C^\infty smoothing for weak solutions of the inhomogeneous Landau equation

TL;DR

This paper proves that weak solutions to the inhomogeneous Landau equation become instantly smooth under certain conditions, using Schauder estimates, with results varying based on the softness of the potentials involved.

Contribution

It introduces a novel approach using Schauder estimates to establish instant regularization of solutions for the Landau equation with different potential regimes.

Findings

Weak solutions become smooth immediately for moderately soft potentials.

Additional velocity moment bounds are required for very soft potentials.

The method relies on iterative Schauder-type estimates.

Abstract

We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.