Conditional appearance of decay for the non-cutoff Boltzmann equation in a domain

Abstract

This work is concerned with the generation of decay estimates in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff on a bounded spatial domain for hard and moderately soft potentials. We work with a suitable notion of weak solutions, provided that mass, energy and entropy density functions are under control. We treat several boundary conditions: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. We show that the solutions generate some amount (up to $d+1$) of pointwise polynomial velocity decay. In the case of moderately soft potentials, we show that it is not possible to generate a decay higher than $d+2$ if the energy is bounded.