Existence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential

TL;DR

This paper proves the global existence, uniqueness, and convergence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential, demonstrating stability towards the Fermi-Dirac equilibrium.

Contribution

It establishes the first rigorous proof of smooth solution existence and convergence for this equation with Coulomb interactions.

Findings

Global-in-time existence and uniqueness of smooth solutions

Solutions converge to Fermi-Dirac equilibrium

Convergence is algebraic near equilibrium

Abstract

In this paper, we prove global-in-time existence and uniqueness of smooth solutions to the homogeneous Landau-Fermi-Dirac equation with Coulomb potential. The initial conditions are nonnegative, bounded and integrable. We also show that any weak solution converges towards the steady state given by the Fermi-Dirac statistics. Furthermore, the convergence is algebraic, provided that the initial datum is close to the steady state in a suitable weighted Lebesgue norm.