Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma

The main concern of this paper is to analyze 
a boundary layer called a sheath that 
occurs on the surface of materials 
when in contact with a multicomponent plasma. 
For the formation of a sheath, 
the generalized Bohm criterion demands that 
ions enter the sheath region with a high velocity. 
The motion of a multicomponent plasma is governed 
by the Euler--Poisson equations, and a sheath is 
understood as a monotone stationary solution to those equations. 
In this paper, we prove the unique existence of 
the monotone stationary solution 
by assuming the generalized Bohm criterion. 
Moreover, it is shown that 
the stationary solution is time asymptotically stable 
provided that an initial perturbation 
is sufficiently small in weighted Sobolev space. 
We also obtain the convergence rate, 
which is subject to the decay rate of the initial perturbation, 
of the time global solution toward the stationary solution.


1.
Introduction. The theory of plasma systems, comprising electrons and several ion species, is important in various fields of plasma technology such as gas discharge, plasma-surface interactions, and nuclear fusion. Consequently, interest in the investigation of multicomponent plasmas has been increasing. In particular, we are concerned with the boundary layer problem for multicomponent plasmas. This problem occurs in plasma devices when the plasma is in contact with any surface. Owing to the difference in the mobility of electrons and ions, the surface has a negative potential with respect to the plasma. The non-neutral potential region between the plasma and the surface is called a sheath (for details, see [4,8,11]).
A plasma system for k components of ions can be described by the Euler equations for the ion density n i (t, x) and ion velocity u i (t, x) of the i-th component: The constants m i > 0 and e i = 0 denote the mass and charge of the i-th ion, respectively. In addition, ε > 0 is the permittivity. We assume that the pressure p i can be described by an adiabatic law, that is, where κ is the Boltzmann constant and C i is a positive constant. Moreover, T i , S i , γ i , and c vi are the temperature, entropy, adiabatic constant, and specific heat at constant volume of the i-th ion, respectively. The electron density n e is assumed to obey the Boltzmann relation n e = n e+ exp e κT e v .
Here, the constants n e+ > 0, e < 0, and T e > 0 are the reference density value, charge, and temperature of the electron, respectively. For the formation of a sheath, Langmuir [7] concluded that the positive ions must enter the sheath region with a high velocity. For the simple case wherein the plasma contains electrons and only one component of mono-valence ions, Bohm [3] derived the original Bohm criterion for the velocity u 1 as κT e + κγ 1 T 1 < m 1 u 2 1 , u 1 < 0.
Riemann [12] extended this criterion to multicomponent plasmas by studying the stationary problem of (1) over a half space. The generalized Bohm criterion is Note that (5a) does not uniquely define a criterion for the velocity of each ion component. This complexity arises from the interaction effects among multiple ions, which is implicitly provided through the electrostatic potential. We call (5b) the supersonic outflow condition. Let us mention mathematical results regarding the formation of a sheath and the original Bohm criterion (4). Ha and Slemrod [5] formulated the problem of sheath formation as a free boundary problem of (1) with k = 1 and T 1 = 0 assuming (4) on the boundary. They discussed the time global solvability but not the asymptotic behavior of the solution. It is reasonable to expect that the asymptotic state is given by a stationary solution, since a sheath is observed as a stationary boundary layer. The papers [1,2,9,13] studied this expectation for initial-boundary value problems of (1) with k = 1. Ambroso, Méhats, and Raviart [2] showed the unique existence of a monotone stationary solution over a one-dimensional bounded domain under assumption (4). The paper [1] numerically showed that the solution to (1) approaches the stationary solution constructed in [2] as the time variable becomes large. Suzuki [13] derived a necessary and sufficient condition, including (4), for the unique existence of a monotone stationary solution over a half space. Moreover, the stability of the stationary solution was shown under a condition slightly stronger than (4) in [13]. Recently, Nishibata, Ohnawa, and Suzuki [9] obtained the stability theorem under (4). These results rigorously clarify that a sheath is regarded as a stationary solution. Furthermore, they ensure the mathematical validity of the original Bohm criterion (4).
The main purpose of the present paper is the rigorous justification of the generalized Bohm criterion (5), because the generalized criterion is more important than the original criterion in plasma technology. More precisely, we extend the existence and stability theorems in [9,13] to (1) for multicomponent plasmas.
Outline of the paper. The remainder of this paper is organized as follows. In Section 2, we formulate an initial-boundary value problem of (1) over a half space and mention our main results on the unique existence and asymptotic stability of the stationary solution. The generalized Bohm criterion provides a sufficient condition for existence. Moreover, we obtain the stability theorem by assuming the additional condition u 1+ = . . . = u k+ . In Section 3, we begin detailed discussions on the proof of the unique existence of the stationary solution. Here, the stationary problem is reduced to a boundary value problem for a scalar equation for the potential. The reduced problem can be solved by a basic ordinary differential equation theory. Section 4 is devoted to the proof of the stability theorem. First, we discuss briefly the unique existence of the time local solution to the initial-boundary value problem. Second, we derive an a priori estimate with suitable weight functions to construct the time global solution. Furthermore, the a priori estimate gives the convergence rate, which is subject to the decay the rate of the initial perturbation, of the global solution toward the stationary solution.
Notation. For a non-negative integer l, H l (Ω) denotes the l-th order Sobolev space in the L 2 sense, equipped with the norm · l . We note H 0 = L 2 and · := · 0 . C k ([0, T ]; H l (Ω)) denotes the space of k-times continuously differentiable functions on the interval [0, T ] with values in H l (Ω). Moreover, the function space X j i is defined as for i = 1, . . . , k to reduce the number of physical constants. By substituting the above values in (1) and applying assumptions (2) and (3), we have The Bohm criterion (5) is rewritten in a form corresponding to (6) as where we have used the ideal gas law p i (n i ) = κn i T i . We discuss the existence and asymptotic stability of the stationary solution to an initial-boundary value problem of (6) over the one-dimensional half space R + := {x > 0}. The initial and boundary data are prescribed as where ρ i+ , u i+ , and φ b are constants. We take the reference point of the potential The quasi-neutrality condition is necessary for the classical solvability of the Poisson equation (6c). We establish the unique existence of the solution (ρ 1 , u 1 , . . . , ρ k , u k , φ) to (6), (8), and (9) in the region where the conditions hold under the same initial assumptions Condition (12), the supersonic outflow condition, ensures that all characteristic speed of the hyperbolic system of 2k equations in (6a) and (6b) are negative. Therefore, one boundary condition (9) is necessary and sufficient for the well-posedness of the initial-boundary value problem of (6), (8), and (9). We use (13) to obtain bounds of the solution φ to the Poisson equation (6c). Note that (11) ensures (13) for the case wherein the plasma consists of electrons, positive ions, and no negative ions, that is, q i > 0 for any i = 1, . . . , k.
In studying the generalized Bohm criterion (7), we must pay attention to (7a). In fact, (7b) is the supersonic outflow condition contained in the formulation of our initial-boundary value problem.

Stationary solution.
In this subsection we discuss the unique existence of the stationary solution (R,φ) := (ρ 1 ,ũ 1 , . . . ,ρ k ,ũ k ,φ) that is time-independent. This solution satisfies together with conditions (8)-(13), that is, Before mentioning our main result on the existence of the stationary solution, let us define the Sagdeev potential V , which plays crucial roles in our analysis, as follows: Here, we define the function f i as and restrict its domain to where M i corresponds to the Mach number at x = ∞ for the i-th ions. Notice that f i is invertible on I i and hence, V is well-defined. The second-order derivative of the Sagdeev potential V atφ = 0 is equal to the left-hand side of (7a). In other words,

MASAHIRO SUZUKI
The unique existence result of the stationary solution is as follows. (15) and There exists a positive constant δ such that if |φ b | ≤ δ, then the stationary problem of (16) and (17) has a unique monotone solution (R,φ) satisfying Moreover, the stationary solution belongs to B ∞ (R + ) and satisfies where c and C are positive constants.
In the above theorem, V (φ b ) ≥ 0 in assertion (ii) is a necessary condition for existence. We derive this condition in Section 3. Suzuki [13] constructed nonmonotone stationary solutions for the case k = 1. Thus, monotonicity is required for uniqueness.
2.3. Asymptotic stability. This subsection is devoted to the discussion of the asymptotic stability of the stationary solution in Theorem 2.1 (i). To this end, we use perturbations from the stationary solution, such that For notational convenience, we use vectors with 2k components: Ψ := t (ψ 1 , η 1 , . . . , ψ k , η k ), q := t (q 1 , 0, . . . , q k , 0). Subtracting (6) from (16), together with the mean-value theorem, yields where ·, · is the inner product in C 2k . Here, the 2k × 2k symmetric matrices A 0 = (a 0 lm ) and A 1 = (a 1 lm ) are defined as if (l, m) = (2i, 2i), 0 otherwise, where i = 1, . . . , k. Note that A 0 and −A 1 are positive definite under conditions (11) and (12). Moreover, b = t (b 1 , . . . , b 2k ) and h = t (h 1 , . . . , h 2k ) denote vectors with 2k components: where i = 1, . . . , k. The scalar function g is The initial and boundary conditions to (24) are derived from (8) and (9): Let us mention difficulties of our stability analysis. Linearizing system (24) around the asymptotic state at x = ∞ by substituting Ψ = 0, σ = 0,R = R + , and R x = 0 into A 0 , A 1 , b, h, and g in (24), we have 1 Notice that the real parts of all spectra of this system are zero under the assumption This creates difficulty for our problem, because a standard energy method is not applicable. The author [9,13] overcomes this issue for the case k = 1 by using the weighted energy method with a weight function (1 + βx) λ or e βx .
Furthermore, we see that this method is applicable in our analysis of the multicomponent plasma as follows. Multiply (26) by e βx/2 and then introduce new unknown functions (Ψ, Σ) := (e βx/2 Ψ, e βx/2 σ) to obtain The spectral analysis for system (28) yields Proposition 2.2 which implies an advantage for the weighted energy method. Note that we cannot have explicit formulas for the spectra of (28) for the multicomponent case, even though there are formulas for the one-component case (see [9]). In the Appendix, Proposition 2.2 will be proved using a combination of several general theories, because a direct calculation using the formulas is not available.

Proposition 2.2.
Let the asymptotic state R + defined in (17a) satisfy (15) and (27). Then the following three assertions are equivalent: (i) The real part of all spectra of (28) in the whole space R is negative for sufficiently small β > 0. (ii) The generalized Bohm criterion (21) holds. (iii) The 2k × 2k symmetric matrix is uniformly positive definite for an arbitrary ξ ∈ R.
Even if we use the weighted energy method, another difficulty in the analysis of the multicomponent plasma still exists. In (24), the interaction effects among multiple ions are provided implicitly through the electrostatic potential. We need delicate estimates of these effects to obtain the stability theorem under the generalized Bohm criterion (21). On the other hand, Proposition 2.2 means that the interaction effects can be handled well in the Fourier space, and we explicitly write these effects in the Fourier space using an algebraic equation. To apply Fourier analysis to our initial-boundary value problem over R + , a suitable extension from the half space R + to the whole space R is required. In Section 4, we prove the stability theorem by combining the weighted energy method and Fourier analysis together with a suitable extension. This new technical method is worth noting in the stability analysis of the multicomponent plasma.
3. Unique existence of the stationary solution. In this section we show the unique existence of a monotone stationary solution. We start from reducing the stationary system (16) to a scalar equation forφ. At the moment, we assume the existence of a monotone solution satisfying (22) to the stationary problem of (16) and (17) and drive a scalar equation whichφ satisfies. Integrating (16a) over (x, ∞) gives Substitute (31) into (16b), divide the result byρ, and then integrate over (x, ∞) to obtain where we have used (17a) and f i is defined in (19). Here, f i is strictly decreasing over the interval I i defined in (20) and strictly increasing over (0, ∞)\I i . We define an inverse function f −1 by restricting the domain of f i to I i since the asymptotic state ρ i+ belongs to I i owing to (K i ρ γi−1 i+ − u 2 i+ ) < 0. Then it holds that . . , k. Substitute these equalities into (16c), multiply the resultant equation byφ x , integrate the result over (x, ∞), and then utilize condition (17a) and lim x→∞φx (x) = 0. Then we have a scalar equation forφ: where V is defined in (18). Note that this equation requires the necessary condition On the other hand, if the boundary value problem of (33) and (17b) has a monotone solutionφ ∈ C 2 (R + ) satisfying then it is immediately seen that , . . . , f −1 k (q kφ ), ,φ is a desired monotone stationary solution to (16) and (17). The uniqueness of a monotone stationary solution to (16) and (17) also follows from the uniqueness of the solution to (33) and (17b). Hence, it is sufficient to show the unique solvability of the boundary value problem of (33) and (17b) in order to prove Theorem 2.1.
In the remainder of this section, we show the next lemma.
Owing to (36), there exists a positive constant δ such that if |φ| ≤ δ, V (φ) is non-negative. We prove the existence of a monotone decreasing solution for the case φ b ∈ (0, δ], since the other case is shown in a similar manner. When φ b ∈ (0, δ], we rewrite the equation (33) into the equivalent equatioñ . A basic ordinary differential equation theory gives the unique solvability of this equation with (17b).
The function is positive on φ = 0 owing to (10). Thus, by retaking δ sufficiently small if necessary, we see that the solutionφ constructed above satisfies (34). One can show (35) by straightforward computations together with (36) and V (0) > 0. The proof of assertion (i) is complete.
4. Asymptotic stability of stationary solution. In this section we discuss the asymptotic stability of the stationary solution of which the asymptotic state R + satisfies (21). The time local solvability of the initial-boundary value problem of (24) and (25) is summarized as follows.
Notice that the time global solution (Ψ, σ) can be constructed by the standard continuation argument using the time local solvability developed in Lemma 4.1 and the a priori estimate in Proposition 4.2. Once the global solution is constructed, it is obvious that the resultant (Ψ, σ) satisfies (37) and (38) for t ∈ [0, ∞). The decay estimate (29) immediately follows from (37) together with (40) in Lemma 4.3. In addition, similarly as in [6,10], applying an induction argument to (38) gives the decay estimate (30). Hence, it suffices to show Proposition 4.2 in order to prove Theorem 2.3. The rest of this section is devoted to the derivation of the a priori estimate. The necessary and sufficient condition for S 0 > 0 is that the determinants D (l) of all leading principal minors of S 0 are positive. Straightforward computations give for i = 1, . . . , k. This means that S 0 is positive definite if and only if P i (u 2 + ) is positive for i = 1, . . . , k. Then we see from S 0 > 0 that P k (u 2 + ) > 0 holds. On the other hand, the positivity P i (u 2 + ) > 0 for i = 1, . . . , k − 1 follows from (15) and P k (u 2 + ) > 0. Hence, the positivity P k (u 2 + ) > 0 also gives S 0 > 0.