Landau equation for very soft and Coulomb potentials near Maxwellians

TL;DR

This paper analyzes the Landau equation for very soft and Coulomb potentials near Maxwellian equilibria, establishing stability, existence, and rapid convergence of solutions in an improved functional framework.

Contribution

It introduces a new method for stability estimates of the linearized operator and extends convergence results to a broader class of initial data.

Findings

Proves stability estimates for the semigroup in large functional spaces

Establishes existence and uniqueness of solutions near equilibrium

Demonstrates faster convergence to Maxwellian equilibrium

Abstract

This work deals with the Landau equation for very soft and Coulomb potentials near the associated Maxwellian equilibrium. We first investigate the corresponding linearized operator and develop a method to prove stability estimates of its associated semigroup in large functional spaces. We then deduce existence, uniqueness and fast decay of the solutions to the nonlinear equation in a close-to-equilibrium framework. Our result drastically improves the set of initial data compared to the one considered by Guo and Strain who established similar results in [21, 37, 38]. Our functional framework is compatible with the non perturbative frameworks developed by Villani, Desvillettes and co-authors [42, 17, 16, 13], and our main result then makes possible to improve the speed of convergence to the equilibrium established therein.