On the semi-classical limit for the Landau-Fermi-Dirac equation

TL;DR

This paper investigates the transition from quantum to classical behavior in the Landau-Fermi-Dirac equation with Coulomb potential, proving solution convergence to a classical Landau equation with a defect measure as quantum effects vanish.

Contribution

It establishes the compactness and convergence of solutions of the Landau-Fermi-Dirac equation to a classical Landau solution with a defect measure in the semi-classical limit.

Findings

Solutions are compact in the semi-classical limit.

Solutions converge to a classical Landau equation with a defect measure.

The approach combines techniques from classical and quantum Landau equations.

Abstract

We study sequences of solutions to the inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential in which the quantum parameter converges to zero. Our main result establishes the compactness of these sequences, which allows us to show that, up to a subsequence, these solutions converge to a renormalized solution of the classical Landau equation with a defect measure, as defined by Villani. To do this, we work in the class of solutions that are obtained through approximation procedures. For these solutions, we were able to show compactness in the vanishing quantum parameter limit through a diagonal argument, which combines techniques from the study of Cauchy problems for both the classical Landau and the Landau-Fermi-Dirac equations.