Asymptotic Behaviour and Self-Similarity for the Three Dimensional Vlasov–Poisson–Fokker–Planck System

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 inL1norm, which means that the effect of the interaction potential vanishes comparatively att→∞. In quantitative terms the result assures that the total nonlinear interaction force decays for large time with a decay rate of ordert−3and the potential energy behaves likeO(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.