A Revisit of the Velocity Averaging Lemma: on the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain

TL;DR

This paper applies the velocity averaging lemma to prove fractional Sobolev regularity for solutions of the stationary linearized Boltzmann equation in bounded convex domains, advancing understanding of kinetic regularity.

Contribution

It introduces a novel approach using velocity averaging to establish fractional regularity for stationary Boltzmann equations in bounded convex domains.

Findings

Regularity in fractional Sobolev space up to order 1^- in space variable.

Method achieves regularity considering incoming data with three iterations.

Advances the theoretical understanding of the Boltzmann equation's regularity properties.

Abstract

In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$.