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On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation. (English) Zbl 1151.35077
Summary: The dynamics of dilute electrons and plasma can be modeled by Vlasov-Poisson-Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is $O\left({t}^{-\infty }\right)$, by using a method developed for the Boltzmann equation without external force in [L. Desvillettes and C. Villani, Invent. Math. 159, No. 2, 245–316 (2005; Zbl 1162.82316)]. In particular, the idea of this method is to show that the solution $f$ cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro-micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 82C40 Kinetic theory of gases (time-dependent statistical mechanics) 82C70 Transport processes (time-dependent statistical mechanics) 78A35 Motion of charged particles