A new regularization possibility for the Boltzmann equation with soft potentials

TL;DR

This paper demonstrates that for a simplified Boltzmann equation with a sufficiently singular velocity cross section, solutions can instantaneously become regular functions, even from singular initial conditions, highlighting a surprising regularization effect.

Contribution

The study proves the first regularization result for the Boltzmann equation with cutoff and singular cross sections, showing singularities can promote regularity.

Findings

Solutions become functions instantaneously under certain singular cross sections

Regularization occurs even with initial conditions that are not highly regular

Singularity in the velocity cross section can facilitate regularization

Abstract

We consider a simplified Boltzmann equation: spatially homogeneous, two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized around its initial condition. We prove that for a sufficiently singular velocity cross section, the solution may become instantaneously a function, even if the initial condition is a singular measure. To our knowledge, this is the first regularization result in the case with cutoff: all the previous results were relying on the non-integrability of the angular cross section. Furthermore, our result is quite surprising: the regularization occurs for initial conditions that are not too singular, but also not too regular. The objective of the present work is to explain that the singularity of the velocity cross section, which is often considered as a (technical) obstacle to regularization, seems on the contrary to help the regularization.