The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Abstract

Abstract. Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate L ∞ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in L ∞ space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori L ∞ estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the L 2 − L ∞ theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted L ∞ theory so that we could deal with the greater difficulties stemmed from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in L ∞ space with the aid of the L 2 theory and a bootstrap argument. These methods in the latter case can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

Effects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rarefied Gas

“…Particularly, [20] found a velocity weight function containing an exponential factor exp{c|v| 2 /(1 + t) q }. We remark that this kind of weight could be useful for simultaneously dealing with the global existence and large-time behavior of solutions for the problem in the torus domain or even in the general bounded domain (for instance, [36]), since the typical function exp{− v γ t − c|v| 2 /(1 + t) q } induces a time-decay rate exp{−λt β ′ } with β ′ = (2 − q|κ|)/(2 + |κ|). Therefore, the large-time behavior of solutions is gained by making the velocity weight in the solution space become lower and lower as time goes on.…”

The Boltzmann equation with large-amplitude initial data in bounded domains

“…For the mathematical study of the Boltzmann equation with boundaries, we would further mention many other contributions in different aspects: [1,2,3,11,14,28,39,40,42,44,58]; See also the books [12,13,43,52] and references therein. Here the book [52] by Sone gives the systematic investigations of the initial and/or boundary value problem on the Boltzmann equation from the numerical point of view.…”

The Boltzmann equation with weakly inhomogeneous data in bounded domain

“…Recently, the first author of the paper developed an L 2 − L ∞ theory to establish the time decay and continuity of the unique global solution of the Boltzmann equation with four basic boundary conditions: in flow, bounce back, specular reflection and diffuse reflection [17]. The result is then extended to the soft potential case by the second author of the paper and Yang [22]. More recently, the W 1,p (1 < p < 2) regularity for the Botlzmann equation in general classes of bounded domain were further proved [18] .…”