Exponential Convergence to the Maxwell Distribution For Spatially Inhomogenous Boltzmann Equations

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.

An error in spectral analysis has been corrected

References in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Asymptotic Stability of the Boltzmann Equation with Maxwell Boundary Conditions
• Stability of global equilibrium for the multi-species Boltzmann equation in $L^{\infty }$ settings