LARGE AMPLITUDE STATIONARY SOLUTIONS OF THE MORROW MODEL OF GAS IONIZATION model

. We consider the steady states of a gas between two parallel plates that is ionized by a strong electric (cid:12)eld so as to create a plasma. We use global bifurcation theory to prove that there is a curve K of such states with the following property. The curve begins at the sparking voltage and either the particle density becomes unbounded or the curve ends at the anti-sparking voltage.

1. The model. This paper is concerned with a model for the ionization of a gas such as air due to a strong applied electric field. The high voltage thereby creates a plasma, which may possess very hot electrical arcs. A century ago Townsend experimented with a pair of parallel plates to which he applied a strong voltage that produces cascades of free electrons and ions. This phenomenon is called the Townsend discharge or avalanche. The collision of gas particles within the plasma is sometimes called the α-mechanism. For more details, we refer the reader to [17].
Many models have been proposed to describe this phenomenon (see [1,8,9,10,12,13,14,15]). In 1985 Morrow [15] was perhaps the first to provide a model of its detailed mechanism. The model consists of continuity equations for the electrons and ions coupled to the Poisson equation for the electrostatic potential. For simplicity in this paper we consider only electrons and positive ions. We do not consider certain much smaller mechanisms such as 'attachment' and 'recombination', which were denoted by η and β in equations (1)-(3) in Morrow's paper. Thus the model 1298 WALTER A. STRAUSS AND MASAHIRO SUZUKI in this paper is as follows: where L is the distance between the planar parallel plates, ρ i is the density of positive ions, ρ e is the electron density, and −Φ is the electrostatic potential. Moreover, k i , k e , a, and b are positive constants. The ion and electron velocities u i and u e are assumed to obey the constitutive velocity relations (1d), which are due to the fact that the ions are much heavier than the electrons. The right sides of (1a) and (1b) come from the α-mechanism. They express the number of ion-electron pairs generated per unit volume by the impacts of the electrons. Specifically, the coefficient α = a exp ( −b|∂ x Φ| −1 ) is the first Townsend ionization coefficient, which can be found in equation (A1) of Morrow's paper.
The interesting article [8] of Degond and Lucquin-Desreux derives the model from the general Euler-Maxwell system by scaling assumptions, in particular by assuming a very small mass ratio between the electrons and ions. In an appropriate limit the Morrow model is obtained at the end of their paper in equations (160) and (163), which we have specialized to assume constant temperature and no neutral particles. We are also ignoring the γ-mechanism, which refers to the secondary emission of electrons caused by the impacts of the ions with the cathode. Now let us consider the structure of the model (1). Substituting the constitutive velocity relations (1d) into the continuity equations (1a) and (1b), we observe that the system is of hyperbolic-parabolic-elliptic type. We may consider the initialboundary value problem for (1) by prescribing the initial and boundary data The boundaries x = 0 and x = L correspond to the anode and cathode, respectively, since −Φ is the electrostatic potential. The boundary condition (1f) means that, at each instant, electrons are absorbed by the anode and ions are repelled from the anode. Due to the assumed lack of a γ-mechanism, ρ e is assumed to vanish at the cathode x = L. Of course the non-negativity of the mass densities ρ i0 and ρ e0 is a natural condition. Suzuki and Tani in [18] gave the first mathematical analysis of this model. Typical shapes of the cathode and anode in physical and numerical experiments are a sphere or a plate. Therefore they proved the time-local solvability of an initial boundary value problem over domains with a pair of boundaries that are plates or spheres. In another paper [19] they did a deeper analysis of problem (1). They proved that there exists a certain threshold of voltage at which the trivial solution (with ρ i = ρ e = 0) goes from stable to unstable. This fact means that gas discharge can occur and continue for a voltage greater than the threshold. The remarkable point is that gas discharge can occur even if γ-mechanism is not taken into account, in contrast to Townsend's theory which required the γ-mechanism for gas discharge to occur.
2. Stationary solutions. In this paper we consider the stationary problem. First of all, there are the trivial solutions ρ i ≡ 0, ρ e ≡ 0, ∂ x Φ ≡ constant. Unless the electric field is strong enough, avalanche does not occur. The critical threshold value of the voltage is called the sparking voltage V * c . The ionization coefficient a in (1a) and (1b) must be large enough, depending on b and L, in order to reach this threshold. In that case it was proven in [19] that (0, 0, V * c ) is a bifurcation point. The local bifurcation theorem is stated below.
Our goal in this paper is to extend the local bifurcation curve to a global one, thereby obtaining a one-parameter family of stationary solutions of large amplitude. In Section 3 we apply a functional-analytical global bifurcation theorem to construct a global curve K of stationary solutions (ρ i , ρ e , V ). This global curve includes solutions with positive densities as well as solutions with negative "densities". In Section 4 we restrict our attention to positive solutions.
Ultimately we prove our main result, namely, that there is a curve K pos ⊂ K along which ρ i > 0 and ρ e > 0 in (0, L) and either ρ i +ρ e is unbounded in L ∞ ([0, L]) or K pos "ends" at a point (0, 0, V # c ), where V # c > V * c is another critical voltage defined below. See Theorem 4.5 for a precise statement.
In Figure 4 in Appendix A, we present the voltage-current curve in a typical laboratory experiment. The sparking voltage in Figure 4 is denoted by V S , and the current is denoted by I := −ρ e u e + ρ i u i . Notice that the current appears to be unbounded.
For mathematical convenience, we rewrite initial-boundary value problem (1) by using the new unknown function which will play an essential role in our analysis. We also decompose the electrostatic potential as x R e with the boundary conditions V (0) = V (L) = 0. Occasionally we will denote it by As a result, we have the following system for stationary solutions: where the nonlinear term f e is defined as

WALTER A. STRAUSS AND MASAHIRO SUZUKI
The graph of g was drawn in [19] depending on the physical parameters a, b, and L. There are two cases, illustrated in Figures 1 and 2. As mentioned above, here and in [19] only the case in Figure 1 is considered, in order that g(V c ) can be sufficiently large. Then the sparking voltage V * c > 0 for the Degond-Lucquin-Desreux-Morrow model is uniquely defined by We also define the anti-sparking voltage V # c > 0 by Note that π 2 /L 2 is the lowest eigenvalue of −∂ 2 x on [0, L]. Local bifurcation was proven in [19]. That is, there is a unique non-trivial stationary solution curve in a neighborhood of the point (ρ i , R e , V c ) = (0, 0, V * c ) where the voltage V c is regarded as the bifurcation parameter. The precise result is summarized as follows. (In [19] the theorem was stated in terms of R i = e −Lx/Vc ρ i instead of ρ i , but for the analysis of global bifurcation it is more convenient to adopt ρ i .) Let I be the interval (0, L) and denote H 1 0l (I) = {v ∈ H 1 (I) : v(0) = 0}. We use s as a parameter along the curve.

Theorem 2.1 (Local Bifurcation). There exists a value
, and for all s ∈ [−s 0 , s 0 ] they solve the stationary problem (4). Furthermore, the solutions have the form The following additional properties of the local curve of solutions were proven in [19]. However, these properties are not used in the rest of the present article.
The main purpose of this article is to prove that there exist many more stationary solutions, including ones of large amplitude. This is accomplished by a global bifurcation technique. We introduce some notation for the stationary system as follows. We write the system (4) as where we denote λ :=V c /L, 3. Global bifurcation. In this section, we apply a functional-analytic global bifurcation theorem to the stationary problem (10). The theory of global bifurcation goes back to Rabinowitz [16,11] using topological degree. A different version using analytic continuation goes back to Dancer [7,4]. The specific version that is most convenient to use here is Theorem 6 in [5], which is the following: (H2) for some λ * ∈ R, N (∂ u F(λ * , 0)) and Y \R(∂ u F(λ * , 0)) are one-dimensional, with the null space generated by u * , which satisfies the transversality condition where ∂ u and ∂ 2 λ,u mean Fréchet derivatives for (λ, u) ∈ O, and N (L) and R(L) denote the null space and range of a linear operator L between two Banach spaces; (C4) K has a real-analytic reparametrization locally around each of its points; (C5) one of the following two alternatives occurs: To apply the theorem to our situation, we define the two spaces and the sets Note that O is an open set and each O j is a closed bounded subset of O. Futhermore, the F j are real-analytic operators because they are polynomials in (λ, ρ i , R e , V ) and their x-derivatives, except for the factor h(∂ x V + λ). However, ∂ x V + λ > 0 in O and the function s → h(s) is analytic for s > 0. Hypothesis (H1) is obvious. The local bifurcation condition (H2) is easily checked in exactly the same way as in [19].
The conditions (H3) and (H4) are validated in the following two lemmas.
Let us first show that the linear operator L 0 has a finite-dimensional nullspace and a closed range. By [20,Theorem 12.12] or [3, Exercise 6.9.1], it is equivalent to prove that L 0 satisfies the estimate for all (S i , S e , W ) ∈ X and for some constant C depending only on (λ, ρ 0 i , R 0 e , V 0 ). Indeed, keeping in mind that ∂ x V 0 + λ ≥ 1/j, we see from (11) and (13) that By writing (12) as Finally, (13) leads to (15)-(17), we find the estimate (14).
Because L 0 has a finite-dimensional nullspace and a closed range, it is called a semi-Fredholm operator. By the proof of Theorem 2.1, we know that at the bifurcation point the nullspace of ∂ (ρi,Re,V ) F(V * c /L, 0, 0, 0) has dimension one and the codimension of its range is is also one, so that its index is zero. Since O is connected and the index is a topological invariant [2, Theorem 4.51, p166], L 0 also has index zero. In particular, this implies that the codimension of L 0 is also finite.
Proof. Let {(λ n , ρ in , R en , V n )} be any sequence in K j . It suffices to show that it has a convergent subsequence whose limit also belongs to K j . By the assumed bound |λ n | + ∥(ρ in , R en , V n )∥ X ≤ j, there exists a subsequence, still denoted by {(λ n , ρ in , R en , V n )}, and (λ,

WALTER A. STRAUSS AND MASAHIRO SUZUKI
Furthermore, Taking the limit and using (18), we see that where the right hand side converges in C 1 ([0, L]). Hence, we see that Taking the limit using (18) in the third equation F 3 (λ n , ρ in , R en , V n ) = 0 immediately leads to The second equation F 2 (λ n , ρ in , R en , V n ) = 0 can be written as Hence ∂ x R en converges in C 1 ([0, L]), which means that R en converges in C 2 ([0, L]).
As we have checked all conditions in Theorem 3.1, the following conclusion is valid.

Moreover, such a curve of solutions to problem (10) having the properties (C1)-(C5) is unique (up to reparametrization).
Conditions (C1)-(C3) are an expression of the local bifurcation, while (C4)-(C5) are assertions about the global curve K. Alternatives (c) and (d) assert that K may be unbounded. Alternative (e) asserts that K may form a closed curve (a 'loop').

Positive densities.
Of course, we should keep in mind that for the physical problem ρ i and R e are densities of particles and so they should be non-negative. In this section we investigate the part of the curve K that corresponds to such densities.
A simple observation is the following proposition, which states that either ρ i and R e remain positive or the curve of positive solutions forms a half-loop going from V * c to V # c . Here V * c and V # c are defined in (6) and (7). The bifurcation diagram of the half-loop case is qualitatively sketched in Figure 3.
It follows that for suitably large n the expressions |λ(s n )|, ∥V (s n )∥ C 2 , and ∥h(∂ x V (s n )+λ(s n ))∥ C 0 are arbitrarily small. We multiply F 2 (λ(s n ), ρ i (s n ), R e (s n ), V (s n )) = 0 by R e (s n ) and integrate by parts over [0, L]. Then using Poincaré's inequality and taking n suitably large, we obtain Since R e vanishes at the endpoints, we conclude that R e (s n ) ≡ 0, which contradicts the assumed positivity. Proof. Suppose that sup s>0 ∥V (s)∥ C 2 is bounded. We take a subsequence as above. For suitably large n, the expressions 1 λ(sn) and ∥(h(∂ x V (s n ) + λ(s n ))/λ 2 (s n )∥ C 0 are arbitrarily small. Write s = s n for brevity. Multiplying F 2 (λ(s), ρ i (s), R e (s), V (s)) = 0 by R e (s)/λ 2 (s) and then integrating by parts over [0, L], we obtain Once again this leads to R e (s) ≡ 0, which contradicts the assumed positivity.

Lemma 4.3. Assume alternative (i) in Proposition 1. If lim
Proof. On the contrary, suppose that sup s>0 {∥ρ i (s) inf We shall show that e. x and j = 1, 2, 3.
Noting that taking the limit n → ∞ in the weak form, and using (23), we have This means that R e ∈ H 1 0 (0, L) is a weak solution to F 2 = 0. Furthermore, the standard theory of elliptic equations ensures that R e ∈ H 1 0 (0, L) ∩ H 2 (0, L) is also a strong solution to We now set x * := inf{x ∈ [0, L]; (∂ x V * + λ * )(x) = 0}. We divide our proof into two cases x * = 0 and x * > 0.
Let us reduce Condition (d) in Theorem 3.4 to a simpler condition. We write the result directly in terms of the ion density ρ i and the electron density ρ e = R e e −λx/2 .
We conclude with the following main result, which asserts that either ρ i and ρ e are positive along all of K with 0 < s < ∞ and ρ i + ρ e is unbounded or else there is a half-loop of positive solutions going from V * c to V # c . Proof. Suppose that (B), which is the same as the second alternative (ii) in Proposition 1, does not hold. Then the first alternative (i) in Proposition 1 must hold. Now in Theorem 3.4 there are five alternatives. Alternative (e) cannot happen because ρ i and R e are negative on part of the loop. Lemmas 4.1 -4.3 assert that any one of (a) or (b) or (c) implies that sup s>0 {∥ρ i (s)∥ C 0 + ∥R e (s)∥ C 2 + ∥V (s)∥ C 2 } is unbounded. Then Lemma 4.4 implies that sup s>0 {∥ρ i (s)∥ C 0 + ∥ρ e (s)∥ C 0 } must also be unbounded. This means that (A) holds.
Appendix A. Voltage-current curve. Figure 4 shows some experimental data together with some comments that appear in [21], where the sparking voltage is denoted as V S . The current appears to be unbounded, which is consistent with Alternative (A) of Theorem 4.5.