On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions

TL;DR

This paper rigorously confirms Boltzmann's hypotheses for the Landau equation with harmonic potential, demonstrating that entropy-invariant solutions are time-periodic Maxwell-Boltzmann distributions and establishing their nonlinear stability with optimal convergence rates.

Contribution

It provides the first rigorous proof that entropy-invariant solutions are time-periodic Maxwell-Boltzmann distributions and proves their nonlinear stability in this context.

Findings

Entropy-invariant solutions are time-periodic Maxwell-Boltzmann distributions.

These distributions are characterized by thirteen conservation laws.

They exhibit nonlinear stability with optimal convergence rates.

Abstract

We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper \cite{Boltzmann} within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each {\it entropy-invariant solution} can be identified as a {\it time-periodic Maxwell-Boltzmann distribution}. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} is nonlinearly stable, including neutral asymptotic stability and Lyapunov stability. Furthermore, the convergence rate is entirely reliant on the thirteen conservation laws and is optimal when compared to the linear scenario.