Hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for gas mixture

TL;DR

This paper rigorously derives the hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for gas mixtures, establishing convergence and solution validity timeframes with novel estimates, applicable to high-altitude ionospheric gas flows.

Contribution

It introduces a new framework for analyzing asymmetric collision effects and provides rigorous convergence results for the hydrodynamic limit of the system.

Findings

Validity time of solutions depends on potential range and expansion order.

Constructs a new weight function for remainder estimates.

Applicable to high-altitude ionospheric gas dynamics.

Abstract

In this paper, we study the hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for a gas mixture in the whole space $(x \in \mathbb{R}^3)$ with the potential range of $\gamma \in\left(-3, 1\right]$. Using the method of Hilbert expansion, we first derive a bi-Maxwellian determined by the Euler-Poisson system of two fluids. To justify the convergence of the solution rigorously as the Knudsen number tends to zero, we sequentially calculate the first $2k-1$ terms of the expansion series $(k \geq 6)$, and then truncate it, and express the solution as the sum of these first $2k-1$ terms and a remainder term. Within the framework of the $L_{x,v}^2-W_{x,v}^{1,\infty}$ interplay established by Guo and Jang \cite{[ininp]Guo2010CMP}, we construct a new weight function to estimate the remainder term in four different cases regarding the potential $\gamma$. Here, the particle masses $m^A, m^B > 0$ and their charges $e^A, e^B$ can be given arbitrarily. This causes the collision operator to exhibit asymmetric effects ($m^A \neq m^B$), rendering the system of equations impossible to decouple. So, it adds difficulties to both $L^2$, $L^{\infty}$ estimates for the remainder. Therefore, we adopt the framework of vector-valued functions and analyze the velocity decay rate of the operator $K_{M,2,w}^{\alpha,c}$ to eliminate the singularity induced by small parameters in characteristic line iterations. Our results show that the validity time of the solution is $O(\varepsilon^{-y})$, where $y$ is $-\frac{2k-3}{2(2k-1)}$ when $-1 \leq \gamma \leq 1$, and it becomes $-\frac{2k-3}{(1-\gamma)(2k-1)}$, when $-3 < \gamma < -1$. These results possess strong physical realism and can be applied to analyze gas flow dynamics in the daytime ionosphere at high altitudes above the Earth.