Local conditional regularity for the Landau equation with Coulomb potential

TL;DR

This paper proves that solutions to the 3D Landau equation with Coulomb potential, which are bounded in a specific Lebesgue space, possess locally Holder continuous derivatives in velocity and time variables.

Contribution

It establishes local regularity results for Villani solutions of the Landau equation with Coulomb interaction, showing Holder continuity of derivatives under certain Lebesgue space conditions.

Findings

Solutions in L_{t}^{ty}L_{v}^{q} with q>3 are Holder continuous in derivatives.

Regularity holds locally in space and time for solutions in specified Lebesgue spaces.

The results advance understanding of regularity for Coulomb-interacting particle systems.

Abstract

This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space L_{t}^{\infty}L_{v}^{q} with q>3 in an open cylinder (0,S)\times B, where B is an open ball of \mathbb{R}^{3}, must have Holder continuous second order derivatives in the velocity variables, and first order derivative in the time variable locally in any compact subset of that cylinder.