Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system. (English) Zbl 1198.35273
Summary: 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 globally in time to the corresponding solution to the Euler-Poisson system, as the mean free path \(\varepsilon \) goes to zero. We use a recent \(L^2\)-\(L^\infty\) framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in \(\varepsilon \) and globally in time.
MSC:
| 35Q83 | Vlasov equations |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 82D10 | Statistical mechanics of plasmas |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 35A35 | Theoretical approximation in context of PDEs |
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