Relaxation in Sobolev spaces and $L^1$ spectral gap of the 1D dissipative Boltzmann equation with Maxwell interactions

Abstract

We study the dynamic relaxation to equilibrium of the 1D dissipative Boltzmann equation with Maxwell interactions in classical $H^s$ Sobolev spaces. In addition, we present a spectral shrinkage analysis and spectral gap estimates for the linearised 1D dissipative Boltzmann operator with such interactions. Based on this study, we explore the convergence in $H^s$ and $L^1$ spaces for the linear and nonlinear models. This study extends classical results found in the literature given for spaces with weak topologies.

The manuscript contains material from our previous contribution arXiv:2211.03446 which has been split into two papers