Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

Abstract

We study the Hilbert expansion for small Knudsen number $\epsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $F_0(t,x,v)=\frac{{\rho }_0(t,x)}{(2\pi {\theta }_0(t,x){)}^{3/2}}{e}^{-|v-u_0(t,x){|}^2/2{\theta }_0(t,x)},{\theta }_0(t,x)=K{\rho }_0^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\le t\le {\epsilon }^{-1/2\frac{2k-3}{2k-2}},$ where ${\rho }_0(t,x)$ and $u_0(t,x)$ satisfy the Euler-Poisson system for monatomic gas $\gamma =5/3$.

References in corpus (1)

• The Vlasov-Maxwell-Boltzmann System in The Whole Space

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