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

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 ($-3 < \gamma < 0$).