About the use of entropy production for the Landau-Fermi-Dirac equation

TL;DR

This paper develops new entropy dissipation estimates for the Landau-Fermi-Dirac equation, which are uniform with respect to the quantum parameter, aiding in understanding its long-term behavior and classical limit.

Contribution

It introduces novel entropy dissipation estimates using a weighted Fisher information, applicable to both quantum and classical Landau equations, with uniform bounds across parameters.

Findings

Estimates are uniform with respect to the quantum parameter.

New a priori estimates are provided for soft potentials.

Results apply to both quantum and classical Landau equations.

Abstract

In this paper, we present new estimates for the entropy dissipation of the Landau-Fermi-Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.