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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
4 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 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
6 F. G o l s e
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
The Boltzmann equation and its hydrodynamic limits 7
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
The Boltzmann equation and its hydrodynamic limits 9
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
10 F. G o l s e
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
The Boltzmann equation and its hydrodynamic limits 11
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
12 F. G o l s e
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
The Boltzmann equation and its hydrodynamic limits 13
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
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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
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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 in M 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