Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum

TL;DR

This paper provides the first rigorous proof that measure-valued solutions to the homogeneous Boltzmann equation with Maxwellian cross sections become smooth over time, confirming Villani's conjecture about the smoothing effect.

Contribution

It introduces a novel time degenerate coercivity estimate using microlocal analysis to prove regularization for measure initial data, extending previous results beyond Dirac masses.

Findings

Proof of smoothing effect for measure initial data

Introduction of a time degenerate coercivity estimate

Optimal regularity description for positive times

Abstract

We justify the Villani conjecture on the smoothing effect for measure value solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure value initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.