Entropy dissipation estimates for the Landau equation in the Coulomb case and applications

TL;DR

This paper derives entropy dissipation bounds for the Landau equation with Coulomb interaction, leading to new regularity and moment propagation results, and extends these methods to soft potentials, offering alternative proofs for recent findings.

Contribution

It introduces a novel entropy dissipation estimate for the Coulomb case and applies it to establish regularity and moment propagation, also extending to soft potentials.

Findings

Weighted H^1 norm bounds for the square root of f

L^1_t(L^3_v) estimates for solutions

Propagation of L^1 moments of any order

Abstract

We present in this paper an estimate which bounds from below the entropy dissipation D(f) of the Landau operator with Coulomb interaction by a weighted H^1 norm of the square root of f. As a consequence, we get a weighted L^1_t(L^3_v) estimate for the solutions of the spatially homogeneous Landau equation with Coulomb interaction, and the propagation of L^1 moments of any order for this equation. We also present an application of our estimate to the Landau equation with (moderately) soft potentials, providing thus a new proof of some recent results of Kung-Chien Wu