Energy method for Boltzmann equation

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Abstract

A basic, simple energy method for the Boltzmann equation is presented here. It is based on a new macro–micro decomposition of the Boltzmann equation as well as the H-theorem. This allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method. As an illustration, we apply the method for the study of the time-asymptotic, nonlinear stability of the global Maxwellian states. Previous energy method, starting with Grad and finishing with Ukai, involves the spectral analysis and regularity of collision operator through sophisticated weighted norms.

Introduction

Consider the Boltzmann equation: ft+ξ·∇xf=Q(f,f)κ,(f,x,t,ξ)∈R×R3×R+×R3,where the positive constant κ is the Knudsen number [1]. For simplicity, we consider the hard sphere model, for which the bilinear collision operator Q(f,g) is of the following form: Q(f,g)(ξ)≡12R3×S2(ξ−ξ)·Ω≥0(−f(ξ)g(ξ)−g(ξ)f(ξ)+f(ξ′)g(ξ′)+g(ξ′)f(ξ′))|(ξ−ξ)·Ω|dξdΩ,where ξ′=ξ−[(ξ−ξ)·Ω]Ω,ξ′=ξ+[(ξ−ξ)·Ω]Ω.

The main purpose of the present paper is to introduce a macro–micro decomposition of the equation. The decomposition is based on the decomposition of the solution into the macroscopic, fluid part, the local Maxwellian M=M(x,t,ξ)=M[ρ,u,θ](ξ), and the microscopic, non-fluid part G=G(x,t,ξ) of the solution: f=M+G.The local Maxwellian is constructed from the fluid variables, the five conserved quantities, the mass density ρ(x,t), momentum m(x,t)=ρu(x,t) and energy E+|u|2/2 of the Boltzmann equation [9]: ρ(x,t)≡∫R3f(x,t,ξ)dξ,mi(x,t)≡∫R3ψif(x,t,ξ)dξfori=1,2,3,ρE+12|u|2(x,t)≡∫R3ψ4f(x,t,ξ)dξ,MM[ρ,u,θ](ξ)≡ρ(2πRθ)3exp|ξ−u|22Rθ.Here θ(x,t) is the temperature and is related to the internal energy E through the gas constant R, E=(3/2), and u(x,t) is the fluid velocity. The five fluid variables are conserved quantities because of the following property of the collision invariants ψα [1]: R3ψαQ(h,g)dξ=0foranyα=0,1,2,3,4and for any functions h,g: ψ0≡1,ψi≡ξifori=1,2,3,ψ412|ξ|2.With respect to the local Maxwellian, we define an inner product in ξ∈R3 as 〈h,g〉≡∫R31Mh(ξ)g(ξ)dξfor functions h, g of ξ. The following functions are orthogonal with respect to this inner product: χ0(ξ;ρ,u,θ)≡1ρM,χi(ξ;ρ,u,θ)≡ξi−uiRθρMfori=1,2,3,χ4(ξ;ρ,u,θ)≡1|ξ−u|2−3M,〈χαβ〉=δαβforα,β=0,1,2,3,4.We define the macroscopic projection P0 and microscopic projection P1 as follows: P0h≡∑α=04〈h,χα〉χα,P1h≡h−P0h.We view the above decomposition of Boltzmann equation as the linearization around the local Maxwellian states so that the linear collision operator L[ρ,u,θ] is L=L[ρ,u,θ]g≡Q(M[ρ,u,θ]+g,M[ρ,u,θ]+g)−Q(g,g).The operator P0 and P1 are projections, that is P0P0=P0,P1P1=P1.A function h(ξ) is called non-fluid if it gives raise to zero conserved quantities, that is R3h(ξ)ψαdξ=0forα=0,1,2,3,4.Note that functions in the range of the microscopic projection P1 are non-fluid. It is clear that for the solution f(x,t,ξ) of the Boltzmann equation: P0f=M,P1f=G.From the decomposition of the solution f=M+G, the Boltzmann equation becomes (M+G)t+ξ·∇x(M+G)=1κ(2Q(G,M)+Q(G,G)).We now decompose the Boltzmann equation. The conservation laws are obtained, as usual, by integrating with respect to ξ of the Boltzmann equation times the collision invariants ψα(ξ): ρt+divm=0,mit+j=13ujmixj+pxi+∫R3ψi(ξ·∇xG)dξ=0fori=1,2,3,ρ|u|22+Et+∑j=13ujρ|u|22+E+pxj+∫R3ψ4(ξ·∇xG)dξ=0.Here p is the pressure for the monatomic gases: p=23ρE.The microscopic equation is obtained by applying the microscopic projection P1 to the Boltzmann equation (1.9). Since the projections are based on local Maxwellian, the projections and partial differentiations in (x,t) may not commute. Nevertheless, we note that Mt, as a function of ξ, is in the space spanned by χα, α=1,2,3,4,5. Thus P0Mt=Mt. We note that P0h=0 if R3αdξ=0.Thus the projection of collision terms under P0 is zero. We also have R3Gtψαdξ=∂tR3αdξ=0.Thus we have P0Gt=0, and so P1(Mt+Gt)=Gt. With these, the microscopic equation is Gt+P1(ξ·∇xG+ξ·∇xM)=1κLG+1κQ(G,G).This decomposition improvises that of [8], where the linearization is about the global Maxwellian. The advantage of the present one is that the nonlinear term Q(G,G) in (1.11) depends only on the microscopic part G. This is convenient for the energy method.

From (1.11) we have G=κL−1(P1ξ·∇xM)+L−1(κ(∂tG+P1ξ·∇xG)−Q(G,G))and substitute this into (1.10) to result in ρt+divm=0,mit+j=13ujmixj+pxi+κ∫R3ψi(ξ·∇xL−1P1ξ·∇xM)dξ+∫R3ψi(ξ·∇xL−1)(κ[Gt+P1ξ·∇xG]−Q(G,G))dξ=0fori=1,2,3,ρ|u|22+Et+∑j=13ujρ|u|22+E+pxj+κ∫R3ψ4(ξ·∇xL−1P1ξ·∇xM)dξ+∫R3ψ4(ξ·∇xL−1)(κ[Gt+P1ξ·∇xG]−Q(G,G))dξ=0.The fluid equations, the Euler and Navier–Stokes equations, are in fact part of the above equations. For instance, when the Knudsen number κ and the microscopic part G are set zero, the system (1.13) becomes the Euler equations as in the Hilbert expansion. When only the microscopic part G is set to be zero in (1.13), it becomes the Navier–Stokes equations as in the Chapman–Enskog expansion. These fluid equations as derived through the Hilbert and Chapman–Enskog expansions are approximations to the Boltzmann equation [3]. Here we derive it as part of the full Boltzmann equation. Nevertheless, our approach is consistent in spirit with these expansions in that the higher order terms beyond first order in the expansions must satisfy a solvability condition, which means that these terms are microscopic.

In the above system, the terms: −κ∫R3ψi(ξ·∇xL−1P1ξ·∇xM)dξ=−κ∫R3ψi(ξ·∇xL[ρ,u,θ]−1P1ξ·∇xM[ρ,u,θ])dξ=−κ∫R3ψi(ξ·∇xL[1,u,θ]−1P1ξ·∇xM[1,u,θ])dξ,i=1,2,3,−κ∫R3ψ4(ξ·∇xL−1P1ξ·∇xM)dξ=−κ∫R3ψ4(ξ·∇xL[ρ,u,θ]−1P1ξ·∇xM[ρ,u,θ])dξ=−κ∫R3ψ4(ξ·∇xL[1,u,θ]−1P1ξ·∇xM[1,u,θ])dξare the viscosity and heat conductivity terms for the Navier–Stokes equations; and they are independent of the density gradient ∇xρ.

The Boltzmann equation as decomposed in , consists of the fluid equations plus the microscopic part. This allows for the use of the ideas from hyperbolic and viscous conservation laws for the energy method. For the conservation laws, there is the basic concept of entropy. For this, we discuss in Section 2 the derivation of the macroscopic entropy based on the H-theorem for the Boltzmann equation. In Section 3 we carry out the energy method for the nonlinear stability of global Maxwellian states. The energy method here is elementary and generalizes that in [8]. For other energy methods making use of the spectral properties of the linearized operator, see [7], [10], [11].

Section snippets

H-theorem

The H-theorem of the Boltzmann equation is based on the observation that R3Q(f,f)logfdξ≤0and that the equality holds only when the solution is a Maxwellian, f=M. The H-theorem is obtained by multiplying the Boltzmann equation by logf and integrating with respect to ξ: R3flogfdξ+∇·∫R3ξflogfdξ=κ∫R3Q(f,f)logfdξ≤0.There are two ways to view this. The first is to ignore the transport term and study the linearized collision operator. The linearized collision operator L of (1.7) is symmetric:

Nonlinear stability of a Maxwellian state

In this section, we will show that the macro–micro decomposition yields elementary energy estimates for stability of a global Maxwellian state. Thus, we assume that the initial value f|t=0 is a small perturbation of a global Maxwellian state M̄. We will show that the macroscopic component M tends to M̄ and the microscopic component G tends to zero as t tends to infinity. There are two steps in the energy estimates. In the first step, the lower order estimate follows from the two versions of the

Acknowledgements

The research of the first author was supported by Institute of Mathematics, Academia Sinica and NSF Grant DMS-9803323. The research of the second author was supported by the Competitive Earmarked Research Grant of Hong Kong CityU 1142/01P #9040648. The research of the third author was supported by the Competitive Earmarked Research Grant of Hong Kong #9040645.

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