Energy method for Boltzmann equation
Introduction
Consider the Boltzmann equation: 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: where
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 , and the microscopic, non-fluid part of the solution: 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]: Here θ(x,t) is the temperature and is related to the internal energy E through the gas constant R, E=(3/2)Rθ, 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]: and for any functions h,g: With respect to the local Maxwellian, we define an inner product in as for functions h, g of ξ. The following functions are orthogonal with respect to this inner product: We define the macroscopic projection and microscopic projection as follows: 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 The operator and are projections, that is A function h(ξ) is called non-fluid if it gives raise to zero conserved quantities, that is Note that functions in the range of the microscopic projection are non-fluid. It is clear that for the solution f(x,t,ξ) of the Boltzmann equation: From the decomposition of the solution , the Boltzmann equation becomes 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): Here p is the pressure for the monatomic gases: The microscopic equation is obtained by applying the microscopic projection 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 , as a function of ξ, is in the space spanned by χα, α=1,2,3,4,5. Thus . We note that if Thus the projection of collision terms under is zero. We also have Thus we have , and so . With these, the microscopic equation is 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 in (1.11) depends only on the microscopic part . This is convenient for the energy method.
From (1.11) we have and substitute this into (1.10) to result in 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 are set zero, the system (1.13) becomes the Euler equations as in the Hilbert expansion. When only the microscopic part 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: are 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 and that the equality holds only when the solution is a Maxwellian, . The H-theorem is obtained by multiplying the Boltzmann equation by and integrating with respect to ξ: 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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