The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Abstract

The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In the paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in [59] that we can be able to develop into the case for the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a nonconstant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system. 10 2.3. Diffusion and heat-conductivity 15 3. Quasineutral Euler equations and rarefaction waves 20 4. Preliminary estimates on two-component collision operator 23 5. Proof of the main result 27 6. A priori estimates on the fluid part 32 6.1. Estimate on zero-order energy 32 6.2. Estimate on first-order dissipation 37 6.3. Estimate on first-order energy 44 7. A priori estimates on the non-fluid part 51 7.1. Estimate on zero-order dissipation 51 7.2. Estimate on high-order energy 55 7.3. Estimate on energy with mixed derivatives 60 References 61 2010 Mathematics Subject Classification. Primary: 35Q20, 76P05; Secondary: 35B35, 35B40.

Stability of the Superposition of a Viscous Contact Wave with Two Rarefaction Waves to the Bipolar Vlasov--Poisson--Boltzmann System

“…Recently, Li-Wang-Yang-Zhong [26] gave a unified micro-macro decomposition to the bipolar VPB system and proved the stability of rarefaction wave and viscous shock profile to the bipolar VPB system by this new decomposition. Then, Duan-Liu [6] generalized the result in [5] to the bipolar VPB system with the disparate mass and Huang-Liu [24] showed the stability of a single viscous contact wave case.…”

BGK model for multi-component gases near a global Maxwellian

“…In [16], the author constructed the global-in-time mild solution near-global equilibrium for the mixture Boltzmann equation. The Vlasov-Poisson-Boltzmann equation was considered in [30] about large time asymptotic profiles when the different-species gases tend to two distinct global Maxwellians. In [32], the existence and uniqueness are constructed in spatially homogeneous settings when an initial data has upper and lower bounds for some polynomial moments.…”