Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded 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 suitable weak solutions, provided that mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The notion of weak solutions relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced through F.~Golse, L.~Silvestre and the first author (2023) in the spatially homogeneous case. We show that the solutions generate some amount (up to $d+1$) of pointwise polynomial velocity decay. In 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.