Classical solutions of the Boltzmann equation with irregular initial data

TL;DR

This paper establishes the existence of classical and weak solutions to the non-cutoff Boltzmann equation with irregular initial data, including bounded measurable data with polynomial decay, and explores conditions for uniqueness and global existence.

Contribution

It constructs classical solutions for large, irregular initial data without strict positivity, and proves weak solutions and uniqueness under certain regularity assumptions.

Findings

Existence of classical solutions with irregular initial data

Weak solutions exist under relaxed decay and positivity conditions

Global existence near equilibrium for polynomial decay initial data

Abstract

This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our result requires no assumption of strict positivity for the initial data, except locally in some small ball in phase space. We also obtain existence results for weak solutions when our decay and positivity assumptions for the initial data are relaxed. Because the regularity of our solutions may degenerate as $t \rightarrow 0$, uniqueness is a challenging issue. We establish weak-strong uniqueness under the additional assumption that the initial data possesses no vacuum regions and is H\"older continuous. As an application of our short-time existence theorem, we prove global existence near equilibrium for bounded, measurable initial data that decays at a finite polynomial rate in velocity.