A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications

TL;DR

This paper introduces a novel monotonicity formula involving entropy and Sobolev seminorms for the Landau equation with Coulomb potential, providing new insights into its regularity and blowup behavior.

Contribution

It presents a new time-dependent functional that decreases along solutions, illuminating the balance between dissipation and nonlinearity in the Landau equation with Coulomb potential.

Findings

The functional decreases over time for solutions.

New regularity and blowup results are derived.

The approach offers a deeper understanding of solution behavior.

Abstract

We describe a time-dependent functional involving the relative entropy and the $\dot{H}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.