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The Vlasov-Poisson-Fokker-Planck system with infinite kinetic energy. (English) Zbl 0926.35004
The author studies solutions of the Vlasov-Fokker-Planck-Poisson system in the three-dimensional case with infinite kinetic energy. The system describes an ensemble of particles which interact via the electrostatic or gravitational field which they create collectively. In the Vlasov-Fokker-Planck equation the transport operator \(\partial_t + v\cdot \partial_x\) and the Fokker-Planck operator simulating frictional and collisional effects act on the particle number density in phase space, in addition to the nonlinear interaction term via the field.
The motivating question for this paper is which of the two effects, dispersion via transport or diffusion via the Fokker-Planck term, dominates the time evolution. The result is that for short times the transport term dominates while after the initial regularization due to dispersion the diffusive term dominates. Exploiting this a global in time existence result for initial data with infinite kinetic energy is established. An important tool in the analysis is a new identity or conservation law of a type which was recently discovered by several authors for other models in kinetic theory.

MSC:
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82D15 Statistical mechanics of liquids
35Q72 Other PDE from mechanics (MSC2000)
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