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

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 $(k\ge 6)$, 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 ($m^A\ne m^B$), 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 $O({\epsilon }^{-y})$, where $y$ is $-\frac{2k-3}{2(2k-1)}$ when $-1\le \gamma \le 1$, 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.

72pages