Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure

TL;DR

This paper proves the existence of global renormalized solutions for the non-cutoff Boltzmann equation with hard potentials, showing the defect measure vanishes, unlike in the soft potential case where the approach fails.

Contribution

It demonstrates that for hard potentials, the defect measure in renormalized solutions is zero, enabling solutions without defect measures, unlike previous results requiring them.

Findings

Defect measure vanishes for hard potentials, enabling standard renormalized solutions.

Counterexample shows approach fails for soft potentials.

Stronger coercivity estimates are key to the proof.

Abstract

The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials ($0 \leq \gamma \leq 1$). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ($-3 < \gamma < 0$).