Stationary Solutions of the Vlasov–Poisson System with Diffusive Boundary Conditions

Abstract

The stationary solutions of the Vlasov–Poisson system for a plasma are studied with general diffusive boundary conditions. The distribution function $f(x,v)$, which depends on the local energy and angular momentum, is determined uniquely under certain integrability and decay assumptions on the diffusive kernels and the particle injection intensities. The resulting nonlinear Poisson equation is then solved for the electric potential $\phi (x)$. We study the existence and uniqueness of its solutions in one and higher dimensions under a variety of settings.

Appendix

Theorem 8

(Noussair 1979) Let $\Omega _{n}$, $n=1,2...$ be a sequence of bounded open sets in ${\mathbb {R}}^{N}$ with boundaries $\partial \Omega _{n}$ of class $C^{2+\alpha }$, $\alpha \in (0,1)$ such that $\Omega _{n}\subset \Omega _{n+1}\subset \Omega$ for all $n$, $\cup _{n=1}^{\infty }\Omega _{n}=\Omega$ and for each $x\in \partial \Omega$, $\left| x\right| \le n$ implies $x\in \partial \Omega _{n}$. Suppose that the boundary value problems

$$\begin{aligned} -L\phi ^{*}&\ge g(\phi ^{*})\text { in }\Omega ,\quad \phi ^{*}\ge c\text { on }\partial \Omega \text {,}\nonumber \\ -L\phi _{*}&\le g(\phi _{*})\text { in }\Omega ,\quad \phi _{*}\le c\text { on }\partial \Omega , \end{aligned}$$

(5.1)

have solutions $\phi ^{*},\phi _{*}\in C^{\alpha }(\Omega )$, where $L$ is a uniformly elliptic operator, $c$ is a constant number. Further assume that $g$ is a function in $C^{\alpha }([a,b])$ $(\alpha \in (0,1))\,(-\infty <a<b<\infty )$ and $g(t_{1})-g(t_{2})\ge -D(t_{1}-t_{2})$ holds for all $a\le t_{2}\le t_{1}\le b$ and some constant $D>0$. Then, the boundary value problem

$$\begin{aligned} L\phi =g(\phi )\text { in }\Omega ,\ \ \phi =c\;on\;\partial \Omega , \end{aligned}$$

has a solution $\phi \in C^{\alpha }(\Omega )$ satisfying $\phi _{*}\le \phi \le \phi ^{*}$.