Self-generating lower bounds and continuation for the Boltzmann equation

TL;DR

This paper establishes instantaneous pointwise lower bounds for solutions to the inhomogeneous Boltzmann equation without cutoff, improving continuation criteria for large-data solutions by removing previous assumptions.

Contribution

It introduces a method to obtain lower bounds that depend solely on initial data and certain upper bounds, enhancing understanding of solution behavior.

Findings

Instantaneous lower bounds for solutions even with vacuum regions

Improved continuation criterion for large-data solutions

Removal of previous assumptions on mass and entropy bounds

Abstract

For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space $\mathbb R^3_x$, we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above.