Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

The dynamics of an electron gas in a constant ion background can be decribed by the Vlasov-Poisson-Boltzmann system at the kinetic level, or by the compressible Euler-Poisson system at the fluid level. We prove that any solution of the Vlasov-Poisson-Boltzmann system near a smooth local Maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the Euler-Poisson system, as the mean free path ε goes to zero. We use a recent L2 − L∞ framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in ε and globally in time.

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