ASYMPTOTIC-BEHAVIOR AND SELF-SIMILARITY FOR THE 3-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 di mensions. We consider both cases, with gravitational and electrostatic interaction, but disregard friction. It is proved that the distributi on of particles tends for large time to the fundamental solution of th e linear operator in L(1) norm, which means that the effect of the int eraction potential vanishes comparatively at t-->infinity. In quantita tive terms the result assures that the total nonlinear interaction for ce decays for large time with a decay rate of order t(-3) and the pote ntial energy behaves like O(t(-3/2)). The asymptotic result is indepen dent of the repulsive or attractive character of the interaction field . The main idea is to use the self-similarity of the fundamental solut ion of the linear part of the equation and the regularity of the Fokke r-Planck operator in order to study the large-time distribution of par ticles. (C) 1996 Academic Press. Inc.