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

TL;DR

This work proves a rigorous hydrodynamic (Knudsen) limit for a two-species Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$ with potential exponent $\gamma\in(-3,1]$, allowing unequal molecular masses and arbitrary charges. Using a Hilbert expansion truncated at order $2k-1$ (with $k\ge6$) and an $L^2$–$L^{\infty}$ hybrid framework, the authors construct a bi-Maxwellian equilibrium governed by an isentropic Euler-Poisson system for two fluids and obtain sharp remainder estimates in both $L^2$ and weighted $L^{\infty}$ norms. A novel velocity-weight function $\omega_{\gamma}$ and a vector-valued collision operator enable control across four regimes: $\gamma=1$, $0\le\gamma<1$, $-1\le\gamma<0$, and $-3<\gamma<-1$, with the strongest results for the hard-sphere case and a shortened validity time for the soft-potential regime. The analysis yields explicit time scales $T_\gamma=O(\varepsilon^{-y})$ with $y$ depending on $\gamma$, and it addresses physically realistic ion-containing mixtures relevant to daytime ionospheric dynamics at high altitudes. This sets a rigorous foundation for multi-species kinetic-to-fluid transitions in ionized gases with self-consistent fields and mass-asymmetry effects.

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 $γ\in\left(-3, 1\right]$. 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 $γ$. 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 $K_{M,2,w}^{α,c}$ 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 $-1 \leq γ\leq 1$, and it becomes $-\frac{2k-3}{(1-γ)(2k-1)}$, when $-3 < γ< -1$. 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.

Figures (2)

• Figure 1: The collision between the nitrosyl ion and the oxygen ion
• Figure 2: The validity time of the solution is $O(\varepsilon^{-y})$

Theorems & Definitions (37)

• Remark 1.1
• Theorem 1.1
• Remark 1.2: $\mathbf{Criteria\,\, for\,\, classification}$
• Remark 1.3: $\mathbf{Different \,\,weight \,\,functions\,\,for\,\,the \,\,remainder \,\,term}$
• Remark 1.4: $\mathbf{Shortening \,\,of \,\,the\,\,validity\,\,time \,\,for \,-3<\gamma<-1}$
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3: [61]Jiang, [50]Wu2023JDE
• Remark 2.4