Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

TL;DR

This paper derives upper bounds and decay estimates for solutions to the inhomogeneous Landau equation with soft potentials, demonstrating propagation of Gaussian bounds over time.

Contribution

It provides the first a priori estimates for solutions with arbitrary initial data under controlled mass, energy, and entropy, specifically for moderately soft potentials.

Findings

Solutions have polynomial decay in velocity.

Gaussian upper bounds are propagated over time.

Estimates hold for arbitrary initial data under certain conditions.

Abstract

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.