On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions
Abstract
We provide the first and rigorous confirmations of the hypotheses by Ludwig Boltzmann in his seminal paper Boltzmann within the context of the Landau equation in the presence of a harmonic potential. We prove that (i) Each entropy-invariant solution can be identified as a time-periodic Maxwell-Boltzmann distribution. Moreover, these distributions can be characterized by thirteen conservation laws, which sheds light on the global dynamics. (ii) Each 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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