The Kinetic and Hydrodynamic Bohm Criteria for Plasma Sheath Formation

Abstract

The purpose of this paper is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criteria which are required for the formation. Bohm originally derived the (hydrodynamic) Bohm criterion from the Euler–Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from a kinetic point of view, and then Riemann derived it from the Vlasov–Poisson system. In this paper, we prove the solvability of boundary value problems of the Vlasov–Poisson system. In the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov–Poisson and Euler–Poisson systems. To clarify the relation, we also study the delta mass limit of solutions of the Vlasov–Poisson system.

Appendices

Reduction

In this section, we reduce the boundary value problem of (1.1a)–(1.1b) with conditions (1.1c), (1.1e), (1.1f), and (1.3) to the boundary value problem (1.1). For simplicity, we treat the reduction only for the completely absorbing boundary, i.e. $f_{b}=\alpha =0$.

Suppose that the former boundary value problem has a solution $(f,\phi )$ with $\partial _{x} \phi (x)<0$. It is sufficient to show that a value $\phi (0)$ is determined a priori by $f_{\infty }$. Indeed we can construct the solution of the former problem by solving the problem (1.1) with $\phi _{b}=\phi (0)$. Let us find an a priori value $\phi (0)$. Following the proof of Lemma 3.1, we see that f must be written by the form (2.9) even for the former problem. Substituting (2.9) into (1.3) with $\alpha =0$ yields the following condition:

$$\begin{aligned} n_{e}(\phi (0)) v_{e}=\int _{\mathbb R^{3}} \xi _{1} f_{\infty }(-\sqrt{\xi _{1}^{2}-2\phi (0)},\xi ')\chi (\xi _{1}^{2}-2\phi (0))\chi (-\xi _{1}) {\text {d}}\xi . \end{aligned}$$

By solving this with respect to $\phi (0)$, we can know the a priori value $\phi (0)$. Consequently, the former problem can be reduced to the problem (1.1) with $\phi _{b}=\phi (0)$.⁸

Estimates of $\rho ^{\pm }$

This section provides the proofs of estimates (2.20) and (2.21). First, we can obtain (2.20) by using the Hölder inequality as:

$$\begin{aligned} |\rho _{i}^{+}(\phi )|&\leqq \Vert f_{\infty }\Vert _{L^{1}(\mathbb R^{3})} +C \int _{\sqrt{2(\phi _b-\phi )\chi (\phi _b-\phi )}}^{\sqrt{2\phi _b}} \frac{\xi _1}{\sqrt{\xi _1^2+2\phi -2\phi _b}} \left( \int _{\mathbb R^2} f_b(\xi )\, \,{\text {d}}\xi ' \right) {\text {d}}\xi _1 \\&\leqq \Vert f_{\infty }\Vert _{L^{1}(\mathbb R^{3})} \\&\quad +C \left( \int _{\sqrt{2(\phi _b-\phi )\chi (\phi _b-\phi )}}^{\sqrt{2\phi _b}} \frac{|\xi _1|^{r'}}{({\xi _1^2+2\phi -2\phi _b})^{r'/2}} {\text {d}}\xi _1 \right) ^{\!\!1/r'} \!\!\Vert f_{b}\Vert _{L^{r}(0,\sqrt{2\phi _{b}};L^{1}(\mathbb R^{2}))} \\&\leqq \Vert f_{\infty }\Vert _{L^{1}(\mathbb R^{3})} + C\Vert f_{b}\Vert _{L^{r}(0,\sqrt{2\phi _{b}};L^{1}(\mathbb R^{2}))} \end{aligned}$$

for $\phi \geqq 0$, where $r'<2$ is the Hölder conjugate of $r>2$. Similarly, we observe that for $\phi \in [-M,0]$,

$$\begin{aligned} |\rho _{i}^{-}(\phi )|&= \int _{{\mathbb R}^3}f_\infty (\xi )\frac{|\xi _1|}{\sqrt{\xi _1^2+2\phi }} \chi (\xi _1^2-4M)\,{\text {d}}\xi \\&\quad +\int _{{\mathbb R}^3}f_\infty (\xi )\frac{|\xi _1|}{\sqrt{\xi _1^2+2\phi }} \{\chi (\xi _1^2+2\phi )-\chi (\xi _1^2-4M)\}\,{\text {d}}\xi \\&\leqq \sqrt{2}\Vert f_{\infty }\Vert _{L^{1}(\mathbb R^{3})} + \int _{-2\sqrt{M}}^{2\sqrt{M}} \frac{|\xi _1|}{\sqrt{\xi _1^2+2\phi }} \chi (\xi _1^2+2\phi ) \ \left( \int _{{\mathbb R}^2}f_\infty (\xi ) {\text {d}}\xi ' \right) {\text {d}}\xi _{1} \\&\leqq \sqrt{2}\Vert f_{\infty }\Vert _{L^{1}(\mathbb R^{3})} +C_{M}\Vert f_{\infty }\Vert _{L^{r}(-2\sqrt{M},2\sqrt{M};L^{1}(\mathbb R^{2}))}. \end{aligned}$$

Thus (2.21) holds. The proofs are complete.

Properties of $V^{\pm }$

We investigate properties of the functions $V^{\pm }$ defined in (2.16) and (2.17).

Lemma C1

(Attractive boundary) Let $\phi _b>0$, $\alpha \ne 1$, and $f_b\in L^1({\mathbb R}_+^3)$ satisfy $f_{b} \geqq 0$ and

$$\begin{aligned} f_b(\xi )=0, \quad 0<\xi _1<c_0 \end{aligned}$$

(C.1)

for some $c_0>0$. Suppose that $f_{\infty } \in L^{1}(\mathbb R^{3})$ satisfies $f_{\infty } \geqq 0$, (2.2), (2.3), and (2.8). Then there exists a positive constant $\delta$ such that if $0<\phi _{b}<\delta$, the function $V^{+}$ belongs to $C^{2}([0,\phi _{b}])$ and satisfies ${d^{2} V^{+}}/{d \phi ^{2}}(0)>0$.

(Repulsive boundary) Let $\phi _b<0$, $f_b\in L^1({\mathbb R}_+^3)$, and $f_{b} \geqq 0$. Suppose that $f_{\infty } \in L^{1}(\mathbb R^{3})$ satisfies $f_{\infty } \geqq 0$, (2.2), (2.4), (2.8), and

$$\begin{aligned} f_\infty (\xi )=0, \quad - c_0<\xi _1<c_0 \end{aligned}$$

for some $c_0>0$. Then there exists a constant $\delta$ such that if $-\delta<\phi _{b}<0$, the function $V^{-}$ belongs to $C^{2}([\phi _{b},0])$ and satisfies ${d^{2} V^{-}}/{d \phi ^{2}}(0)>0$.

Proof

We show only the assertion for the attractive boundary, since the other one can be shown similarly. Recalling the definition of $\rho _{i}^{+}$ in (2.18), we see from (C.1) that the second term vanishes for sufficiently small $\phi _{b}$. Therefore, we can complete the proof by following the same argument as in the last paragraph of the proof of Lemma 3.1, and also noting that

$$\begin{aligned} \frac{d^{2} V^{+}}{d \phi ^{2}}(0) =-\int _{\mathbb R^{3}}\xi _{1}^{-2}f_{\infty }(\xi ) {\text {d}}\xi +1>0, \end{aligned}$$

where we have used (2.8) in deriving the last inequality.$\quad \square$