HYDRODYNAMIC LIMIT FOR THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: ANALYSIS OF THE TWO-DIMENSIONAL CASE

We consider the hydrodynamic limit for the VPFP system in dimension two, dealing with general initial data having finite mass, energy and entropy. The limit equation consists in a drift-diffusion equation, where the drift velocity is defined by means of the Poisson relation. Our result is twofold. In the case of repulsive (electrostatic) forces, we prove the convergence globally in time in a weak L1 setting. Considering attractive (gravitational) forces, the same result applies provided a certain scaling parameter is large enough. This is precisely the assumption which prevents the formation of Dirac masses in finite time in the limit equations, as recently shown by Dolbeault–Perthame.

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