A new regularization mechanism for the Boltzmann equation without cut-off

TL;DR

This paper introduces a novel regularization mechanism for the non cut-off Boltzmann equation, providing new a priori estimates in Hölder and L-infinity spaces, applicable to both homogeneous and inhomogeneous cases.

Contribution

It develops a new regularization approach for the Boltzmann equation without cut-off, extending regularity results and a priori estimates to broader settings.

Findings

Derived Hölder space a priori estimates for the non cut-off Boltzmann equation.

Established L-infinity a priori bounds for the inhomogeneous case under bounded macroscopic quantities.

Extended regularity results to general integro-differential equations related to the Boltzmann equation.

Abstract

We apply recent results on regularity for general integro-differential equations to derive a priori estimates in H\"older spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in $L^\infty$ which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.