Gaussian bounds for the inhomogeneous Landau equation with hard potentials

TL;DR

This paper establishes Gaussian upper and lower bounds for solutions of the inhomogeneous Landau equation with hard potentials, showing regularity and continuation depend only on basic physical quantities.

Contribution

It provides the first Gaussian bounds for solutions with arbitrary initial data in the hard potential regime, unlike the soft potential case.

Findings

Solutions satisfy pointwise Gaussian bounds in velocity.

Weak solutions are infinitely differentiable in all variables.

Continuation depends solely on mass, energy, and entropy.

Abstract

We consider weak solutions of the spatially inhomogeneous Landau equation with hard potentials ($\gamma \in (0,1]$), under the assumption that mass, energy, and entropy densities are under control. In this regime, with arbitrary initial data, we show that solutions satisfy pointwise Gaussian upper and lower bounds in the velocity variable. This is different from the behavior in the soft potentials case ($\gamma <0$), where Gaussian estimates are known not to hold without corresponding assumptions on the initial data. Our upper bounds imply weak solutions are $C^\infty$ in all three variables, and that continuation of solutions is governed only by the mass, energy, and entropy.