Exponential Convergence to the Maxwell Distribution For Spatially Inhomogenous Boltzmann Equations

TL;DR

This paper proves that solutions to the spatially inhomogeneous Boltzmann equations with hard sphere potentials converge exponentially fast to Maxwellian equilibrium distributions under certain regularity and localization conditions.

Contribution

It confirms Villani's conjecture by demonstrating exponential convergence in $L^{1}$ space for solutions with specific smoothness and localization properties.

Findings

Solutions converge exponentially fast to Maxwellians.

Convergence holds in weighted $L^{1}$ space.

Results confirm a conjecture by C. Villani.

Abstract

We consider the rate of convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogenous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3)$. We prove a conjecture of C. Villani: assume the solution is sufficiently localized and sufficiently smooth, then the solution, in $L^{1}$-space, converges to a Maxwellian, exponentially fast in time.