Cauchy problem and exponential stability for the inhomogeneous Landau equation

TL;DR

This paper studies the inhomogeneous Landau equation on the torus, establishing exponential decay for the linearized problem and proving exponential stability of solutions near equilibrium for various potential types.

Contribution

It provides the first comprehensive analysis of exponential decay and stability for the inhomogeneous Landau equation across different potential regimes.

Findings

Exponential decay estimates for the linearized Landau semigroup.

Construction of solutions close to equilibrium using linear decay.

Proof of exponential stability with near-optimal rate.

Abstract

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove a exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.