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arXiv · 2407.07659

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

Chuqi Cao·Ling-Bing He·Jie Ji
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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.

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Chuqi Cao, Ling-Bing He, Jie Ji. 2024-08-14. On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions. https://arxiv.org/abs/2407.07659

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