About the Landau-Fermi-Dirac equation with moderately soft potentials

TL;DR

This paper investigates the Landau-Fermi-Dirac equation with moderately soft potentials, establishing key properties of solutions, including regularity, moment bounds, and convergence to equilibrium, with results applicable to the classical Landau equation.

Contribution

It provides new uniform estimates and relaxation results for solutions, extending techniques like De Giorgi level set analysis and entropy inequalities to this quantum kinetic context.

Findings

Uniform in time moment bounds

Generation of $L^{p}$ norms and Sobolev regularity

Algebraic relaxation towards Fermi-Dirac equilibrium

Abstract

We present in this document some essential properties of solutions to the homogeneous Landau-Fermi-Dirac equation for moderately soft potentials. Uniform in time estimates for statistical moments, $L^{p}$-norm generation and Sobolev regularity are shown using a combination of techniques that include recent developments concerning level set analysis in the spirit of De Giorgi and refined entropy-entropy dissipation functional inequalities for the Landau collision operator which are extended to the case in question here. As a consequence of the analysis, we prove algebraic relaxation of non degenerate distributions towards the Fermi-Dirac statistics under a weak non saturation condition for the initial datum. All quantitative estimates are uniform with respect to the quantum parameter. They therefore also hold for the classical limit, that is the Landau equation.