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Asymptotic behaviour and selfsimilarity for the three dimensional Vlasov-Poisson-Fokker-Planck system. (English) Zbl 0873.35066
Summary: The aim of this work is to study the asymptotic behaviour of global in time solutions of the Vlasov-Poisson-Fokker-Planck system in three dimensions. We consider both cases, with gravitational and electrostatic interaction, but disregard friction.
It is proved that the distribution of particles tends for large time to the fundamental solution of the linear operator in $$L^1$$ norm, which means that the effect of the interaction potential vanishes comparatively at $$t\to\infty$$. In quantitative terms the result assures that the total nonlinear interaction force decays for large time with a decay rate of order $$t^{-3}$$ and the potential energy behaves like $$O(t^{-3/2})$$. The asymptotic result is independent of the repulsive or attractive character of the interaction field. The main idea is to use the self-similarity of the fundamental solution of the linear part of the equation and the regularity of the Fokker-Planck operator in order to study the large-time distribution of particles.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 35B40 Asymptotic behavior of solutions to PDEs 78A35 Motion of charged particles
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