# Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials

## TL;DR

This paper investigates the long-term behavior of the homogeneous Landau-Fermi-Dirac equation with hard potentials, establishing exponential convergence to equilibrium and uniform estimates that hold across quantum and classical regimes.

## Contribution

It provides uniform in time estimates for moments and Sobolev regularity, proving exponential relaxation to Fermi-Dirac equilibrium for general initial data, independent of the quantum parameter.

## Key findings

- Exponential relaxation to Fermi-Dirac statistics.
- Uniform estimates valid for general initial data.
- Results recover classical Landau equation in the limit.

## Abstract

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for rather general initial datum. An important feature of the estimates is the independence with respect to the quantum parameter. Consequently, in the classical limit the same estimates are recovered for the Landau equation.