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

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.