Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules

TL;DR

This paper proves that weak solutions to the fully nonlinear non-cutoff homogeneous Boltzmann and Kac equations for Maxwellian molecules become Gevrey regular immediately after positive time, confirming long-standing conjectures about their smoothing properties.

Contribution

It establishes Gevrey regularity for solutions of the non-cutoff Boltzmann and Kac equations with Maxwellian molecules, a significant advancement in understanding their regularity properties.

Findings

Weak solutions become Gevrey regular immediately after positive time.

The result applies to initial data with finite mass, energy, and entropy.

Confirms the conjecture about the smoothing effect of the non-cutoff Boltzmann operator.

Abstract

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.