Q E 1/2R ,E 1/2R + 2E 1/2B εk 1F ,R 30 E ε ε k ε 30 k=1 31 31 32 k+l−4 −1/2 3 −1/2 32 + ε E B(Fk,Fl) − ε E (∂t + v ·∇x)F6 33 k+l7 33 34 34 1 35 + Rε(∂t + v ·∇x) ln E, (9.7) 35 36 2 36 37 where E is the local Maxwellian whose parameters solve (9.2) and the quadratic operator 37 38 38 QM is deﬁned by 39 39 √ √ 40 −1/2 40 QM (φ, φ) = M B Mφ, Mφ . 41 41 42 42 The last term on the right-hand side of (9.7) is the most annoying one, since it grows like 43 43 44 |∇ | 44 1 3 xθ 45 |v| Rε; 45 2 θ 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 106

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1 in particular, it cannot be controlled by the damping part of the linearized collision integral, 1 2 since aM (|v|) = O(|v|) as |v|→+∞. 2 3 To overcome this difficulty, R. Caflisch introduced a new Maxwellian state M defined 3 4 by 4 5 5 6 1 −|v|2/2θˆ 6 M(v) = M ˆ (v) = e , 7 (1,0,θ) (2πθ)ˆ 3/2 7 8 8 9 where 9 10 10 ˆ 11 θ = 2θL∞ . 11 12 12 13 Hence, there exists a constant C, that depends on ρL∞ , uL∞ and 1/θL∞ such that 13 14 14 15 E CM. 15 16 16

17 Next, one decomposes Rε as 17 18 √ √ 18 19 19 Rε = Erε + Mqε, 20 20 21 21 where the new unknowns rε and qε are governed by the coupled system 22 22 23 1 1 23 (∂ + v ·∇ )r =− L r + 1| | σ K q , 24 t x ε ε E ε v c ε M ε 24 25 25 −1/2 1/2 1 26 (∂ + v ·∇ )q =−M r (∂ + v ·∇ )E − (a − 1| | K )q 26 t x ε ε t x ε M v >c M ε 27 (9.8) 27 6 28 28 + −1/2B k−1 1/2 + 29 2M ε Fk, M (σ rε qε) 29 = 30 k 1 30 31 2 31 + 2QM(σ rε + qε,σrε + qε) + ε s, 32 32 33 where 33 34 34 35 E 35 σ = , 36 M 36 37 37 38 where c is a truncation parameter to be chosen below, and the source term s is 38 39 39 −1/2 k+l−6 −1/2 40 s = M ε B(Fk,Fl) − εM (∂t + v ·∇x)F6. 40 41 k+l7 41 42 42 43 This system is solved for the initial condition 43 44 44

45 rε|t=0 = qε|t=0 = 0 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 107

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1 by a fixed point argument in the norm 1 2 2 3 = +| | r · 3 φ r,s sup 1 ξ f(,ξ) H s (R3) 4 ξ∈R3 4 5 5 6 for s>3/2. We refer to [22] for the complete proof. 6 7 7 8 REMARKS. Several remarks are in order. 8 9 1. The construction in Caflisch’s theorem leads to a solution of the Boltzmann equation 9 10 that exists and approximates the Maxwellian built on the solution of the compressible 10 11 Euler system for as long as the solution of that system exists and remains smooth. 11 12 This is very satisfying: should there be a blow-up in finite time in the solution of the 12 13 Boltzmann equation, it cannot happen before the onset of singularities in the Euler 13 14 system. 14 15 2. There is however a rather unpleasant feature in Caflisch’s construction: the solution 15 16 of the Boltzmann equation so constructed is, in general, not everywhere nonnegative, 16 17 and therefore loses physical meaning. This is most easily seen on the initial data for 17 18 18 Fε in the form (9.4): in Caflisch’s paper, Rε|t=0 = 0, so that, in the particular case 19 19 of Maxwell’s molecules, Fε|t=0 is a polynomial in v that is not (at least in general) 20 everywhere nonnegative. It could be that an improvement of Caflisch’s ansatz with 20 21 initial layers as in [73] helps avoiding this; however, there is no mention of this diffi- 21 22 culty in either [22]or[73]. 22 23 3. The interested reader is invited to compare Caflisch’s theorem with an earlier re- 23 24 sult by Nishida [103], who obtained the compressible Euler limit of the Boltzmann 24 25 25 equation by an abstract Cauchy–Kowalewski argument (in the style of Nirenberg and 26 26 Ovsyannikov, see, for instance, [102]). Nishida’s result is as follows: consider the 27 27 scaled Boltzmann equation (9.1) with an initial data F in analytic in x with enough 28 28 decay in v that is a perturbation of some absolute Maxwellian. Then, there exists 29 29 a family of solutions of the scaled Boltzmann parametrized by ε>0 that lives on 30 30 some interval of time independent of ε>0. This family of solutions converges to 31 31 the Maxwellian built on an analytic solution of the compressible Euler system in the 32 32 vanishing ε limit. However, the lifespan of Nishida’s family of solutions of the scaled 33 33 Boltzmann equation is not known to coincide with the blow-up time of the limiting 34 34 smooth solution of the compressible Euler system. See also [118] for an improved 35 35 variant of Nishida’s result. 36 36 37 37 38 38 39 10. The relative entropy method: Application to the incompressible Euler limit 39 40 40 41 As explained in Section 9, all methods based on asymptotic expansions in the Knudsen 41 42 number apply only to situations where the solutions of both the Boltzmann equation and the 42 43 limiting hydrodynamic equation are smooth. The present section introduces a new method 43 44 for cases where only the solution of the target equation (i.e., the hydrodynamic model) is 44 45 smooth. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 108

108 F. G o l s e

1 We shall explain how this method applies to the incompressible Euler limit of the Boltz- 1 2 mann equation. Consider the Boltzmann equation in the incompressible Euler scaling 2 3 3 1 4 + ·∇ = B ∈ R∗ × T3 × R3 4 ε∂t Fε v xFε q (Fε,Fε), (t, x, v) + , 5 ε 5 (10.1) 6 in 6 Fε|t=0 = M 1,εu , 1 , 7 7 8 where q>1 and uin ≡ uin(x) is a divergence-free vector ﬁeld on T3. Here, the collision 8 9 kernel b is supposed to satisfy assumption (3.55) (i.e., to come from a hard cut-off poten- 9 10 tial) as well as the additional condition 10 11 11 b(z,ω) 12 inf > 0. (10.2) 12 13 (z,ω)∈R3×S2 |(z/|z|) · ω| 13 14 14 15 Throughout this section, we denote by M the centered reduced Gaussian distribution 15 16 in v, i.e., 16 17 17 1 −|v|2/2 18 M(v)= M( , , )(v) = e . 18 1 0 1 (2π)3/2 19 19 20 Next, we introduce a new concept of limit that is especially well adapted to all incom- 20 21 pressible hydrodynamic limits of the Boltzmann equation. 21 22 22

23 DEFINITION 10.1 (Bardos, Golse and Levermore [10]). A family gε ≡ gε(x, v) of 23 24 1 T3 × R3; 24 Lloc( M dx dv) is said to converge entropically to g at rate ε if 25 3 3 25 • 1 + εgε 0a.e.onT × R for each ε>0, 26 • → 1 T3 × R3; 26 gε g weakly in Lloc( M dx dv), 27 • and 27 28 28 1 1 29 H M( + εg )M → g(x,v)2 x v 29 2 1 ε d d 30 ε 2 T3×R3 30 31 31 as ε → 0. 32 32 33 33 After these preliminaries, we can state the incompressible Euler limit theorem. 34 34 35 35 THEOREM 10.2 (Saint-Raymond [110]). Assume that uin ∈ H 3(T3) is a divergence-free 36 vector ﬁeld, and let u be the maximal solution of the incompressible Euler equations 36 37 37 38 38 ∂t u + u ·∇xu +∇xp = 0, divx u = 0, 39 39 | = in 40 u t=0 u 40

41 3 41 on [0,T)× T . For ε>0, let Fε be a renormalized solution relative to M of the scaled 42 42 Boltzmann equation (10.1). Then, for each t ∈[0,T), 43 43 44 F (t,x,v)− M(v) 44 ε → u(t, x) · v 45 εM(v) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 109

The Boltzmann equation and its hydrodynamic limits 109

1 entropically at rate ε as ε → 0. 1 2 2 3 The relative entropy method was used for the ﬁrst time to derive incompressible hy- 3 4 drodynamic models from the Boltzmann equation in Chapter 2 of [18], [54] and in [88] 4 5 – however, the results obtained in these references were incomplete since the proofs used 5 6 additional controls not known to be satisﬁed by renormalized solutions of the Boltzmann 6 7 equation. 7 8 Of course, if u is a 2D–3C solution of the incompressible Euler equations (see Sec- 8 9 tion 7.5), then T =+∞and the incompressible Euler limit is global. 9 10 Let us now describe the main ideas in the relative entropy method. 10 11 First, assuming that Fε is a classical solution to the scaled Boltzmann equation (10.1), 11 12 we compute 12 13 13 14 14 d 1 |M =−1 : − ⊗2 15 H(Fε (1,εu,1)) D(u) (v εu) Fε dv dx 15 dt ε2 ε2 T3 R3 16 16 1 17 + ∇xp · (v − εu)Fε dv dx. 17 18 ε T3 R3 18 19 19 Now, this identity is not known to be true if F is a renormalized solution. What is known 20 ε 20 instead is the following variant of it (see Theorem 8.10); for each t ∈[0,T), one has 21 21 22 22 1 1 23 H(Fε|M(1,εu,1))(t) + trace mε(t) 23 ε2 ε T3 24 24 t 25 1 ⊗ 25 − D(u) : (v − εu) 2F dv dx ds (10.3) 26 2 ε 26 ε 0 T3 R3 27 27 1 t 1 t 28 28 + ∇xp · (v − εu)Fε dv dx ds − D(u) : mε(s) ds. 29 ε 0 T3 R3 ε 0 T3 29 30 30 31 The key argument is the following lemma. 31

32 32 ∈[ 33 LEMMA 10.3. For each T 0,T)there exists CT 1 such that 33 34 34 1 t 35 D(u) : (v − εu)⊗2F v x s 35 2 ε d d d 36 ε 0 T3×R3 36 37 t 37 CT D(u) ∞ H(F |M )(s) s + ( ) 38 2 L ε (1,εu,1) d o 1 38 ε 0 39 39 40 uniformly in t ∈[0,T] as ε → 0. 40 41 41 42 Deﬁne 42 43 43 44 44 = 1 |M + 1 ; 45 Xε(t) H(Fε (1,εu,1))(t) trace mε(t) 45 ε2 ε T3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 110

110 F. G o l s e

1 it follows from (10.3) and Lemma 10.3 that, for each T ∈[0,T)and each t ∈[0,T ], one 1 2 has 2 3 3 t 4 4 Xε(t) CT D(u) L∞([0,T ]×T3) Xε(s) ds 5 0 5 6 1 t 6 7 + ∇xp · (v − εu)Fε dv dx ds + o(1)L∞([0,T ]). 7 ε T3 R3 8 0 8 9 Hence 9 10 10 t 11 1 11 CT D(u) ∞ [ ]×T3 Xε(t) e L ( 0,T ) ∇xp · (v − εu)Fε dv dx ds 12 ε 0 T3 R3 12 13 13 + o(1)L∞([0,T ]). (10.4) 14 14 15 With this inequality, one concludes as follows. The sequence of ﬂuctuations 15 16 16 17 17 1 ∗ ∞ 1 2 (F − M) is relatively compact in w − L R+; w − L 1 +|v| dx dv 18 ε ε 18 19 19 20 so that, modulo extraction of a subsequence 20 21 21 22 ∞ 1 3 22 Fε dv → 1inL R+; L T 23 R3 23 24 24 and 25 25 26 26 1 → ∗ − ∞ R ; − 1 T3 27 vFε dv U in w L + w L . 27 ε R3 28 28 29 Because of the local conservation of mass that is satisﬁed by the renormalized solution Fε, 29 30 one has 30 31 31 32 divx U = 0. 32 33 33 34 Hence 34 35 35 1 t t 36 ∇xp · (v − εu)Fε dv dx ds → ∇xp · (U − u) dx ds = 0 36 37 ε 0 T3 R3 0 T3 37 38 so that 38 39 39 40 40 Xε(t) → 0asε → 0 for each t ∈ 0,T . 41 41 42 By convexity and weak limit, one has 42 43 43 44 1 1 44 (U − u)(t)2 lim H(F |M )(t) 45 L2(T3) 2 ε (1,εu,1) 45 2 ε→0 ε dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 111

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1 for each t ∈[0,T ]. Therefore, passing to the limit in (10.4) shows that U = u, as an- 1 2 nounced. 2 3 We shall not describe the proof of Lemma 10.3, which is really technical. In this proof, 3 4 one has to master the difficulties created by the high velocity tails of the number density; 4 5 this is done by using decay estimates due to Grad and Caflisch [23] on the gain term of the 5 6 linearized collision operator, we refer to [110] for a complete proof. 6 7 But the difficulties in the proof of Lemma 10.3 are special to the Boltzmann equation; 7 8 the main line of the relative entropy method is as described above. This method is due to 8 9 Yau [122] who used it for the hydrodynamic limit of Ginzburg–Landau models. 9 10 10 11 11 12 11. Applications of the moment method 12 13 13 14 The moment method is based on compactness results which are used to pass to the limit 14 15 in the local conservation laws of mass momentum and energy as the Knudsen number 15 16 vanishes. This method does not require estimates other than the natural bounds on mass, 16 17 energy, entropy and entropy production. On principle, it could therefore be used when both 17 18 the solutions of the scaled Boltzmann equation and of the limiting hydrodynamic equations 18 19 are not known to be regular. 19 20 First, we state the various theorems on hydrodynamic limits that can be proved in this 20 21 way. 21 22 22 23 23 24 11.1. The acoustic limit 24 25 25 26 We start from the Boltzmann equation in the acoustic scaling posed in the periodic box 26 27 27 1 ∗ 3 3 28 ∂ F + v ·∇ F = B(F ,F ), (t, x, v) ∈ R+ × T × R , 28 t ε x ε ε ε ε 29 (11.1) 29 | = in 30 Fε t=0 Fε . 30 31 31 32 Assume that b satisfies the weak cut-off assumption (8.1) as well as the bound 32 33 33 2 β 3 34 0 < b(z,ω)dω Cb 1 +|z| a.e., z ∈ R , (11.2) 34 35 S2 35 36 36 for some β ∈[0, 1]. 37 37 We assume that 38 38 39 39 in = 40 Fε dx dv 1, 40 T3×R3 41 41 42 in = 42 vFε dx dv 0, (11.3) T3×R3 43 43 44 44 1| |2 in = 3 45 v Fε dx dv . 45 T3×R3 2 2 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 112

112 F. G o l s e

1 Set M to be the centered, reduced Gaussian distribution 1 2 2 3 1 −| |2 3 M(v)= e v /2, (11.4) 4 (2π)3/2 4 5 5 6 so that 6 7 7 8 8 M dx dv = 1, 9 R3 9 10 10 11 vM dx dv = 0, 11 R3 12 12 1 3 13 |v|2M dx dv = . 13 14 R3 2 2 14 15 15 16 The acoustic limit of the Boltzmann equation (11.1) is given by the following theorem. 16 17 17

18 THEOREM 11.1 (Golse and Levermore [53]). Let δε > 0 be such that 18 19 19 20 β/2 1/2 20 δε → 0 and δε| ln δε| = o ε 21 21 22 as ε → 0. Assume that 22 23 23 24 in − 24 Fε (x, v) M in in in 1 2 25 → ρ (x) + u (x) · v + θ (x) |v| − 3 25 δ M 2 26 ε 26 27 27 entropically at rate δε. For each ε>0, let Fε be a renormalized solution relative to M of 28 28 the scaled Boltzmann equation (11.1) with initial data F in. 29 ε 29 Then, for each t 0, the family 30 30 31 − 31 Fε(t,x,v) M 1 2 32 → ρ(t,x)+ u(t, x) · v + θ(t,x) |v| − 3 32 33 δεM 2 33 34 34 entropically at rate δ , where (ρ,u,θ)is a solution of the acoustic system 35 ε 35 36 36 + = 37 ∂t ρ divx u 0, 37 38 ∗ 3 38 ∂t u +∇x(ρ + θ)= 0,(t,x)∈ R+ × T , 39 39 40 3 40 ∂t θ + divx u = 0, 41 2 41 42 42 43 with initial data 43 44 44 | = in in in 45 (ρ,u,θ) t=0 ρ ,u ,θ . 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 113

The Boltzmann equation and its hydrodynamic limits 113

1 Notice that our assumptions on the family of initial data imply that 1 2 2 3 ρin,uin,θin ∈ L2 T3 , 3 4 4 5 and that 5 6 6 7 7 ρin(x) dx = 0, uin(x) dx = 0 and θ in(x) dx = 0. 8 T3 T3 T3 8 9 9 10 In particular, the existence and uniqueness theory for the acoustic system described in 10 11 Section 7.1.2 applies here, and the solution (ρ,u,θ)satisﬁes 11 12 12 2 3 13 (ρ,u,θ)∈ C R+; L T 13 14 14 15 and 15 16 16 17 17 ρin(x) dx = 0, uin(x) dx = 0 and θ in(x) dx = 0. 18 T3 T3 T3 18 19 19 20 20 21 11.2. The Stokes–Fourier limit 21 22 22 23 We start from the Boltzmann equation in the Stokes scaling posed in the periodic box 23 24 24

25 1 ∗ 3 3 25 ε∂ F + v ·∇ F = B(F ,F ), (t, x, v) ∈ R+ × T × R , 26 t ε x ε ε ε ε 26 (11.5) 27 in 27 F | = = F . 28 ε t 0 ε 28 29 29 30 Assume that the collision kernel b comes from a hard cut-off potential, i.e., that it sat- 30 ∈[ ] in 31 isfies (5.2) for some α 0, 1 . Assume further that, for each ε>0, the initial data Fε 31 32 satisfies the relations (11.3). 32 33 33 34 THEOREM 11.2 (Golse and Levermore [53]). Let δε > 0 be such that 34 35 35 α 36 δε → 0 and δε| ln δε| = o(ε) 36 37 37 38 as ε → 0. Assume that 38 39 39 F in(x, v) − M 1 40 ε → uin(x) · v + θ in(x) |v|2 − 5 40 41 δεM 2 41 42 42 in 43 entropically at rate δε, where u satisfies 43 44 44 in 45 divx u = 0 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 114

114 F. G o l s e

1 and where M is the centered, reduced Gaussian distribution (11.4). For each ε>0, let 1 2 Fε be a renormalized solution relative to M of the scaled Boltzmann equation (11.5) with 2 3 in 3 initial data Fε . 4 Then, for each t 0, the family 4 5 5 6 − 6 Fε(t,x,v) M → · + 1 | |2 − 7 u(t, x) v θ(t,x) v 5 7 δεM 2 8 8

9 entropically at rate δε, where (u, θ) is a solution of the Stokes–Fourier system 9 10 10 11 ∗ 3 11 ∂t u +∇xp = ν xu, divx u = 0,(t,x)∈ R+ × T , 12 12 13 ∂t θ = κ xθ, 13 14 14 15 with initial data 15 16 16 in in 17 (u, θ)|t=0 = u ,θ . 17 18 18 19 The viscosity and heat conductivity are given by 19 20 20 21 21 1 2 22 ν = A : AM dv, κ = B · BM dv, 22 10 R3 15 R3 23 23 24 see also formula (6.9) for expressions of these quantities in terms of the functions a and b 24 25 deﬁned in (6.12). 25 26 26 27 27 We recall that A and B are deﬁned in terms of 28 28 29 29 1 2 1 2 30 A(v) = v ⊗ v − |v| I, B = |v| − 5 v 30 3 2 31 31 32 by 32 33 33 34 34 LM A = A, A ⊥ Ker LM , 35 35 36 LM B = B, B ⊥ Ker LM . 36 37 37 38 Our assumptions on the family of initial data imply that 38 39 39 40 uin,θin ∈ L2 T3 , 40 41 41 42 while 42 43 43 44 44 in = in = 45 u (x) dx 0 and θ (x) dx 0. 45 T3 T3 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 115

The Boltzmann equation and its hydrodynamic limits 115

1 Hence, the existence and uniqueness theory for the Stokes–Fourier system described in 1 2 Section 7.1.1 applies here, and the solution (u, θ) satisfies 2 3 3 2 3 ∞ ∗ 3 4 (u, θ) ∈ C R+; L T ∩ C R+ × R 4 5 5 6 and 6 7 7 8 uin(x) dx = 0 and θ in(x) dx = 0. 8 9 T3 T3 9 10 10 11 11 12 11.3. The Navier–Stokes–Fourier limit 12 13 13 14 We start from the Boltzmann equation in the Navier–Stokes scaling, posed in the Euclidean 14 15 space, with Maxwellian equilibrium at infinity 15 16 16 17 1 ∗ 3 3 17 ε∂t Fε + v ·∇xFε = B(Fε,Fε), (t, x, v) ∈ R+ × R × R , 18 ε 18 19 Fε(t,x,v)→ M |x|→+∞, (11.6) 19 20 in 20 F | = = F , 21 ε t 0 ε 21 22 22 M 23 where is the centered, reduced Gaussian distribution (11.4). 23 Assume that the collision kernel b comes from a hard cut-off potential, i.e., that it satis- 24 ∈[ ] 24 25 fies (5.2) for some α 0, 1 . 25 26 26 27 THEOREM 11.3 (Golse and Saint-Raymond [61,62]). Assume that 27 28 28 F in(x, v) − M 1 29 ε → uin(x) · v + θ in(x) |v|2 − 5 29 30 εM 2 30 31 31 in 32 entropically at rate ε, where u satisfies 32 33 33 in = 34 divx u 0. 34 35 35 36 For each ε>0, let Fε be a renormalized solution relative to M of the scaled Boltzmann 36 in 37 equation (11.5) with initial data Fε . 37 38 Then the family 38 39 39 40 1 1 1 2 40 vFε(t,x,v)dv, |v| − 1 Fε(t,x,v)− M dv 41 ε R3 ε R3 3 41 42 42 1 R × R3 → 43 is weakly relatively compact in Lloc( + ), and each of its limit points as ε 0 is 43 44 44 2 3 2 1 3 45 (u, θ) ∈ C R+,w− L R ∩ L R+; H R , 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 116

116 F. G o l s e

1 a solution of the Navier–Stokes–Fourier system 1 2 2 ∗ 3 3 ∂t u + divx(u ⊗ u) +∇xp = ν xu, divx u = 0,(t,x)∈ R+ × T , 3 4 4 ∂t θ + divx(uθ) = κ xθ, 5 5 6 with initial data 6 7 7 8 in in 8 (u, θ)|t=0 = u ,θ . 9 9 10 The viscosity and heat conductivity are given by 10 11 11 12 12 = 1 : = 2 · 13 ν A AM dv, κ B BM dv 13 10 R3 15 R3 14 14 15 again, see (6.9) for expressions of these quantities in terms of the functions a and b defined 15 16 in (6.12). Moreover, this solution (u, θ) satisfies, for each t>0, the inequality 16 17 17 t 18 1 2 + 5 2 + |∇ |2 + |∇ |2 18 u(t) L2 θ(t) L2 ν xu κ xθ dx ds 19 2 4 0 R3 19 20 20 1 in2 5 in2 21 u + θ . 21 2 L2 4 L2 22 22 23 In particular, if θ in = 0, this theorem shows that any weak limit point of 23 24 24 25 1 25 26 vFε(t,x,v)dv 26 ε R3 27 27 28 1 R × R3 → 28 in Lloc( + ) as ε 0 is a Leray solution of the Navier–Stokes equations with initial 29 data uin. 29 30 This theorem explains the following observation by Lions: “[...] the global existence of 30 31 [renormalized] solutions [...] can be seen as the analogue for Boltzmann’s equation to the 31 32 pioneering work on the Navier–Stokes equations by J. Leray” (see [84], p. 432). 32 33 33 34 34 35 11.4. Sketch of the proof of the Navier–Stokes–Fourier limit by the moment method 35 36 36 37 The proof of the Navier–Stokes–Fourier limit theorem above involves many ideas devel- 37 38 oped in a sequence of papers over the past 15 years: 38 39 • the BGL program was defined in [10]; this reference provided the general entropy and 39 40 entropy production estimates used to control the number density fluctuation and its 40 41 distance to local equilibrium; as a result, the evolution Stokes and the steady Navier– 41 42 Stokes motion equations were derived under the assumption that the renormalized 42 43 solutions to the Boltzmann equation considered satisfy the local conservation of mo- 43 44 mentum as well as a nonlinear compactness estimate (for the Navier–Stokes limit) 44 45 that will be described below in more details; 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 117

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1 • under the same assumptions as in [10], Lions and Masmoudi [87] were able to de- 1 2 rive the evolution Navier–Stokes motion equations, by a kind of “compensated com- 2 3 pactness” argument bearing on fast oscillating acoustic waves; they also introduced 3 4 a slightly modified notion of renormalized solution which led them to a complete 4 5 derivation of the evolution Stokes motion equation; 5 6 • in [11], it was observed for the first time that the local conservation law of momen- 6 7 tum could be proved in the hydrodynamic limit, thereby relieving the need for as- 7 8 suming that local conservation law at the level of the renormalized solutions of the 8 9 Boltzmann equation; this led to a complete proof of the acoustic limit for bounded 9 10 collision kernels; a more complete understanding of how the local conservation laws 10 11 of both momentum and energy could be proved in the hydrodynamic limit was even- 11 12 tually reached in [53]; the latter reference provided an essentially optimal derivation 12 13 of the Stokes motion and energy equations as well as a derivation of the acoustic sys- 13 14 tem that allowed for the most general hard cut-off potentials; however, the acoustic 14 15 system was established only under some unphysical restriction on the scaling of the 15 16 number density fluctuation; 16 17 • in [107,109], Saint-Raymond gave a complete derivation of both the Navier–Stokes 17 18 motion and energy equations for the BGK model with constant relaxation time; her 18 19 proof was based on obtaining for the first time some weaker analogue of the nonlinear 19 20 compactness assumption used in [10]; 20 21 • finally, a complete derivation of the incompressible Navier–Stokes motion and heat 21 22 equations from the Boltzmann equation was proposed for the first time in [61]for 22 23 bounded collision kernels (such as occurring in the case of cut-off Maxwell mole- 23 24 cules); this reference used all the methods constructed in the previous works men- 24 25 tioned above, together with a new velocity averaging method specific to the L1 case 25 26 and that amplified Saint-Raymond’s observation in [107]; this result was later ex- 26 27 tended to all hard cut-off potentials (including hard spheres) in [62]. 27 28 It is this last reference that we describe below; although its scope is more general than 28 29 that of [61], it involves a new idea for handling unbounded collision kernels that actually 29 30 simplifies the discussion in [61]. 30 31 31 32 32 11.4.1. A priori estimates. In this subsection, we quickly list the a priori estimates on 33 33 the family F of solutions of the Boltzmann equation that are uniform in ε>0. As can be 34 ε 34 seen from Theorem 8.10, the only such estimate comes from the DiPerna–Lions variant of 35 35 Boltzmann’s H -theorem, 36 36 37 t 37 1 38 H F (t) M + F F − F F ∗ 38 ε 2 ε ε∗ ε ε 39 4ε 0 R3 R3×R3×S2 39 40 40 × FεFε∗ 41 ln b dv dv∗ dω dx ds 41 FεFε∗ 42 42 in 43 H Fε M . (11.7) 43 44 44 45 We shall further transform (11.7) with the two following inequalities. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 118

118 F. G o l s e

∗ 1 Pointwise inequalities. For each ξ and η ∈ R+, 1 2 2 3 1 3 ξ − 1 2 (ξ ln ξ − ξ + 1) (11.8) 4 4 4 5 5 6 while 6 7 7 √ 1 8 ξ − η 2 (ξ − η)(ln ξ − ln η). (11.9) 8 9 4 9 10 10 − 11 Since (Fε M)/(εM) converges entropically at rate ε, one has 11 12 12 in in 2 13 H Fε M C ε . (11.10) 13 14 14 15 This bound and the relative entropy inequality (11.7) entail the following entropy bound 15 16 for each t>0 16 17 17 in 2 18 H Fε(t) M C ε (11.11) 18 19 19 20 and the following entropy production bounds 20 21 21 22 +∞ 22 − FεFε∗ 23 FεFε∗ FεFε∗ ln b dv dv∗ dω dx dt 23 0 R3 R3×R3×S2 FεFε∗ 24 24 in 4 25 4C ε . (11.12) 25 26 26 27 Introducing the relative number density and relative number density ﬂuctuations, 27 28 28 29 F F − M 29 G = ε and g = ε , (11.13) 30 ε M ε εM 30 31 31 32 the two pointwise inequalities (11.8) and (11.9) convert the entropy and entropy production 32 33 bounds into the two uniform a priori estimates 33 34 34 35 2 1 in 2 35 Gε(t) − 1 dx C ε (11.14) 36 R3 4 36 37 37 38 and 38 39 39 +∞ 40 − 2 in 4 40 GεGε∗ GεGε∗ dx dt C ε . (11.15) 41 0 R3 41 42 42 43 The importance of the two a priori bounds above (11.14) and (11.15) in the derivation of the 43 44 Navier–Stokes limit cannot be overestimated. In fact, various analogues of these bounds 44 45 were used earlier in the context of nonlinear diffusion limits, see [12] and especially [59]. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 119

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1 The importance of the entropy and entropy dissipation bounds for hydrodynamic limits 1 2 of diffusive type was noticed. We recall that 2 3 3 4 4 φ= φ(v)M(v)dv, 5 R3 5 6 6 7 and we further introduce the notation 7 8 8 9 9 Φ = Φ(v,v∗,ω)dµ(v, v∗,ω), 10 R3×R3×S2 10 11 11 12 where 12 13 13 14 dµ(v, v∗,ω)= b(v − v∗, ω)M(v) dvM(v∗) dv∗ dω. 14 15 15 16 In the sequel, we outline the main ideas in the derivation of the Navier–Stokes motion 16 17 equation in Theorem 11.3, since the derivation of the heat equation is essentially analogous. 17 18 Of course, this proof more or less follows the formal argument presented in Section 6.1. 18 19 However, several key properties of the solutions to the scaled Boltzmann equation used in 19 20 this formal argument – such as, for instance, the local conservation laws of momentum and 20 21 energy – are not known to be satisfied by renormalized solutions. Hence the proof sketched 21 22 below differs noticeably from the formal argument in several places, yet the general idea 22 23 remains essentially the same. 23 24 24 25 25 11.4.2. Normalizing functions. As explained in Section 8.2, the Boltzmann equation can 26 26 be equivalently renormalized with any admissible nonlinearity whose derivative saturates 27 27 the quadratic growth of the collision integral. 28 28 Throughout the proof of the Navier–Stokes limit theorem, we shall essentially use two 29 29 kinds of normalizing nonlinearities 30 30 • compactly supported nonlinearities that coincide with the identity near the reference 31 31 Maxwellian state, and 32 32 • variants of the maximal, i.e., square-root renormalization. 33 33 34 Nonlinearities of the first kind are used to define the renormalized form of the Boltzmann 34 35 equation in which one passes to the vanishing ε limit, while the square-root normaliza- 35 36 tion is used to establish compactness properties of the family of solutions to the scaled 36 37 Boltzmann equation. 37 38 The first kind of normalizing nonlinearities is defined through the class of bump func- 38 ∈ ∞ R 39 tions γ C ( +) such that 39 40 40 41 γ |[0,3/2] ≡ 1,γ|[2,+∞) ≡ 0,γis nonincreasing on R+. (11.16) 41 42 42 43 The Boltzmann equation is then renormalized with the nonlinearity 43 44 44 45 Γ(Z)= (Z − 1)γ (Z); 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 120

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1 later on, we denote 1 2 2 3 d 3 γ(Z)ˆ = (Z − 1)γ (Z) = Γ (Z). (11.17) 4 dZ 4 5 5 6 The scaled Boltzmann equation renormalized with Γ is put in the form 6 7 7 1 1 8 ∂ (g γ ) + v ·∇ (g γ ) = γˆ Q (G ,G ), 8 t ε ε x ε ε 3 ε M ε ε (11.18) 9 ε ε 9 10 where we have denoted 10 11 11 12 12 γε = γ(Gε), γˆε =ˆγ(Gε), 13 13 14 and where Q designates the Boltzmann collision integral intertwined with the multiplica- 14 15 tion by M (see (6.6)), 15 16 16 17 Q(G, G) = M−1B(MG, MG). 17 18 18 19 Later on, we shall pass to the limit in the momentum equation deduced from (11.18). 19 20 The second class of normalizing nonlinearities that we shall use to establish compactness 20 21 21 properties of the number density fluctuations Gε is defined as 22 22 23 23 Γζ (Z) = ζ + Z, ζ > 0, 24 24 25 where the parameter ζ will later be adapted to ε. 25 26 26 27 27 11.4.3. Governing equations for moments of gε. As explained in Theorem 8.10, renor- 28 malized solutions to the Boltzmann equation satisfy the local conservation of mass (i.e., the 28 29 29 continuity equation); in terms of the number density fluctuation gε, this local conservation 30 law is expressed as 30 31 31

32 ε∂t gε+divxvgε=0. (11.19) 32 33 33 34 Now, the entropy bound (11.11) implies that 34 35 35 36 +| |2 − 1 ; 1 36 1 v gε is relatively compact in w Lloc dt dx L (M dv) . (11.20) 37 37 38 Before saying a few words on (11.20), let us explain how we use it. Modulo extraction of 38 39 a subsequence, one has 39 40 40 41 → − 1 ; 1 +| |2 41 gε g in w Lloc dt dx L 1 v M dv 42 42 43 and hence 43 44 44 45 → → − 1 45 gε g and vgε vg in w Lloc(dt dx). dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 121

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1 Passing to the limit as ε → 0 in (11.19) leads to 1 2 2

3 divxvg=0, 3 4 4 5 so that, denoting 5 6 6 7 u =vg 7 8 8 9 the relation above is the incompressibility condition in the Navier–Stokes equations, i.e., 9 10 10 11 11 divx u = 0. 12 12 13 Let us go back to (11.20) and explain how it follows from the entropy bound (11.11), 13 14 see [10] for more details on this. Deﬁne 14 15 15 16 h(z) = (1 + z) ln(1 + z) − z, z > −1. 16 17 17 18 In terms of h, the entropy bound (11.11) is expressed as 18 19 19 20 20 1 in 21 h εgε(t) dx C . 21 ε2 R3 22 22 23 If the entropy bound (11.11) was equivalent to an Lp(M dv dx) bound for some p>1, 23 24 2 ∞ r 24 Hölder’s inequality would imply that (1 +|v|) gε is bounded in L (dt; L (M dv dx)) for 25 some r>1, since (1 +|v|2) ∈ Lq (M dv) for each q ∈ (1, +∞). However, the entropy 25 26 p 26 control (11.11) on gε is weaker than an L (M dv dx) bound. But (11.20) follows from a 27 careful use of Young’s inequality 27 28 28

29 α 1 ∗ 29 p|z| h(εz) + h (p), p > 0,z>−1, 0 <ε<α, 30 ε2 α 30 31 31 32 where 32 33 33 ∗ p 34 h (p) = e − p − 1 34 35 35 36 designates the Legendre dual of h. The compactness (11.20) follows from replacing z 36 37 1 +| |2 → 37 with gε and p with 4 (1 v ) in the inequality above, letting then α 0 in the inequality 38 so obtained. 38 39 Let us now explain how the motion equation in the Navier–Stokes system is derived 39 40 from the Boltzmann equation. This is of course the main part in the proof, and it involves 40 41 several technicalities. 41 42 In particular, we shall need truncations in the velocity variable at a level that is tied to ε. 42 43 For each function ξ ≡ ξ(v) and each K>6, we deﬁne 43 44 44 45 = 45 ξKε (v) ξ(v)1|v|2K| ln ε|. (11.21) dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 122

122 F. G o l s e

1 Multiplying each side of the scaled, renormalized Boltzmann equation (11.18) by each 1 2 2 component of vKε and averaging in v leads to 3 3 4 1 1 2 4 ∂t vgεγε+divx Fε(A) +∇x |v| gεγε = Dε(v), (11.22) 5 ε 3 Kε 5 6 6 7 where Fε(A) is the truncated, renormalized traceless part of the momentum flux 7 8 8 1 9 = 9 Fε(A) AKε gεγε (11.23) 10 ε 10 11 11 while Dε(v) is the momentum conservation defect 12 12 13 13 1 D (v) = v γˆ G G ∗ − G G ∗ . (11.24) 14 ε ε3 Kε ε ε ε ε ε 14 15 15 16 Notice that truncating large velocities in the number density, or large values thereof (which 16 17 is what the renormalization procedure does) break the symmetries in the collision integral 17 18 leading to the local conservation of momentum (see Proposition 3.1): this accounts for 18 → → → 19 the defect Dε(v) on the right-hand side of (11.22). As ε 0, vKε v while Gε 1so 19 20 that γˆε → 1; hence, the missing symmetries are restored in the integrand defining Dε(v). 20 21 Hence, one can hope that Dε(v) → 0asε → 0. 21 22 In fact, the strategy for establishing the Navier–Stokes limit theorem consists of the 22 23 following three steps. 23 24 24 25 Step 1. Prove that, modulo extraction of a subsequence 25 26 26 → = − 1 27 vgεγε vg u in w Lloc(dt dx), 27 28 28 29 while 29 30 3 30 P vg γ →u in C R+; D R , 31 ε ε 31 32 32 where P denotes the Leray projection, i.e., the orthogonal projection on divergence-free 33 33 vector fields in L2(R3). 34 34 35 35 Step 2. Likewise, prove that 36 36 37 → 1 37 Dε(v) 0inLloc(dt dx). 38 38 39 Step 3. Finally prove that 39 40 40 41 ∗ 3 41 P divx Fε(A) → P divx(u ⊗ u) − ν xu in D R+ × R . 42 42 43 Once these three steps are completed, one applies P to both sides of (11.22), which gives 43 44 44 45 ∂t P vgεγε+P divx Fε(A) = P Dε(v). (11.25) 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 123

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1 Taking limits in each term as ε → 0 shows that u satisﬁes the Navier–Stokes motion equa- 1 2 tion. As for the initial condition, observe that it is guaranteed by the uniform convergence 2 3 in t, i.e., by the second statement in Step 1. 3 4 4 5 11.4.4. Vanishing of the momentum conservation defect. We start with Step 2, i.e., we 5 6 explain how to prove the following proposition. 6 7 7 8 8 PROPOSITION 11.4. Under the same assumptions as in Theorem 11.3, 9 9 10 1 10 Dε(v) → 0 in L (dt dx). 11 loc 11 12 12 First, we start from the elementary formula 13 13 14 14 − GεGε∗ GεGε∗ 15 15 16 = − + 16 GεGε∗ GεGε∗ GεGε∗ GεGε∗ 17 17 = − 2 + − 18 GεGε∗ GεGε∗ 2 GεGε∗ GεGε∗ GεGε∗ 18 19 19 20 and split the momentum conservation defect as 20 21 21 22 = 1 + 2 22 Dε(v) Dε(v) Dε(v) 23 23 24 with 24 25 25 26 26 1 1 2 D (v) = v γˆ G G ∗ − G G ∗ 27 ε ε3 Kε ε ε ε ε ε 27 28 28 29 and 29 30 30 31 2 = 2 ˆ − 31 Dε(v) vKε γε GεGε∗ GεGε∗ GεGε∗ . 32 ε3 32 33 33 1 → 1 34 That Dε(v) 0inLloc(dt dx) follows from the entropy production estimate (11.15). Set- 34 35 ting 35 36 36 37 = 1 − 37 Ξε GεGε∗ GεGε∗ GεGε∗ 38 ε2 38 39 39 2 40 we further split Dε(v) as 40 41 41 42 2 2 2 42 D (v) =− v1| |2 γˆεΞε + vγˆε 1 −ˆγε∗γˆ γˆ ∗ Ξε 43 ε ε v >Kε ε ε ε 43 44 44 1 45 + (v + v∗)γˆ γˆ ∗γˆ γˆ ∗Ξ . 45 ε ε ε ε ε ε dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 124

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1 The first term is easily mastered by the entropy production estimate (11.15) and the fol- 1 2 lowing classical estimate on the tail of Gaussian integrals 2 3 3 4 4 −|v|2/2| |a = (a+N)/2−1 −R/2 →+∞ 5 e v 1|v|2>R dv O R e as R . 5 RN 6 6 7 Observe that the integrand in the third term has the same symmetries as the original col- 7 8 lision integrand (before truncation in |v| and renormalization). It is also mastered by a 8 9 combination of the entropy production estimate (11.15) with the Gaussian tail estimate 9 10 above. 10 11 The most difficult part in the analysis of the momentum conservation defect is by far the 11 12 2 1 → 12 second term in the decomposition of Dε(v) above. That it vanishes in Lloc(dt dx) as ε 0 13 ultimately relies upon the following estimate. 13 14 14 15 15 Nonlinear compactness estimate. 16 16 17 √ 17 G − 1 2 18 1 +|v| α ε is uniformly integrable on [0,T]×K × R3 (11.26) 18 19 ε 19 20 20 21 for the measure dt dxM dv, for each T>0 and each compact K ⊂ R3, where α is the 21 22 relative velocity exponent that appears in the hard cut-off assumption (3.55) on the collision 22 23 kernel b. 23 24 2 → 1 24 We shall not give further details on the proof that Dε(v) 0inLloc(dt dx), which is 25 based on the above nonlinear compactness estimate together with the entropy production 25 26 bound (11.15). 26 27 Let us however say a few words on the nonlinear compactness estimate itself. The rela- 27 28 ∞ 2 28 tive entropy bound (11.11) is essentially as good as an L (dt; L (M dv dx)) bound on gε 29 29 on the set of (t,x,v)’s such that Gε(t,x,v)= O(1). Elsewhere, it essentially reduces to 30 an O(ε) bound in L∞(dt; L1(M dv dx)), which is quite not enough for the Navier–Stokes 30 31 limit. This is why the first works on this limit assumed some variant of this nonlinear 31 32 compactness estimate. For instance, in either [10]or[87], it was assumed that 32 33 33 34 2 34 2 gε 3 35 +|v| [ ,T]×K × R , 35 1 + is uniformly integrable on 0 (11.27) 36 1 Gε 36 37 37 whereas all that was known on this quantity was the estimate 38 38 39 39 g2 40 +| |2 ε = | | 1 ; 1 40 1 v O ln ε in Lloc dt dx L (M dv) 41 1 + Gε 41 42 42 43 (see [10]). This led to a decomposition of the number density fluctuation as 43 44 44 45 = + ! 45 gε gε εgε, dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 125

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1 where the “good” part of the fluctuation is 1 2 2 g 3 = ε = ∞ ; 2 3 gε O(1) in L dt L (M dv dx) , 4 0.5 + 0.5Gε 4 5 5 while the “bad” part is 6 6 7 2 7 ! gε ∞ 1 8 g = = O(1) in L dt; L (M dv dx) . 8 ε 1 + G 9 ε 9 10 In later works – for instance in [107,109] and [61] – this decomposition was slightly mod- 10 11 ∈ ∞ R∗ 11 ified as follows. Pick a bump function γ Cc ( +) such that 12 12 13 13 | ≡ ⊂ 1 3 14 γ [ 3 , 5 ] 1, supp(γ ) , and 0 γ 1, 14 4 4 2 2 15 15 16 and define 16 17 17 1 − γ(G ) 18 = ! = ε 18 gε gεγ(Gε), gε gε. 19 ε 19 20 It was proved in [61] that 20 21 21 22 22 g 2 is uniformly integrable on [0,T]×K × R3 (11.28) 23 ε 23 24 for the measure dt dxM dv, while 24 25 25 26 1 26 g! = O in L1 dt dx; L1(M dv) . (11.29) 27 ε ln | ln ε| loc 27 28 28 29 Observe the difference between these last two controls and (11.27): with the new defin- 29 2 30 ition of g and g!, it is no longer true that |g |2 Cg!, while ( gε )2 gε , so that 30 ε ε ε ε 1+Gε 1+Gε 31 (11.27) actually entailed that the square of the good part in the old flat-sharp decomposi- 31 32 tion is uniformly integrable, even with a quadratic weight in v. In fact, the techniques in 32 33 [61] did not allow adding a quadratic weight in v as in (11.27), so that this compactness 33 34 assumption remained unproved; fortunately, it was possible to complete the proof of the 34 35 Navier–Stokes limit for cut-off Maxwell molecules with only the bounds (11.28)–(11.29), 35 36 and the weighted estimate 36 37 37 38 2 38 s ε 1 1 1 +|v| 1 − γ(Gε) = O √ in L dt dx; L (M dv) . (11.30) 39 ln | ln ε| loc 39 40 40 41 This control shows that the set where the bad part of the number density fluctuation domi- 41 42 nates is small in weighted v-space. There is a definite lack of symmetry between the con- 42 43 trols (11.29), bearing on large values of gε, and (11.30), bearing on large |v|’s. This lack 43 44 of symmetry is remedied in the most recent variant (11.26) of these nonlinear compactness 44 45 estimates (see [62]), we shall return to this when sketching the proof of (11.26). 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 126

126 F. G o l s e

1 11.4.5. The asymptotic momentum flux. With the vanishing of conservation defects 1 2 (Step 2 in the proof of the Navier–Stokes limit) settled in the previous section, we turn 2 3 our attention to Step 3, i.e., passing to the limit in the divergence of the momentum flux 3 4 modulo gradients. This is by far the most difficult part of our analysis, and does require 4 5 several preparations. In the present subsection, we reduce the momentum flux to some as- 5 6 ymptotic normal form, to which we eventually apply compactness results to be described 6 7 later. 7 8 8 9 9 LEMMA 11.5. Let Π be the L2(M dv)-orthogonal projection on Ker L; then, under the 10 same assumptions as in Theorem 11.3, 10 11 11 √ 12 − 2 12 Gε 1 1 13 Fε(A) = A Π − 2 A QM Gε, Gε + o(1) 1 , 13 2 Lloc(dt dx) 14 ε ε 14 15 15 16 where we recall that the tensor field A is defined by 16 17 17 18 1 2 18 A ⊥ Ker LM and LM A = A = v ⊗ v − |v| I. 19 3 19 20 20 21 The proof of this lemma is based upon splitting the momentum flux as 21 22 22 − 23 = 1 Gε 1 23 Fε(A) AKε γε 24 ε ε 24 25 √ √ 25 G − 1 2 2 G − 1 26 = ε + ε 26 AKε γε AKε γε 27 ε ε ε 27 28 = 1 + 2 28 Fε(A) Fε(A), 29 29 30 30 as a consequence of the elementary identity 31 31 32 32 1 1 33 (Gε − 1) = Gε − 1 Gε + 1 33 34 ε ε 34 35 1 2 2 35 = Gε − 1 + Gε − 1 . 36 ε ε 36 37 37 38 Then, one applies the following corollary of the nonlinear compactness estimate (11.26). 38 39 39 40 COROLLARY 11.6. Under the same assumptions as in Theorem 11.3, 40 41 √ √ 41 42 G − 1 G − 1 42 ε − Π ε → 0 in L2 dt dx; L2 1 +|v|α M dv 43 ε ε 43 44 44 45 as ε → 0. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 127

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1 1 With the corollary above, one can show that the term Fε(A) in the decomposition of the 1 2 momentum ﬂux is asymptotically close to 2 3 √ 3 4 G − 1 2 4 A Π ε 5 ε 5 6 6 √ 7 7 (notice that the high velocity truncation is disposed of since Π( Gε − 1)/ε has at most 8 | |→+∞ 2 8 polynomial growth in v as v ). That the second term Fε(A) is close to 9 9 10 1 10 A Q G , G 11 ε2 M ε ε 11 12 12 13 uses Lemma 6.2. 13 14 Next, we explain how Lemma 11.5 is used in the proof of the Navier–Stokes limit. To 14 15 begin with, since 15 16 √ 16 17 Gε − 1 1 17 gεγε, 18 ε 2 18 19 19 20 one has 20 21 √ 21 − 2 22 Gε 1 ⊗ −1 2 22 A Π vKε gεγε vKε gεγε vKε gεγε I. 23 ε 3 23 24 24 25 On the other hand, the entropy production estimate (11.15) implies that, modulo extraction 25 26 of a subsequence, one has 26 27 27 1 28 G G − G G ∗ → q 28 2 ε ε∗ ε ε 29 ε 29 30 30 in w − L2(dt dx dµ). Passing to the limit in the scaled, renormalized Boltzmann equa- 31 31 tion (11.18) entails the relation 32 32 33 33 34 qb(v − v∗,ω)Mdv∗ dω 34 R3×S2 35 35 36 1 36 = v ·∇xg = A :∇xu + terms that are odd in v. 37 2 37 38 38 39 Eventually we arrive at the following asymptotic form of the momentum ﬂux. 39 40 40 41 PROPOSITION 11.7. Under the same assumptions as in Theorem 11.3, one has 41 42 42 1 2 43 Fε(A) =vK gεγε⊗vK gεγε− vK gεγε I 43 ε ε 3 ε 44 44 T 45 − ν ∇xu + (∇xu) + o(1) 1 , 45 Lloc(dt dx) dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 128

128 F. G o l s e

1 where 1 2 2 = = − 1 ; 1 3 u vg and g lim gε in w Lloc dt dx L (M dv) . 3 ε→0 4 4 5 5 11.4.6. Strong compactness of vKε gεγε . In order to pass to the limit in the quadratic 6 6 term vKε gεγε and to conclude that 7 7 8 1 2 8 P divx vK gεγε⊗vK gεγε− vK gεγε I 9 ε ε 3 ε 9 10 10 1 2 11 → P divx u ⊗ u − |u| I 11 12 3 12 13 13 R∗ × R3 → 14 in the sense of distributions on + as ε 0, the weak convergence properties of gε 14 established so far are clearly insufﬁcient. One needs instead some strong compactness 15 15 16 properties on the family vKε gεγε . 16 17 17 (a) Strong compactness in the x-variable. Velocity averaging is the natural way to obtain 18 18 compactness in the space variable x for kinetic equations in the parabolic scaling (11.6). 19 19 For the purpose of studying the compactness of vK gεγε in the x-variable, we use the 20 ε 20 following variant of the L2-based velocity averaging theorem. 21 21 22 2 ; 2 | |2 22 LEMMA 11.8. Let φε be a bounded family in Lloc(dt dx L (M dv)) such that φε is 23 ∗ 3 3 23 locally uniformly integrable on R+ × R × R for the Lebesgue measure. Assume that 24 24 25 25 (ε ∂ + v ·∇ )φ is bounded in L1 (dt dx dv). 26 t x ε loc 26 27 2 2 27 Then, for each ψ ∈ L (M dv), the family φεψ is relatively compact in L (dt dx) with 28 loc 28 respect to the x-variable, meaning that, for each T>0 and each compact K ⊂ R3, one 29 29 has 30 30 31 31 + − 2 → 32 φεψ (t, x y) φεψ (t, x) dt dx 0 32 [0,T ]×K 33 33 34 as y → 0 uniformly in ε. 34 35 35 1 36 See [61] for the proof, which is somewhat similar to the L case of velocity averaging 36 37 recalled in Section 8.2.2 (especially Proposition ??). 37 38 Now, we apply the lemma above to 38 39 √ 39 c 40 ε + Gε − 1 40 φε = 41 ε 41 42 42 since 43 43

44 1 Q(Gε,Gε) 44 (ε ∂ + v ·∇ )φ = √ = O(1) 1 45 t x ε 2 c L (dt dx dv) 45 ε 2 ε + Gε loc dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 129

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1 for c ∈ (1, 2), by the entropy production estimate (11.15). Since 1 2 √ 2 c 3 ε + Gε − 1 1 3 gεγε, 4 ε 2 4 5 5 6 applying the velocity averaging lemma above leads to the following compactness (“in the 6 7 x-variable”) result. 7 8 8 9 PROPOSITION 11.9. Under the same assumptions as in Theorem 11.3, for each T>0 9 10 and K ⊂ R3 compact, one has 10 11 11 12 + − 2 → 12 vKε gεγε (t, x y) vKε gεγε (t, x) dt dx 0 13 [0,T ]×K 13 14 14 15 uniformly in ε as y → 0. 15 16 16 17 (b) Strong compactness in the t-variable. It remains to obtain compactness in the time 17 18 18 variable. As we shall see, the solenoidal part of vKε gεγε is strongly compact in the 19 t-variable, but its orthogonal complement, which is a gradient ﬁeld, is not because of high 19 20 frequency oscillations in t. 20 21 21 22 PROPOSITION 11.10. Under the assumptions of Theorem 11.3, modulo extraction of a 22 23 subsequence, one has 23 24 24 25 → 25 P vKε gεγε u 26 26 27 R ; − 2 2 → 27 in C( + w Lx) and in Lloc(dt dx) as ε 0. 28 28

29 PROOF. Indeed, Proposition 11.9 and the translation invariance of the Leray projection P 29 30 imply that 30 31 31 32 32 + − 2 → 33 P vKε gεγε (t, x y) P vKε gεγε (t, x) dt dx 0 (11.31) 33 [0,T ]×K 34 34 35 uniformly in ε as y → 0. On the other hand, the conservation law (11.25) implies that 35 36 36 37 37 · = 1 38 ∂t P vKε gεγε ξ dx O(1) in Lloc(dt), (11.32) 38 R3 39 39 40 for each compactly supported, solenoidal vector ﬁeld ξ ∈ H 3(R3), since we know from 40 41 1 41 Lemma 11.5 and the bounds (11.14) and (11.15), that Fε(A) is bounded in Lloc(dt dx). 42 Also, 42 43 43 √ 44 √ 44 Gε − 1 45 g γ 1 + 2 45 ε ε ε dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 130

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1 so that (11.14) implies that 1 2 2 = R ; 2 R3 3 vKε gεγε O(1) in B + L , (11.33) 3 4 4 5 where B(X,Y) denotes the class of bounded maps from X to Y . 5 3 6 Since the class of H , compactly supported solenoidal vector ﬁelds is dense in that of 6 3 7 all H solenoidal vector ﬁelds (see Appendix A of [86]), (11.33) and (11.32) imply that 7 8 2 3 8 P vK gεγε is relatively compact in C R+; w − L R , (11.34) 9 ε 9 10 by a variant of Ascoli’s theorem that can be found in Appendix C of [86]. 10 11 2 11 As for the Lloc(dt dx) compactness, observe that (11.34) implies that 12 12 13 2 13 P vKε gεγε "χδ is relatively compact in Lloc(dt dx), 14 14

15 where χδ designates any mollifying sequence and " is the convolution in the x-variable 15 16 only. Hence 16 17 17 · → · 18 P vKε gεγε P vKε gεγε "χδ Pu Pu"χδ 18 19 19 − 1 → 20 in w Lloc(dt dx) as ε 0. By (11.31), 20 21 21 P v g γ "χ → P v g γ 22 Kε ε ε δ Kε ε ε 22 23 in L2 (dt dx) uniformly in ε as δ → 0. With this, we conclude that 23 24 loc 24 25 2 2 1 25 P vK gεγε →|Pu| in w − L (dt dx) 26 ε loc 26 27 → 2 27 which implies that P vKε gεγε Pustrongly in L (dt dx). 28 loc 28 29 Next, consider 29 30 30 31 ∇ = − 31 xπε vKε gεγε P vKε gεγε . 32 32 33 Since 33 34 34 → − 2 = 35 vKε gεγε u in w Lloc(dt dx) and divx u 0 35 36 36 37 one has 37 38 2 38 ∇xπε → 0inw − L (dt dx) (11.35) 39 loc 39 40 as ε → 0. Decompose then 40 41 41 42 ⊗ 42 P divx vKε gεγε vKε gεγε 43 43 = P divx P vK gεγε⊗P vK gεγε + P divx P vK gεγε⊗∇xπε 44 ε ε ε 44 45 + ∇ ⊗ + ∇ ⊗∇ 45 P divx xπε P vKε gεγε P divx( xπε xπε). dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 131

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1 By Proposition 11.10, the ﬁrst term converges to P divx(u ⊗ u) in the sense of distribu- 1 2 tions, while the second and third terms converge to 0 in the sense of distributions because 2 3 of (11.35). 3 4 As for the last term, let ζδ = ξδ "ξδ "ξδ, where ξδ is an approximate identity. One can 4 5 prove that 5 6 6 7 1 2 1 1 3 7 ε∂t ζδ "x ∇xπε +∇xζδ "x |v| gεγε → 0inL R+; H R , 8 3 Kε loc loc 8 9 9 1 2 5 1 1 3 10 ε∂ ζ " |v| g γ + ζ " π → L R+; H R 10 t δ x Kε ε ε x δ x ε 0inloc loc 11 3 3 11 12 12 as a result of (11.22), the vanishing of momentum and energy conservation defects 13 13 (see Proposition 11.4 for the momentum, and proceed analogously for the energy) 14 1 14 and the fact that Fε(A) is bounded in L (dt dx) (see Lemma 11.5, and the bounds 15 loc 15 (11.14) and (11.15)). From the above system, Lions and Masmoudi observed in [87] that 16 16 17 17 divx(∇xζδ "x πε ⊗∇xζδ "x πε) 18 18 19 2 19 = 1∇ |∇ |2 − 5 1| |2 + 20 x xζδ "x πε ζδ "x v K gεγε o(1)L1 (dt dx). 20 2 3 3 ε loc 21 21 22 Together with the uniform compactness “in the x-variable” proved in Proposition 11.9, this 22 23 implies that 23 24 24 25 P divx(∇xπε ⊗∇xπε) → 0 25 26 26 27 in the sense of distributions. Collecting the observations above, we have just proved the 27 28 following proposition. 28 29 29 30 PROPOSITION 11.11. Under the assumptions of Theorem 11.3, modulo extraction of a 30 31 subsequence, one has 31 32 32 33 ⊗ → ⊗ 33 P divx vKε gεγε vKε gεγε P divx(u u) 34 34 ∗ 3 35 in the sense of distributions on R+ × R as ε → 0. 35 36 36 37 With this proposition, we have completed all three steps in the proof of the Navier– 37 38 Stokes limit (Theorem 11.3). 38 39 39 40 40 41 11.5. The nonlinear compactness estimate 41 42 42 43 It only remains to prove the nonlinear compactness estimate (11.26), on which the two most 43 44 important steps in the proof of the Navier–Stokes limit – i.e., the vanishing of conservation 44 45 defects and the limiting form of the momentum ﬂux – are based. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 132

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1 This nonlinear compactness estimate results from new ideas on velocity averaging in L1; 1 2 consistently with our presentation of this subject in Section 8.2.2, we shall describe these 2 3 ideas on the steady transport equation. 3 4 We start with a definition of the notion of partial uniform integrability in a product space. 4 5 5 N 6 DEFINITION 11.12. Let µ and ν be positive Borel measures on R ; we say that a fam- 6 1 RN ×RN ; 7 ily φε of elements of L ( x y dµ(x) dν(y)) is uniformly integrable in the variable y if 7 8 8 9 9 sup φε(x, y) dν(y) dµ(x) → 0 10 RN ν(A)<η A 10 11 11 12 as η → 0, uniformly in ε. 12 13 13 14 EXAMPLE. Assume that ν is a finite measure; then, for each p>1, any bounded family 14 1 p 15 in L (dµ(x); L (dν(y)) is uniformly integrable in y. Indeed, if φε is any such family, one 15 16 has 16 17 17 18 1/p 18 sup φε(x, y) dν(y) dµ(x) η φε(x, ·) q dx, N N L (dν) 19 R ν(A)<η A R 19 20 20 = − 21 where p p/(p 1), by applying Hölder’s inequality to the inner integral. 21 22 22 As usual, we shall say that a family φ ∈ L1(RN × RN ; dµ(x) dν(y)) is locally uni- 23 ε x y 23 ⊂ RN × RN 24 formly integrable in y if, for each compact K , the family 1K φε is uniformly 24 25 integrable in y. 25 26 With this notion of partial uniform integrability, we can formulate an important improve- 26 1 27 ment of the L -variant of velocity averaging stated in Proposition 8.6 and Theorem 8.7. 27 28 28 HEOREM f 29 T 11.13 (Golse and Saint-Raymond [60]). Let ε be a bounded family in 29 L1 (RN × RN ; dx dy) such that 30 loc 30 • ·∇ 1 RN × RN ; 31 the family v xfε is bounded in Lloc( dx dy), and 31 • 32 the family fε is locally uniformly integrable in the variable v. 32 33 Then 33 34 (1) the family fε is locally uniformly integrable (in both variables (x, v)), and 34 ∈ ∞ RN 35 (2) for each compactly supported φ L ( ), the family of averages 35 36 36 37 fε(x, v)φ(v) dv 37 RN 38 38 39 1 RN ; 39 is relatively compact in Lloc( dx). 40 40 41 This result amplifies an earlier remark by Saint-Raymond who observed in [107] that, 41 42 1 ∞ 42 under the extra assumption that fε is bounded in Lx(Ly ), the family of averages 43 43 44 44 45 fε(x, v)φ(v) dv 45 RN dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 133

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1 is locally uniformly integrable. 1 2 2 3 SKETCH OF THE PROOF OF THEOREM 11.13. We shall explain how to prove Saint- 3 4 Raymond’s result under the assumptions of the theorem above. 4 5 5 6 Step 1. Let χ ≡ χ(t,x,v) be the solution of the free transport equation 6 7 7 N N 8 ∂t χ + v ·∇xχ = 0,(x,v)∈ R × R ,t >0, 8 9 N N 9 χ(0,x,v)= 1A(x), (x, v) ∈ R × R . 10 10 11 11 Clearly, χ(t,x,v)= 1A(x − tv) for each t>0; it can therefore be put in the form 12 12 13 = ∈ RN × RN 13 χ(t,x,v) 1At,x (v), t > 0,(x,v) , 14 14 15 where, for each t>0, 15 16 16 N 17 At,x = v ∈ R x − tv ∈ A . 17 18 18 19 N 19 Further, At,x is measurable and, for each t>0 and x ∈ R , one has 20 20 21 1 |A| 21 |At,x|= χ(t,x,v)dv = 1A(x − tv)dv = 1A(z) dz = . 22 RN RN tN RN tN 22 23 23 24 Step 2. Without loss of generality, assume that fε and φε in the statement of the theorem 24 N N 25 are nonnegative, and that all the fε’s are supported in the same compact K of R × R . 25 26 Then 26 27 27

28 fε(x, v)φ(v) dv dx 28 RN 29 A 29 30 30 = fε(x, v)φ(v) dv dx 31 N 31 R At,x 32 32 t 33 33 − χ(s,x,v)v ·∇xfε(x, v)φ(v) dx dv ds (11.36) 34 0 RN ×RN 34 35 35 36 as can be seen by integrating by parts the second integral on the right-hand side of the 36 37 equality above. 37 38 Pick η>0 arbitrarily small; the second integral on the right-hand side of (11.36) satisﬁes 38 39 t 39 40 ·∇ ·∇ ∞ 40 χ(s,x,v)v xfε(x, v)φ(v) dx dv ds t v xfε L1 φ L 41 0 RN ×RN 41 42 42 and therefore can be made less than η by choosing 43 43 44 η 44 0

134 F. G o l s e

1 For this t>0, the ﬁrst integral on the right-hand side of (11.36) satisﬁes 1 2 2 3 3 fε(x, v)φ(v) dv dx → 0 4 N 4 R At,x 5 5 N 6 as |A|→0 uniformly in ε, since fε is uniformly integrable in v and |At,x|=|A|/t ,as 6 7 established in Step 1. 7 8 Therefore, for each η>0, there exists α>0 such that |A| <αimplies that 8 9 9 10 10 fε(x, v)φ(v) dv dx 2η 11 A RN 11 12 12 13 uniformly in ε>0, which entails that the family of averages 13 14 14

15 fε(x, v)φ(v) dv 15 N 16 R 16 17 is uniformly integrable. 17 18 18 19 The nonlinear compactness estimate (11.26) will of course be obtained from state- 19 20 ment (1) in Theorem 11.13. In fact, we ﬁrst observe that, for each c ∈ (1, 2), 20 21 21 √ √ 22 c + − 2 c + − 22 δ ε Gε 1 ε Gε 1 23 φ = γ εδ 23 ε ε ε 24 24 25 satisﬁes 25 26 26 27 δ = ∞ 1 27 φε O(1) in Lt L (M dv dx) , 28 28 29 while 29 30 30 31 + ·∇ δ = 1 31 (ε ∂t v x)φε O(1) in Lloc(dt dxMdv). 32 32 33 We next let δ → 0 and remove the εc from under the square root (in that order) so that 33 34 √ 34 2 35 G − 1 35 φδ ε . 36 ε ε 36 37 37 38 δ 38 In order to apply Theorem 11.13 to φε , it remains to prove that this family is uniformly 39 integrable in the v-variable. In fact, we prove the following results. 39 40 40 41 PROPOSITION 11.14. Under the assumptions of Theorem 11.3, for each T>0 and each 41 42 compact K ⊂ R3, the family 42 43 √ 43 44 G − 1 2 44 1 +|v| α ε 45 ε 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 135

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1 is uniformly integrable on [0,T]×K × R3 for the measure dt dxMdv.(We recall that α 1 2 is the relative velocity exponent in the hard cut-off assumption on the collision kernel b, 2 3 see (3.55).) 3 4 4 5 This proposition improves upon the result in [61], that applied to cut-off Maxwell mole- 5 6 cules only (i.e., to the case where α = 0). Its proof is fairly technical, so that we shall just 6 7 sketch the main idea in it. 7 8 Start from the identity 8 9 √ 9 10 Gε − 1 10 LM 11 ε 11 √ √ 12 12 Gε − 1 Gε − 1 1 13 = εQM , − QM Gε, Gε . (11.37) 13 14 ε ε ε 14 15 15 Next, we recall the Bardos–Caﬂisch–Nicolaenko spectral gap for LM in weighted space 16 16 (see Theorem 3.11) 17 17 18 α 2 ⊥ 18 φLM φ C0 1 +|v| φ ,φ∈ (Ker LM ) 19 19 20 together with the following continuity estimate for Q (see [57]) 20 21 21 22 Q 22 M (φ, φ) L2((1+|v|)−αM dv) C φ L2(M dv) φ L2((1+|v|)αM dv). 23 23 24 Using both estimates in the identity above leads to the following control 24 25 √ √ √ 25 26 − − − 26 − Gε 1 Gε 1 − Gε 1 27 1 O(ε) Π 27 ε L2(M dv) ε ε L2((1+|v|)αM dv) 28 √ 28 2 29 Gε − 1 29 O(ε) 2 + O(ε) . 30 Lt,x 30 ε L2(M dv) 31 31 32 This control suggests that 32 33 √ √ 33 34 G − 1 G − 1 34 ε is close to its hydrodynamic projection Π ε 35 ε ε 35 36 36 2 37 precisely in that weighted L space that√ appears in the statement of Proposition 11.14. 37 − 38 Gε 1 38 Since the hydrodynamic projection Π ε is as regular in v as one can hope for (being 39 a quadratic polynomial in v), this eventually entails uniform integrability in v once the 39 40 difﬁculties related to the (t, x) dependence in the estimate above have been handled. As 40 41 we already indicated above, the remaining part of the proof is too technical to be described 41 42 here, and we refer the interested reader to [62] for a complete argument. 42 43 Instead of giving more details on this proof, we comment on the differences between 43 44 the nonlinear compactness estimate (11.28)–(11.30) obtained in [61] in the case of cut- 44 45 off Maxwell molecules, and the most recent variant of such controls obtained in [62], 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 136

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1 namely (11.26). In fact, the main difference between these controls lies in the method for 1 2 obtaining uniform integrability in v. 2 3 In [61], the decomposition (11.37) was replaced with 3 4 4 + + 5 A A 5 = − (MGε,MGε) + (MGε,MGε) 6 Gε Gε (11.38) 6 MGε MGε 7 7 8 (in fact, with a more technical variant of (11.38) involving several truncations), where 8 + 9 A was the gain part of a ﬁctitious collision operator 9 10 10 11 + 11 A (φ, φ) = φ φ∗ cos(v − v∗,ω) dv∗ dω. 12 R3×S2 12 13 13 14 Roughly speaking the term 14 15 15 + 16 A (MG ,MG ) 16 G − ε ε 17 ε 17 MGε 18 18 19 was easily controlled by the entropy production bound, while the term 19 20 20 + 21 A (MGε,MGε) 21 22 22 MGε 23 23 24 improved the regularity and decay in v due to estimates by Grad [63] and Caﬂisch [23]– 24 25 that are linear analogues of the result by Lions on the smoothing properties of the gain part 25 26 in the Boltzmann collision integral (Lemma 3.10). 26 27 However, the main drawback of the decomposition (11.38) was that it naturally involved 27 28 the quantity 28 29 29 30 1 − γε 30 gεγε + 31 ε 31 32 32 33 (see (2.17) in [61]), thereby leading to a dissymmetry between the roles of large v’s and 33 34 large values of gε in the estimates (11.28)–(11.30). This, as we already mentioned, was an 34 α 35 obstruction in upgrading (11.28)–(11.29) with a (1 +|v|) weight as in (11.26). 35 36 The method used in [62] essentially differs from that of [61]; to begin with, we do not 36 37 attempt to establish that the number density Fε is close in some sense to its associated local 37

38 equilibrium MFε (i.e., the Maxwellian with same local density, energy and momentum 38 39 as Fε), an idea that is very natural in the context of the BGK model of the Boltzmann 39 40 equation, see [107,109]. 40 41 Instead, we√ seek to prove that some variant of the number density ﬂuctuation, namely 41 − Gε 1 1 42 the quantity ε 2 gε, becomes close to its associated inﬁnitesimal Maxwellian, i.e., 42 2 43 its L (M dv)-orthogonal projection on Ker LM . This new approach naturally leads to using 43 44 the Dirichlet form of the linearized collision operator, whose relative coercivity transver- 44 α 45 sally to Ker LM provides exactly the (1 +|v|) weight needed in the proof. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 137

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1 12. Conclusions and open problems 1 2 2 3 This survey of the Boltzmann equation and its hydrodynamic limits remains incomplete 3 4 in several respects. As already mentioned in the Introduction, we have chosen to empha- 4 5 size global theories (such as the DiPerna–Lions theory for the Boltzmann equation, or the 5 6 Leray theory for the Navier–Stokes equations) for the evolution problem posed on a spa- 6 7 tial domain without boundaries. More realistic formulations of the hydrodynamic limits of 7 8 the Boltzmann equation would certainly involve boundary value problems. At the formal 8 9 level, the influence of boundaries on hydrodynamic limits is discussed at length in Sone’s 9 10 authoritative monograph [115]. 10 11 More specifically, hydrodynamic limits of boundary value problems may, in some par- 11 12 ticular cases, involve boundary layer equations whose mathematical study is a subject of 12 13 its own and has inspired a considerable amount of literature. These boundary layers are 13 14 meant to forget the specific dependence of the boundary data upon the velocity variable so 14 15 as to match it with the inner form of the number density, which, as explained in this survey, 15 16 is a local Maxwellian or infinitesimal Maxwellian in the hydrodynamic limit. This is obvi- 16 17 ously not the place for a description of that theory, to begin with because these boundary 17 18 layer are essentially steady (instead of evolution) problems. The interested reader could 18 19 start with Chapter 5, Section 5 of [27] and then get acquainted with the mathematical the- 19 20 ory of half-space problems as exposed in [6,26,31,57,58,93] and [119]. These works deal 20 21 with linearized, or at best, weakly nonlinear problems. Nonlinear boundary layer equations 21 22 with nonzero net mass flux at the boundary – typically in the case of a gas condensing on, 22 23 or evaporating from a solid boundary – have been studied in depth numerically by Sone, 23 24 Aoki and their collaborators, see Chapter 7 of [115] and the references therein. At the for- 24 25 mal level, their theory is expected to agree with a general theory of boundary conditions 25 26 for the Euler system of compressible fluids, yet to be formulated at the time of this writing. 26 27 In some cases however, boundary layer are not leading order effects. This is the case 27 28 of the incompressible Stokes or Navier–Stokes limit of the Boltzmann equation for a gas 28 29 in a container with specular or diffuse reflection, or Maxwell’s accommodation condition 29 30 at the boundary; see [96], which extends to such a boundary value problems the results 30 31 in [53]. The analogue of the DiPerna–Lions theory for the boundary value problem is due 31 32 to Mischler [97], this is by no means a straightforward extension of [37]. It requires rather 32 33 subtle analytical tools because natural boundary conditions for the Boltzmann equation – 33 34 except in the case of a specular reflection of gas molecule at the boundary – do not inter- 34 35 act well with the renormalization procedure which is essential in controlling the collision 35 36 integral. 36 37 Let us conclude with a few open problems, in addition to those already mentioned in the 37 38 body of this article. 38 39 It would be of considerable interest to derive the global BV solutions constructed 39 40 by T.-P. Liu from the Boltzmann equation. As in the case of the incompressible Euler 40 41 limit of the Boltzmann equation, the entropy production bound entailed by Boltzmann’s 41 42 H -theorem does not balance the action of the streaming operator on the number density: 42 43 the compactness of hydrodynamic moments of the number density is probably to be sought 43 44 in some stability property of BV solutions of the compressible Euler system. Most likely, 44 45 such a theory should use Bressan’s remarkable results in that direction (see [19,20]). 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 138

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1 As an indication of the level of difficulty of this problem, let us simply mention that the 1 2 hydrodynamic limit of the Boltzmann equation leading to a single Riemann problem has 2 3 not been established so far, except in the case of a single, weak shock wave. Shock pro- 3 4 files for the Boltzmann equation were formally discussed in [25]; for the hard sphere gas, 4 5 a complete construction of these profiles in the weakly nonlinear regime can be found in 5 6 a series of important papers by Nicolaenko [99–101]. All these constructions were based 6 7 on a deep understanding of the algebra of the linearized Boltzmann equation in connection 7 8 with the underlying saddle point structure of subsonic states in gas dynamics. They sug- 8 9 gested some topological structure that remains to be unraveled and could ultimately lead to 9 10 a construction of shock profiles of arbitrary strength. The extension to other cut-off mole- 10 11 11 cular interactions is due to Caflisch and Nicolaenko [24]. However, all these constructions 12 12 led to solutions of the Boltzmann equation whose positivity was not established. This latter 13 13 problem was solved only very recently by Liu and Yu [90]. 14 14 Another open problem would be to improve Theorem ??, by relaxing the unphysical 15 15 δ 16 assumption made on the size of the number density fluctuations ε to reach the physically 16 → → 17 natural condition that δε 0asε 0. This would probably require more information 17 18 on the local conservation laws of momentum and energy for renormalized solutions of the 18 19 Boltzmann equation. Such information would most likely be an important prerequisite for 19 20 making progress on the compressible Euler limit. 20 21 Finally, we have only considered evolution problems in this survey. Although steady 21 22 problems are beyond the scope of the present book, they are perhaps even more important 22 23 in some applications (such as aerodynamics). For instance, it is well known that, for any di- 23 2 3 24 vergence free force field f ≡ f(x)∈ L (Ω; R ), the steady incompressible Navier–Stokes 24 3 25 equations in a smooth, bounded open domain Ω ⊂ R 25 26 26 27 27 −ν xu = f −∇xp − (u ·∇x)u, divx u = 0,x∈ Ω, 28 (12.1) 28 29 u|∂Ω = 0 29 30 30 31 31 has at least one classical solution u ≡ u(x) ∈ H 2(Ω, R3), obtained by a Leray–Schauder 32 32 fixed point argument (see, for instance, [74]). Unfortunately, the parallel theory for the 33 33 34 Boltzmann equation is not as advanced: for one thing, at the time of this writing, there is no 34 35 analogue of the DiPerna–Lions result for the steady Boltzmann equation with a prescribed 35 36 external force field. The reader interested in those matters is advised to read the survey 36 37 article by Maslova [94]; see also her book [95]; among classical references on the subject, 37 38 there are some papers by Guiraud (see [65–67]), and more recent work by Arkeryd and 38 39 Nouri (see, for instance, [3]). See also [117] for the case of a gas flow modeled by the 39 40 Boltzmann equation around a convex body. Finally, a rather exhaustive description of such 40 41 steady problems and their hydrodynamic limits (at the formal level) may be found in Sone’s 41 42 book [115]. 42 43 In spite of the difficulties inherent to the steady Boltzmann equation, the fact that the 43 44 solutions of (12.1) are more regular than in the case of the evolution problem could be of 44 45 considerable help, at least in the context of the hydrodynamic limit. 45 dafermos2 v.2005/04/29 Prn:8/06/2005; 8:46 F:dafermos203.tex; VTEX/Lina p. 139

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