Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces

Abstract

In this paper, we present an $L^p$-stability theory for thespace-inhomogeneous Boltzmann equation with cut-off and inversepower law potentials, when initial data are sufficiently small anddecay fast enough in phase space. For moderately soft potentials, weshow that classical solutions depend Lipschitz continuously on theinitial data in weighted $L^p$-norm. In contrast for hardpotentials, we show that classical solutions depend Höldercontinuously on the initial data. Our stability estimates are basedon the dispersion estimates due to time-asymptotic linear Vlasovdynamics.