On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials

TL;DR

This paper studies how solutions to the homogeneous Landau equation with soft potentials approach equilibrium, establishing polynomial and exponential convergence rates using entropy methods and spectral analysis.

Contribution

It introduces new a priori estimates and decay estimates for the linearized Landau operator, achieving optimal exponential convergence rates.

Findings

Polynomial convergence to equilibrium proven via entropy methods.

Exponential convergence with optimal spectral gap rate established.

New decay estimates for the semigroup in weighted L^p-spaces.

Abstract

We investigate in this work the rate of convergence to equilibrium of solutions to the spatially homogeneous Landau equation with soft potentials. Firstly, we prove a polynomial in time convergence using an entropy method with some new a priori estimates. Finally we prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearized operator, combining new decay estimates for the semigroup generated by the linearized Landau operator in weighted $L^p$-spaces togheter with the polynomial decay described above.