Global regularity estimates for the Boltzmann equation without cut-off

TL;DR

This paper establishes smoothness and decay estimates for solutions to the inhomogeneous Boltzmann equation without cut-off, assuming bounds on mass, energy, and entropy, advancing understanding of its regularity properties.

Contribution

It provides the first comprehensive $C^ abla$ a priori estimates and decay results for solutions without cut-off, under minimal assumptions.

Findings

Solutions are $C^ abla$ smooth given bounds on mass, energy, and entropy.

Decay estimates for large velocities are proven for all derivatives.

The results hold for the inhomogeneous Boltzmann equation without cut-off.

Abstract

We derive $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.