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On classical solutions of Vlasov-Poisson Fokker-Planck systems. (English) Zbl 0674.60097
The theory developed for collisionless Vlasov-Poisson models is extended to include Vlasov dynamics for models with collision effects. For such models, the motion of a particle is resolved into the influence of self- consistent forces and dynamical friction forces, due to the interaction with the surrounding medium or fluid, and into a fluctuating part characteristic of Brownian motion.
We derive global existence of classical solutions of arbitrary data when the spatial, or momentum, dimension is less than or equal to two, but local existence for arbitrary data in all other dimensions. This is effected, as in the collisionless cases, by obtaining a-priori estimates on the charge or mass density. The key to developing our theory lies in a procedure for constructing a fundamental solution to a class of degenerate, linear parabolic equations.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
82B05 Classical equilibrium statistical mechanics (general)
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