Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

TL;DR

This paper proves that solutions to the 2D homogeneous Boltzmann equation with certain initial conditions instantly gain regularity in Sobolev spaces, using Malliavin calculus for jump processes.

Contribution

It establishes instant regularization properties for the 2D Boltzmann equation with very hard potentials, a novel result in kinetic theory.

Findings

Solutions belong to Sobolev space $H^r$ instantly for some $r eq 0$

Regularization depends on the parameters of the equation

Uses Malliavin calculus for jump processes in the proof

Abstract

We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some regularization properties: for a class of very hard potentials, the solution instantaneously belongs to $H^r$, for some $r\in (-1,2)$ depending on the parameters of the equation. Our proof relies on the use of a well-suited Malliavin calculus for jump processes.