The initial boundary value problem for the Boltzmann equation with soft potential

TL;DR

This paper establishes global existence and exponential decay of solutions for the Boltzmann equation with soft potential under diffuse and specular boundary conditions, overcoming new mathematical challenges posed by the collision kernel's singularity.

Contribution

It introduces new $L^2$-$L^$ analytical techniques and a time-velocity weighted $L^$ framework to handle boundary effects and kernel singularities in soft potential Boltzmann equations.

Findings

Proved global existence and exponential decay for diffuse reflection boundary conditions.

Developed a new weighted $L^$ theory for specular reflection boundary conditions.

Extended $L^2$-$L^$ methods to handle singular collision kernels in bounded domains.

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 $L^\infty$ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in $L^\infty$ 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 {\it a priori} $L^\infty$ 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 $L^2-L^\infty$ 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 $L^\infty$ 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 $L^\infty$ 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.