Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials

TL;DR

This paper proves that solutions to the homogeneous Landau equation with hard potentials converge exponentially fast to equilibrium, improving previous polynomial convergence results by establishing optimal spectral gap-based rates.

Contribution

It introduces new decay estimates for the linearized Landau operator's semigroup, enabling the proof of exponential convergence to equilibrium with optimal rates.

Findings

Exponential convergence to equilibrium established

Optimal rate given by spectral gap

Improved from previous polynomial convergence

Abstract

This paper deals with the long time behaviour of solutions to the spatially homogeneous Landau equation with hard potentials . We prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearized operator. This result improves the polynomial in time convergence obtained by Desvillettes and Villani \cite{DesVi2}. Our approach is based on new decay estimates for the semigroup generated by the linearized Landau operator in weighted (polynomial or stretched exponential) $L^p$-spaces, using a method develloped by Gualdani, Mischler and Mouhot \cite{GMM}.