The Boltzmann Equation with Large-amplitude Initial Data in Bounded Domains

TL;DR

This paper proves the global existence and exponential convergence to equilibrium of large-amplitude solutions to the Boltzmann equation in bounded domains with diffuse reflection boundaries, even allowing initial vacuum states.

Contribution

It establishes the first global solutions for large initial data with vacuum in bounded domains using novel estimates and bootstrap methods.

Findings

Solutions become close to Maxwellians over time

Initial vacuum states disappear after some positive time

Large-amplitude solutions converge exponentially to equilibrium

Abstract

The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in $L^2$. This yields the disappearance of any initial vacuum and the exponential convergence of large-amplitude solutions to equilibrium in large time. The isothermal diffuse reflection boundary condition plays a vital role in the analysis. The most key ingredients in our strategy of the proof include: (i) $L^2_{x,v}$--$L^\infty_xL^1_v$--$L^\infty_{x,v}$ estimates along a bootstrap argument; (ii) Pointwise estimates on the upper bound of the gain term by the product of $L^\infty$ norm and $L^2$ norm; (iii) An iterative procedure on the nonlinear term.