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

TL;DR

This paper establishes decay estimates in velocity for solutions of the non-cutoff Boltzmann equation in bounded domains, considering various boundary conditions and potential types, with specific decay bounds depending on potential softness.

Contribution

It introduces a new framework of weak solutions based on Truncated Convex Inequalities and derives velocity decay estimates for different boundary conditions and potential classes.

Findings

Solutions exhibit up to d+1 polynomial velocity decay.

For moderately soft potentials, decay cannot exceed d+2 if energy is bounded.

Decay estimates depend on boundary conditions and potential softness.

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.