Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition

TL;DR

This paper proves that solutions to the 3D homogeneous Boltzmann equation with measure initial data become finite entropy functions instantly, under certain conditions on the potential type and initial moments.

Contribution

It establishes instant regularization and finiteness of entropy for solutions with measure initial data in the Boltzmann equation, extending previous results to broader initial conditions.

Findings

Solutions become Besov space functions immediately.

Solutions have finite entropy instantly.

Weak solutions are supported on all of ^3 immediately.

Abstract

We consider the $3D$ spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial condition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume additionally that the initial condition has a moment of sufficiently high order ($8$ is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak solution is immediately supported by ${\mathbb {R}}^3$.