数理解析研究所講究録
第 1905 巻 2014 年 180-187

180

Asymptotic stability of stationary solutions
to the Euler-Poisson equations
for a multicomponent plasma
鈴木政尋
MASAHIRO SUZUKI$*$

東京工業大学大学院情報理工学研究科
GRADUATE SCHOOL 0F INFORMATION SCIENCE AND ENGINEERING,
ToKYO INSTITUTE OF TECHNOLOGY

1Introduction

This short paper is concerned with boundary layers of a multicomponent plasma which
consists of electrons and several positive ion species. The motion of the multicomponent
plasma governed by the Euler equations for the ion density
and the ion velocityof the i-th component:$iS$$u_{i}$$\rho_{i}$$(\rho_{i})_{t}+(\rho_{i}u_{i})_{x}=0$(l.la),$i=1$ , . . . ,$m_{i}(u_{i})_{t}+ \frac{m_{i}}{2}(u_{i}^{2})_{x}+\frac{1}{\rho_{i}}(p_{i}(\rho_{i}))_{x}=e_{i}\phi_{x},$coupled with the Poisson equation for the electrostatic potential$\epsilon_{0}\phi_{xx}=\sum_{i=1}^{k}e_{i}\rho_{i}-e_{0}\rho_{0}(\phi)$.

$-\phi$$k$,(l.lb):(l.lc)The positive constantsand denote the mass and the charge of the i-th ion, respectively. In addition,is permittivity. The pressure is assumed to be a function of theelectron density given by$m_{i}$$e_{i}$$p_{i}$$\epsilon_{0}$$\rho_{i}$$p_{i}(\rho_{i})=\kappa T_{i}\rho_{i},$lmasahiro@is.titech.ac.jp

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where is the Boltzmann constant and is the temperature of the i-th ion. We assume
that the electron density
obeys the Boltzmann relation, that is,$T_{i}$$\kappa$$\rho_{0}$$\rho_{0}(\phi)=\rho_{0+}\exp(-\frac{e_{0}\phi}{\kappa T_{0}})$,where the positive constants
anddenote the reference density value, the charge
and the temperature of the electron, respectively.
We study an initial-boundary value problem to (1.1) over a one-dimensional half space$\mathbb{R}_{+}:=\{x>0\}$ , where the initial and the boundary data are prescribed as$\rho_{0+},$$(\rho_{i}, u_{i})(0, x)=(\rho_{i0}, u_{i0})(x)$,$T_{0}$$e_{0}$$\lim_{xarrow\infty}(\rho_{i0}, u_{i0})(x)=(\rho_{i+}, u_{i+})$$\phi(t, 0)=\phi_{b}$Here,$\rho_{i+},$$x=\infty$$u_{i+}$

and$\phi_{b}$,$i=1$ , . . . ,

$k$

,

.

(1.2)

(1.3)

are constants. We take a reference point of the potential$\phi$

at, that is,$\lim_{xarrow\infty}\phi(t, x)=0.$

In order to solve the Poisson equation (l.lc) in classical sense, the quasi-neutrality condition is required:$\sum_{i=0}^{k}e_{i}\rho_{i+}-e_{0}\rho_{0+}=0$

.

(1.4)

The solution of this initial-boundary value problem is constructed in the region where
the positivity of the density (1.5) and the supersonic outflow condition (1.6) hold, that is,$\inf_{x\in\pi_{+}}\rho_{i}>0$$\inf_{x\in \mathbb{R}_{+}}(m_{i}u_{i}^{2}-\kappa T_{i})>0,$

for $i=1,$$\sup u_{i}<0$

$k$

,

(1.5)

for $i=1,$

$k$

.

(1.6)

$x\in\pi_{+}$

Therefore we suppose that the initial data satisfies the same conditions:$\inf_{x\in \mathbb{R}_{+}}\rho_{i0}>0, \inf_{x\in\pi_{+}}(m_{i}u_{i0}^{2}-\kappa T_{i})>0, \sup_{x\in \mathbb{R}+}u_{i0}<0$$\rho_{i+}>0, m_{i}u_{i+}^{2}-\kappa T_{i}>0, u_{i+}<0$

.

,

(1.7)
(1.8)

Before we close this section, we briefly discuss about the physical background of ourproblem and the related mathematical works. A boundary layer problem occurs in plasma
devices when the plasma contacts with a surface. Due to the diﬀerence of the mobilities
of electrons and positive ions, the surface has a negative potential with respect to theplasma. The non-neutral potential region between the plasma and the surface is called asheath. The readers are referred to [4,5,7]. For the sheath formation, physical observation

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requires that the positive ions must enter the sheath region with a high velocity. Bohm
in [3] treated a simple case when the plasma contains electrons and only one componentof mono-valence ions and derived the original Bohm criterion for the velocity :$u_{1}$$\kappa T_{0}+\kappa T_{1}<m_{1}u_{1+}^{2}, u_{1+}<0$

.

(1.9)

By studying the stationary problem of the system (1.1), Riemann in [8] obtained thegeneralized Bohm criterion for the multicomponent plasma. This criterion claims thatthe velocity of positive ions should satisfy (1.8) and$B_{+}:=- \sum_{i=1}^{k}\frac{e_{i}^{2}\rho_{i+}}{m_{i}u_{i+}^{2}-\kappa T_{i}}+\frac{e_{0}^{2}\rho_{0+}}{\kappa T_{0}}>0$.

(1.10)Let us mention mathematical results which study the sheath formation and the original
Bohm criterion (1.9). Ambroso, M\’ehats and P.-A. Raviart in [2] showed the existence ofthe monotone stationary solution to (1.1) with $k=1$ under (1.9) over a one-dimensionalbounded domain. Later Ambroso in [1] numerically showed that the solution to (1.1)
approaches the stationary solution as time tends to infinity in the same setting as in [2].
Suzuki in [9] interpreted the sheath to be a monotone stationary solution to (1.1) with
$k=1$ over a one-dimensional half space and showed that the Bohm criterion is suﬃcientfor the unique existence of the monotone stationary solution. In [6], the asymptotic
stability of the stationary solution is proved under (1.9). Consequently, these results
ensure the mathematical validity of the original Bohm criterion (1.9).
In this short paper, we briefly study the rigorous justification of the generalized Bohm
criterion (1.10). More precisely, we introduce the existence and the stability theoremson the stationary solution to the system (1.1) for the multicomponent plasma. For thedetailed discussion on this research, please see the paper [10].

2Unique existence of the stationary solution

This section is devoted to the discussion on the unique existence of the monotone
is a solution to (1.1)
stationary solution. The stationary solution
independent of the time variable . Hence, it verifies$(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k},\tilde{\phi})$$t$$(\tilde{\rho}_{i}\tilde{u}_{i})_{x}=0$,$\frac{m_{i}}{2}(\tilde{u}_{i}^{2})_{x}+\frac{\kappa T_{i}}{\tilde{\rho}_{i}}(\tilde{\rho}_{i})_{x}=6_{i}\tilde{\phi}_{x},$(2.1a)$i=1$ , . . . ,$\epsilon_{0}\tilde{\phi}_{xx}=\sum_{i=1}^{k}e_{i}\tilde{\rho}_{i}-e_{0}\rho_{0+}\exp(-\frac{e_{0}\tilde{\phi}}{\kappa T_{0}})$$k$,(2.1b)(2.1c)

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and the conditions $(1.2)-(1.6)$ :$\lim_{xarrow\infty}(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k})(x)=(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+})$$\tilde{\phi}(0)=\phi_{b}, \lim_{xarrow\infty}\tilde{\phi}(x)=0$$\inf_{x\in\pi_{+}}\tilde{\rho}_{i}>0,$,(2.2b)$i=1$ , . . . ,$\sup\tilde{u}_{i}<0,$$\inf_{x\in \mathbb{R}+}(m_{i}\tilde{u}_{i}^{2}-\kappa T_{i})>0,$(2.2a),$k$.

(2.2c)$x\in\pi_{+}$The key of the proof of the existence theorem is reduction of the system (2.1) to ascalar equation for . Assuming the existence of the monotone solution satisfying$\tilde{\phi}$(2.3)$(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k})\in C^{1}(\mathbb{R}_{+}) , \tilde{\phi}\in C(\overline{\mathbb{R}_{+}})\cap C^{2}(\mathbb{R}_{+})$

to the stationary problem (2.1) and (2.2), we drive the scalar equation which
Integrating (2.1a) overgives$\tilde{\phi}$satisfies.

$(x, \infty)$

for $i=1$ , . . . , .

(2.4)

$k$

$\tilde{u}_{i}(x)=\frac{\rho_{i+}u_{i+}}{\tilde{\rho}_{i}(x)}$Substitute (2.4) in (2.1b), divide the result by$e_{i}\tilde{\phi}(x)=f_{i}(\tilde{\rho}_{i}(x))$

where$f_{i}$$\tilde{\rho}$and then integrate over$(x, \infty)$

to obtain

for $i=1$ , . . . , ,

(2.5)

$k$

is defined by$f_{i}( \tilde{\rho}_{i}):=\kappa T_{i}\log\tilde{\rho}_{i}+m_{i}\frac{\rho_{i+}^{2}u_{i+}^{2}}{2\tilde{\rho}_{i}^{2}}-\kappa T_{i}\log\rho_{i+}-m_{i}\frac{u_{i+}^{2}}{2}.$By restricting the domain $D(f_{i})$ into
is invertible. Then it holds that$(0, \rho_{i+}M_{i+}$$\tilde{\rho}_{i}(x)=f_{i}^{-1}(e_{i}\tilde{\phi}(x))$], where$M_{i+}^{2}=m_{i}u_{i+}^{2}/\kappa T_{i}$, we see thatfor $i=1$ , . . . , .

(2.6)

$k$

Substitute (2.6) in (2.1c), multiply the resultant equation by$(x, \infty)$ and then use the condition (2.2a) andequation for :$\tilde{\phi}_{x}$$f_{i}$, integrate the result overto obtain the scalar$\lim_{xarrow\infty}\tilde{\phi}_{x}(x)=0$$\phi$$\frac{\epsilon_{0}}{2}(\tilde{\phi}_{x})^{2}=V(\tilde{\phi}) , V(\tilde{\phi}):=\int_{0}^{\tilde{\phi}}\sum_{i=1}^{k}e_{i}f_{i}^{-1}(e_{i}\eta)-e_{0}\rho_{0+}\exp(-\frac{e_{0}\eta}{\kappa T_{0}})d\eta$

,

(2.7)

where is called as the Sagdeev potential in plasma physics. This equation requires thenecessary condition
On the other hand, if the problem (2.7) and (2.2b) has a monotone solution$C()\cap C^{2}(\mathbb{R}_{+})$ , then we can easily check that$V$

$V(\phi_{b})\geq 0.$$\tilde{\phi}\in$$(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k},\tilde{\phi}):=(f_{1}^{-1}(e_{1}\tilde{\phi}),$$\frac{\rho_{1+}u_{1+}}{f_{1}^{-1}(e_{1}\tilde{\phi})},$...,$f_{k}^{-1}(e_{k}\tilde{\phi}),$$\frac{\rho_{k+}u_{k+}}{f_{k}^{-1}(e_{k}\tilde{\phi})},\tilde{\phi})$184
is a monotone stationary solution to (2.1) and (2.2). The uniqueness of the monotone
stationary solution to (2.1) and (2.2) also follows from the uniqueness of the solution to(2.7) and (2.2b).
Hence, it is suﬃcient to show the unique solvability of the problem (2.7) and (2.2b).
We can solve this problem by virtue of the standard ODE theory. The unique existence
theorem is as follows.

Theorem 2.1. Let the asymptotic state$(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+})$satisfy (1.4) and (1.8).

(i) Suppose $B+>$ O. Then there exists a certain positive constant such that if, the stationary problem (2.1) and (2.2) has a unique monotone stationary solution
verifying (2.3).
(ii) Suppose $B_{+}=$ O. Then there exists a certain positive constant such that if
and $V(\phi_{b})\geq 0$ , the stationary problem (2.1) and (2.2) has a unique monotone
verifying (2.3).
stationary solution, no stationary solution verifying (2.3) exists. If(iii) Suppose $B+<$ O. If$\phi_{b}=0$ , a constant sateis the uniquestationary solution.
$\delta$$|\phi_{b}|\leq\delta$$(\tilde{\rho}_{1},\tilde{u}_{1}, .

.

.

, \tilde{\rho}_{k},\tilde{u}_{k},\tilde{\phi})$$\delta$$|\phi_{b}|\leq\delta$$(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k},\tilde{\phi})$$\phi_{b}\neq 0$$(\tilde{\rho}_{1},\tilde{u}_{1}, \ldots,\tilde{\rho}_{k},\tilde{u}_{k},\tilde{\phi})=(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+}, 0)$

The author in [9] constructed non-monotone stationary solutions for the case $k=1$ . Thus
the monotonicity is necessary to show the uniqueness.

3

Asymptotic stability of the stationary solution

Before stating our stability theorem, let us mention diﬃculties of our stability analysis.
For notational convenience, we introduce the perturbation from the asymptotic stateas$(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+}, 0)$$\psi_{i}:=\rho_{i}-\rho_{i+},$$i=1$ , . . . , $k,$$\eta_{i}:=u_{i}-u_{i+},$$\sigma:=\phi-\tilde{\phi}.$Linearizing the system (1.1) around the asymptotic state gives$\psi_{it}+u_{i+}\psi_{ix}+\rho_{i}+\eta_{ix}=0$,(3.1a)$i=1$ , . . . ,$\eta_{it}+u_{i+}\eta_{ix}+\frac{\kappa T_{0}}{m_{i}\rho_{i+}}\psi_{ix}=\frac{e_{i}}{m_{i}}\sigma_{x},$$\epsilon_{0}\sigma_{xx}-\frac{e_{0}^{2}\rho_{0+}}{\kappa T_{0}}\sigma=\sum_{i=1}^{k}e_{i}\psi_{i}$

$k$

,

.

(3.1b)(3.1c)

Notice that the real part of all spectra of this system is zero under the assumption$u_{+}=u_{1+}=\cdots=u_{k+}$

.

(3.2)

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This causes our problem to be diﬃcult since standard methods are not applicable. For
overcoming this issue, we employ the weighted Sobolve space with a weight function$(1+\beta x)^{\lambda}$or$e^{\beta x}.$We briefly discuss about eﬀectiveness of the weighted Sobolve space in our analysis
and introduce new unknown function
under the criterion (1.10). Multiply (3.1) by. Moreover, rewriting the system (3.1) forandand gives$e^{\beta x/2}$$P_{i}$$:=e^{\beta x/2}\psi_{i},$$Q_{i}:=e^{\beta x/2}\eta_{i}$$R:=e^{\beta x/2}\sigma$$P_{i},$$R$$Q_{i}$$P_{it}+u_{i+}P_{ix}+ \rho_{i+}Q_{ix}-\frac{\beta}{2}(u_{i+}P_{i}+\rho_{i+}Q_{i})=0$,$Q_{it}+u_{i+}Q_{ix}+ \frac{\kappa T_{0}}{m_{i}\rho_{i+}}P_{ix}-\frac{\beta}{2}(u_{i+}Q_{i}+\frac{\kappa T_{0}}{m_{i}\rho_{i+}}P_{i})=\frac{e_{i}}{m_{i}}R_{x}-\frac{\beta e_{i}}{2m_{i}}R,$(3.3a)$i=1$ , . . . , $h,$

(3.3b)$\epsilon_{0}(Q_{xx}-\beta Q_{x}+\frac{\beta^{2}}{4}Q)-\frac{e_{0}^{2}\rho_{0+}}{\kappa T_{0}}Q=\sum_{i=1}^{k}e_{i}P_{i}.

(33c)$

By applying spectral analysis to the system (3.3), we have
Proposition 3.1. Let the asymptotic state
Then the following two conditions are equivalent:$(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+})$(i) The real part of all spectra of (3.3) in the whole space
O.
small

$\mathbb{R}$satisfy (1.8) and (3.2).

is negative for suficiently$\beta>$(ii) The generalized Bohm criterion (1.10) holds.

Although our problem is the boundary value problem, Proposition 3.1 implies thatthe weighted Sobolve space is useful in our stability analysis. The stability theorem issummarized in Theorem 3.2. The proof is based on the combination of the weighted
energy method and Fourier analysis.
Theorem 3.2. Let the asymptotic state(1.4), (1.8), (1.10) and $u_{1+}=\cdots=u_{k+}.$$(\rho_{1+}, u_{1+}, \ldots, \rho_{k+}, u_{k+})$satisfy the conditions

belong to the Sobolve space(i) Suppose thatandfor $i=1$ , . . . , , where is some positive constant. Then there exist positive constants
and such that if$e^{\alpha x/2}(\rho_{i0}-\tilde{\rho}_{i})$$k$$\beta(\leq\alpha)$$e^{\alpha x/2}(u_{i0}-\tilde{u}_{i})$$\alpha$$\delta$$| \phi_{b}|+\sum_{i=1}^{k}\Vert(e^{\beta x/2}(\rho_{i0}-\tilde{\rho}_{i}), e^{\beta x/2}(u_{i0}-\tilde{u}_{i}))\Vert_{H^{2}}\leq\delta,$$H^{2}(\mathbb{R}_{+})$

186
the initial-boundary value problem (1.1) $-(1.3)$ has a unique solution
satisfying

for $i=1$ , . . . ,$(e^{\beta x/2}( \rho_{i}-\tilde{\rho}_{i}), e^{\beta x/2}(u_{i}-\tilde{u}_{i}))\in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{2-j})$$e^{\beta x/2}( \phi-\tilde{\phi})\in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{4-j})$$(\rho_{1}, u_{1}, \ldots, \rho_{k}, u_{k}, \phi)$

$k,$

.

Moreover it verifies the decay estimate$\sup_{x\in \mathbb{R}+}|(\rho_{1}-\tilde{\rho}_{1}, u_{1}-\tilde{u}_{1}, \ldots, \rho_{k}-\tilde{\rho}_{k}, u_{k}-\tilde{u}_{k}, \phi-\tilde{\phi})(t)|\leq Ce^{-\gamma t},$where positive constants and are independent of the time variable(ii) Let and satisfy
Supposeandandbelong to the Sobolve spacefor $i=1$ , . . . , , where is somepositive constant. Then there exist positive constants
and such that if
$C$

$t.$

$\gamma$$\lambda\geq 2$$\lambda$$\nu$$\nu\in(0, \lambda].$$(1+\alpha x)^{\lambda/2}(u_{i0}-\tilde{u}_{i})$$that (1+\alpha x)^{\lambda/2}(\rho_{i0}-\tilde{\rho}_{i})$$k$$H^{2}(\mathbb{R}_{+})$$\beta(\leq\alpha)$$\alpha$$\delta$$| \phi_{b}|+\sum_{i=1}^{k}\Vert((1+\alpha x)^{\lambda/2}(\rho_{i0}-\tilde{\rho}_{i}), (1+\alpha x)^{\lambda/2}(u_{i0}-\tilde{u}_{i}))\Vert_{H^{2}}\leq\delta,$

the initial-boundary value problem (1.1) $-(1.3)$ has a unique solution
satisfying$((1+ \alpha x)^{\lambda/2}(\rho_{i}-\tilde{\rho}_{i}), (1+\alpha x)^{\lambda/2}(u_{i}-\tilde{u}_{i}))\in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{2-j})$$(1+ \alpha x)^{\lambda/2}(\phi-\tilde{\phi})\in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{4-j})$$(\rho_{1}, u_{1}, \ldots, \rho_{k}, u_{k}, \phi)$

for $i=1$ , .. . ,

$k,$

.

Moreover it verifies the decay estimate$\sup_{x\in \mathbb{R}+}|(\rho_{1}-\tilde{\rho}_{1}, u_{1}-\tilde{u}_{1}, \ldots, \rho_{k}-\tilde{\rho}_{k}, u_{k}-\tilde{u}_{k}, \phi-\tilde{\phi})(t)|\leq C(1+\beta t)^{-\lambda+\zeta}$

for an arbitrary$\zeta\in[v, \lambda]$, where the positive constant $C$ is independent of the time variable$t.$

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[3] D. BOHM, Minimum ionic kinetic energy for a stable sheath, in The characteristics ofelectrical discharges in magnetic fields, A. Guthrie and R.K.Wakerling eds., McGrawHill, New York, 1949, pp. 77-86.
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[5] M. A. LIEBERMAN AND A. J. LICHTENBERG, Principles of Plasma Discharges andMaterials Processing, Second ed., Wiley-Interscience, 2005.

[6] S. NISHIBATA, M. OHNAWA AND M. SUZUKI, Asymptotic stability of boundary
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$D$: Appl. Phys.

[8] K.-U. RIEMANN, The Bohm Criterion and Boundary Conditions for a Multicomponent System, IEEE Rans. Plasma Sci. 23 (1995), pp. 709-716.
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[10] M. SUZUKI, Asymptotic stability of a boundary layer to the Euler-Poisson equations
for a mpulticomponent plasma, to appear.