Entropy dissipation estimates for the Boltzmann equation without cut-off

TL;DR

This paper establishes a lower bound on entropy dissipation for the Boltzmann collision operator across various potentials, enhancing understanding of solution regularity without cut-off assumptions.

Contribution

It provides a novel entropy dissipation estimate applicable to a broad class of potentials, including very soft potentials, and applies it to weak solutions of the Boltzmann equation.

Findings

Lower bound on entropy dissipation for a wide range of potentials

Weighted $L^p$-Norm estimates for solutions

Enhanced understanding of solution regularity without cut-off

Abstract

We prove a bound on the entropy dissipation for the Boltzmann collision operator from below by a weighted $L^p$-Norm. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. As an application, we study weak solutions to the spatially homogeneous Boltzmann equation and prove a weighted $L^1_t(L^p_v)$ estimate.