Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

TL;DR

This paper demonstrates that weak solutions to the non-cutoff homogeneous Boltzmann equation with Debye-Yukawa interactions exhibit strong smoothing effects, akin to a logarithmic heat equation, despite weak coercivity estimates.

Contribution

It establishes the smoothing property of solutions for the Boltzmann equation with a logarithmic singularity in the collision kernel, extending understanding of regularity in such kinetic models.

Findings

Weak solutions become smooth over time

Smoothing effect analogous to a logarithmic heat equation

Weak coercivity still yields strong regularity

Abstract

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.