Decay and Continuity of Boltzmann Equation in Bounded Domains

TL;DR

This paper develops a mathematical framework to analyze how solutions to the Boltzmann equation decay over time and remain continuous in bounded domains with various boundary conditions, highlighting exponential decay and boundary regularity.

Contribution

It introduces a new $L^{2}$ decay theory and combines it with $L^{ olinebreak{}^{ ext{infinity}}}$ decay analysis for the linearized Boltzmann equation in bounded domains with different boundary conditions.

Findings

Exponential decay in $L^{ ext{infinity}}$ norm for hard potentials near Maxwellian.

Continuity of solutions in convex domains away from grazing velocities.

Applicability to multiple boundary interaction types, including inflow and reflection.

Abstract

Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{\infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $% L^{\infty}$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.