Smoothing effect for Boltzmann equation with full-range interactions

TL;DR

This paper establishes new bounds for the collision operator in the Boltzmann equation with full-range interactions, demonstrating immediate smoothing of solutions and deriving stronger entropy dissipation estimates.

Contribution

It provides the first constructive bounds for the collision operator in fractional Sobolev norms for full-range potentials, leading to regularity and entropy dissipation results.

Findings

Solutions become immediately smooth in all variables.

New bounds for the collision operator in fractional Sobolev norms.

Stronger global entropy dissipation estimate.

Abstract

In this work, we are concerned with the regularities of the solutions to Boltzmann equation with the physical collision kernels for the full range of intermolecular repulsive potentials, $r^{-(p-1)}$ with $p>2$. We give the new and constructive upper and lower bounds for the collision operator in terms of standard fractional Sobolev norm. As an application, we prove that the strong solutions obtained by Desvillettes \& Mouhot \cite{dm} to homogeneous Boltzmann equation and classical solutions obtained by Gressman-Strain \cite{gs1,gs2} or Alexandre-Morimoto-Ukai-Xu-Yang \cite{amuxy3,amuxy5} for the inhomogeneous Boltzmann equation become immediately smooth with respect to all variables. And as another application, we obtain the global entropy dissipation estimate which is a little stronger than the one of Alexandre-Desvillettes-Villani-Wennberg \cite{advw}.