Boltzmann Equation with a Large Potential in a Periodic Box

TL;DR

This paper proves the stability and well-posedness of Maxwellian solutions to the Boltzmann equation with a large external potential in a periodic domain, extending understanding of kinetic equations under strong external influences.

Contribution

It establishes the $L^{ ext{infinity}}$ stability of Maxwellian solutions with large potentials in a periodic box, a problem previously open for large amplitude external fields.

Findings

Proves $L^{ ext{infinity}}$ stability of Maxwellian with large potential.

Establishes well-posedness of Boltzmann equation under large external potential.

Extends stability results to high-dimensional periodic domains.

Abstract

The stability of the Maxwellian of the Boltzmann equation with a large amplitude external potential $\Phi$ has been an important open problem. In this paper, we resolve this problem with a large $C3-$potential in a periodic box $\mathbb{T}^d$, $d \geq 3$. We use [1] in $L^p-L^{\infty}$ framework to establish the well-posedness and the $L^{\infty}-$stability of the Maxwellian $\mu_E(x,v)=\exp\{-\frac{|v|^2}{2}-\Phi(x)\}$.