Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

Abstract

We study the Hilbert expansion for small Knudsen number $\varepsilon$ 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)},\text{\ }\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 $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $\rho_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson system for monatomic gas $\gamma=5/3$

Global Strong Solutions of the Vlasov–Poisson–Boltzmann System in Bounded Domains

“…For any large m ≫ 1 we define k ̺,m (v, u) = 1 |v−u|≥ 1 m ,|v|≤m k ̺ (v, u), (8.17) such that sup v´R3 |k ̺,m (v, u) − k ̺ (v, u)|du 1 m , and |k ̺,m (v, u)| m 1. Furthermore we split the time interval as, for each ℓ, l 18)…”

“…18), (4.23), and (4.24), the contribution of |Γ(∂f, f )| + |Γ gain (f, ∂f )| + |K∂f | of (4.43) in (4.44) G is bounded by…”

Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

“…Recently, a new model called the INSF system with viscous heating was derived by Bardos-Levermore-Ukai-Yang [4]. The aim of the present paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in a periodic box via an L 2 − L ∞ method developed in [12,13,21,22,23,24]. We now outline a few key points of the paper which are distinct to some extent with the previous work by Bardos-Levermore-Ukai-Yang [4]:…”

Stability of contact discontinuity for the Navier–Stokes–Poisson system with free boundary

“…On the other hand, the NSP system at the fluid level can be justified by taking the hydrodynamical limit of the Vlasov-type Boltzmann equation by the Chapman-Enskog expansion, cf. [4,17,18,19]. In recent years, there have been a great number of mathematical studies of the NSP system.…”