ON THE STABILITY OF HIGH LEWIS NUMBER COMBUSTION FRONTS.

We consider wave fronts that arise in a mathematical model for high Lewis number combustion processes. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possess essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis numbers to the spectral stability of the front in the case of infinite Lewis number. We discuss the implication for nonlinear stability of fronts with high Lewis numbers. This work builds on the ideas developed by Gardner and Jones [12] and generalized in the papers by Bates, Fife, Gardner and Jones [3, 4].

1. Introduction 1.1. Model. We consider a well-known model for the propagation of combustion waves in the case of premixed fuel, with no heat loss, in one spatial dimension x ∈ R. The system describing evolution of the temperature u and concentration of the fuel y reads u t = u xx + yΩ(u), y t = εy xx − βyΩ(u).
The reaction rate has the form of an Arrhenius law without ignition cut off: Ω(u) = e −1/u for u > 0 and Ω(u) = 0 otherwise. The system has two parameters. One is the exothermicity parameter β > 0 which is the ratio of of the activation energy to the heat of the reaction. The other is the reciprocal of the Lewis number ε = 1/Le > 0. Therefore, ε represents the ratio of the fuel diffusivity to the heat diffusivity. The system has been studied for various parameter regimes. Of interest to us are traveling wave solutions to (1) in two cases. One is the system (1) with ε = 0 (Le = ∞). Its physical prototype is the combustion of solid fuels. The other is the case of 0 < ε 1, i.e., when Le is very large but finite. This situation is also physical: (1) then describes burning of very high density fluids at high temperatures. Moreover, even during the burning of solid fuels some liquefaction of the fuel might occur in the reaction zone, thus causing a non-zero value of 1/Le. Our main goal is to relate the stability of the combustion wave of (1) with ε > 0 to the stability of the combustion wave of (1) with ε = 0.

1.2.
Existence of the front. We concentrate our attention on the traveling wave solutions of front type. In particular, we are interested in fronts of (1) that asymptotically connect (2) (u, y) = (1/β, 0) at − ∞, and (u, y) = (0, 1) at ∞, and approach these rest states at exponential rates. The boundary conditions represent the physical state where the fuel is completely burnt, i.e., y = 0, and the maximal temperature u = 1 β is reached, and the state when none of the fuel is yet burnt and the temperature u is still zero.  The traveling waves are sought as solutions of u + cu = −yΩ(u), εy + cy = βyΩ(u), where ε ≥ 0. There is a linear algebraic relation satisfied along any solution of (3), coming from an invariant of the equations, βu + βcu + εy + cy = 0. Thus, the quantity βu + βcu + εy + cy is conserved along trajectories. To satisfy boundary conditions (2), it must equal to c: (4) βu + βcu + εy + cy = c.
It is easy to see from (4), that, if 0 ≤ ε < 1, then there are no standing traveling waves (c = 0) satisfying (2). We fix the direction of propagation of the front by choosing c > 0. In Sect. 4.1, we show that both u and y which solve the traveling wave equation (3) are positive and monotone: u is decreasing and y is increasing. For (3) with no additional assumptions on β > 0, and when ε = 0, the existence and uniqueness of a front that converges to its rest states exponentially fast has been proved in [4,18]. There are also solutions with algebraic rates of decay, but these are considered to be of little interest [18]. The front has also been observed numerically in [19].
To prove the existence and uniqueness of the front in the case of 0 < ε 1, geometric singular perturbation theory seems to be a natural approach. More precisely, the system has a slow-fast structure. In the case of ε = 0 the flow is restricted to a two dimensional invariant manifold. The manifold is normally hyperbolic and attracting, therefore, by Fenichel's First theorem [7], it perturbs to an attracting manifold invariant for the flow with ε > 0. For the reduced problem, the lower dimension of the problem can be used to show that the front in the ε = 0 case is realized as a transversal intersection of relevant invariant manifolds. Transversality can be proved by Melnikov integral calculation [2], or by following the blueprint provided by the proof of the existence and uniqueness of subsonic detonation waves [8]. In [2], the formula for the corrective term for the velocity of propagation of the perturbed front has also been obtained.
An essentially different approach based on Leray-Schauder degree theory has been used in [3] to prove the existence and uniqueness of the front for when 0 < ε < 1. The traveling wave equation (3), in an appropriate scaling, falls in a class of equations described in [3].
1.3. Stability. Concerning the stability properties of the combustion front, the question we want to address here is about the relationship between two cases: ε = 0 and ε > 0. More precisely, do combustion fronts with 0 < ε 1 inherit stability properties of the front with ε = 0? We denote by (u f , y f ) the front, and by c f the corresponding value of its speed, i.e., (u f , y f ), c f refers to (u 0 , y 0 ), c 0 when ε = 0, and (u ε , y ε ), c ε when ε > 0. The linearization of (1) with c = c f about (u f , y f ) is given by The eigenvalue problem reads A traveling wave is called spectrally stable if the spectrum of the linearization of the system about the traveling wave is contained in the left half-plane {λ ∈ C : Re λ ≤ 0}. Generally speaking, the spectral stability need not imply the linear stability of the traveling wave, i.e., the decay of the solutions of the linearization of PDE about the traveling wave (with the exception of the single mode due to the translation invariance).
If the linearized operator is sectorial, the linear stability is guaranteed if there exists B > 0 such that the spectrum belongs to the half-plane {λ ∈ C : Re λ < −B} with the exception of a simple eigenvalue at zero, caused by translation symmetry. In Section 2.1 we show that the essential spectrum of the linearization of (1) about the front reaches the imaginary axis for any ε ≥ 0. Nevertheless, for this problem, it is possible to shift the essential spectrum to the open left halfplane using exponential weights. The translational eigenvalue, λ = 0, is simple, as it has been shown analytically in [9] using Evans function analysis for the case ε = 0. This result is also suggested by a numerical investigation, for both ε = 0 and nonzero, see [10,2,17]. If we assume that there are no other isolated eigenvalues in the closed right half-plane, then the front is spectrally stable in some exponentially weighted space. For fixed ε > 0 the linearized operator is the perturbation of the Laplacian by lower order derivatives and bounded operators, and, thus, it is also sectorial [11,16]. Therefore, the linear stability of the front in appropriate exponentially weighted spaces follows from the spectral stability. In this paper, the nonlinear stability is not discussed.
To discuss the point spectrum spectrum and its robustness under perturbations, we use the concept of the Evans function. Evans function is an analytic function of complex variable λ, defined for λ to the right of the essential spectrum [1]. Zeroes of the Evans function coincide with the eigenvalues of the linearized operator with the order of a zero being equal to the multiplicity of the eigenvalue. In Section 2.2, using the same approach as in [13], we show that the Evans function for our system can be analytically continued across the boundary of the essential spectrum. We will call this analytic continuation of the Evans function the extended Evans function. Embedded in the essential spectrum, eigenvalues are still zeroes of the extended Evans function, but not necessarily vice versa.
A. There is at least one zero of the Evans function, i.e., isolated eigenvalue, in the open right half plane of the complex plane: the front is truly unstable. B. There are no zeroes of the Evans function to the right of the imaginary axis. Moreover, with the exception of the simple zero at the origin, the extended Evans function has no zeroes on the imaginary axis: the most stable case. C. There are no zeroes of the Evans function to the right of the imaginary axis, but there are zeroes on the imaginary axis in addition to the zero at the origin: marginal case. The main result of this paper is that cases A and B with ε = 0 are robust under singular perturbations with ε > 0 sufficiently small. For brevity, in what follows, we denote the extended Evans function E 0 when ε = 0, and E ε when ε > 0. Theorem 1. If the front (u 0 , y 0 ) is unstable then there exists ε 0 > 0 such that for any 0 < ε < ε 0 the front (u ε , y ε ) is also unstable.
Theorem 2. Assume the front (u 0 , y 0 ) is spectrally stable, and, moreover, the statement B about E 0 is true. There exists ε 0 > 0 such that for any 0 < ε < ε 0 the statement B true for E ε ; therefore, the front (u ε , y ε ) is also spectrally stable.
Theorems 1 and 2 are consequences of the following key proposition about the zeroes of the Evans function.
Proposition 3. Assume that λ 0 is a zero of order m of E 0 . There exists ε 0 > 0, independent on λ, such that for any ε (0 < ε < ε 0 ), in a neighborhood of λ 0 of order ε, there are exactly m zeroes (counting multiplicity) of E ε .
This statement is proved in Section 3. These analytic results confirm the expectations based on the numerical analysis in [2] where a count of the number of eigenvalues in various bounded domains of the complex plane has been performed and compared for ε = 0 and ε > 0. Here we have also used the fact that λ = 0 is a simple eigenvalue when ε = 0, and, because of Proposition 3, persists as a simple eigenvalue when ε > 0. Therefore no eigenvalue can cross into the right half-plane through the origin when the problem is perturbed by introducing non-zero ε.
There is numerical evidence [10] that the stability of the front (0 ≤ ε 1) depends on the parameter β. It has been shown numerically that there exist β 0 = O(1) such that for β < β 0 the linearization about the front does not possess any unstable eigenvalues. For larger values of β a pair of complex conjugate eigenvalues crosses the imaginary axis from left to right, causing a socalled pulsating instability. The occurrence of Hopf bifurcation, with the speed of the front as the bifurcation parameter, has been also noted in [17].
The numerical observation of pulsating instability is not unexpected. Its presence has been proved under the assumption of high energy activation [14]. In this situation, the existence of two complex conjugate, purely imaginary eigenvalues has been obtained. But in some way, the behavior of the unstable eigenvalues as functions of β, when β is large, has not been completely captured by the numerics. In other words, it is not clear from the numerics what happens with the unstable eigenvalues for β 1. Based on our estimates in Sect. 4.2, we claim that with further increase of β, at some point, the real parts of the unstable eigenvalues start to decrease. The eigenvalues approach the imaginary axis. This observation is in agreement with the stability analysis performed in high energy activation limit in [14]. The investigation of bifurcations as large β varies will be addressed in a different paper.

Main technique: Stability Index Bundles.
To prove our results we apply the technique developed in [1], where the stability index was introduced. The stability index is a topological invariant which counts the number of eigenvalues inside a given closed contour. If there exists a contour K enclosing all of the unstable eigenvalues, then the spectral stability of the wave can be concluded from the stability index of the contour. More precisely, this topological invariant is the first Chern number of the Stability Index Bundle. Stability Index Bundle, also called augmented unstable bundle [1], is a bundle with fibers formed by certain invariant manifolds in the phase space of the linearization (5) and the base given by a compactification of an infinite cylinder R × K (for the real space variable and the complex parameter λ) capped at ±∞ by the contour K combined with its interior.
The first Chern number of the Stability Index Bundle coincides with the winding number of Evans function over the contour K and therefore counts eigenvalues inside K, because they are zeroes of the Evans function. The detailed construction of the bundle is presented in Sect. 3 and the description of the contour K is discussed in Sect. 2.2.
Because of the slow-fast structure of the eigenvalue problem (5), generally speaking, the Stability Index Bundle can be decomposed into the Whitney sum of the associated slow and fast subbundles [15]. In this particular case we were able to prove that the slow bundle is, in fact, the full unstable bundle. The topological nature of the stability index makes it robust under small perturbations. Therefore the Chern number of the full system (0 < ε 1) is equal to the Chern number of the reduced system (ε = 0) and the statement of the theorem follows. The construction of the Stability Index Bundle is performed in the framework of exterior powers Λ k (C) (see [6,20] The plan of the paper is as follows. In Sect. 2 we construct the extended Evans function. To do so we need to know the location of the essential spectrum. The Stability Index Bundle is defined by its fibers and the base. In Sect. 3 we define the fibers. The base of the bundle is defined in Sect. 4 by means of a construction of a contour in the complex plane that leads us to the proof of Theorems 1 and 2. We call such contour the Index Contour. The existence of Index Contour is guaranteed by the bound on the modulis of the unstable eigenvalues, and properties of zeroes of an analytic function, here, the extended Evans function. Monotonicity of the front (u f , u f ) is used to obtain an estimate on the real parts of the unstable eigenvalues.
Acknowledgment. The authors wish to thank P. Simon for pointing to relevant literature. This work was supported by the NSF grant DMS-0410267.
ON THE STABILITY OF HIGH LEWIS NUMBER COMBUSTION FRONTS. 5

Evans Function
The Evans function for (1) was successfully used to study the discrete spectrum of the problem with fixed ε [2,9,10,19]. We use Evans function to construct the base of the Stability Index Bundle. To include the translational eigenvalue at the origin, embedded in the essential spectrum, in the consideration, we want to to analytically continue the Evans function across the right boundary of the essential spectrum. This section consists of two parts. First, we find the location of the essential spectrum, then define the extended Evans function.
2.1. Essential spectrum. In this section we find the location of the essential spectrum, which is the complement of the point spectrum in the spectrum.
Case ε > 0. The eigenvalue problem (5), written as a first order ODE, on the fast scale η = ξ/ε readsṗ The right-hand side of this system is an action of the matrix The eigenvalues of M ± (λ, ε) are called the spatial eigenvalues as opposed to the temporal eigenvalues λ. The eigenvalues of M − (λ, ε) are When λ crosses the imaginary axis from right to left, κ − 4 crosses the imaginary axis from right to left. The boundaries of the essential spectrum due to the behavior at −∞ are curves (7) {λ = −εν 2 + ciν; ν ∈ R} ∪ {λ = −ν 2 + ciν; ν ∈ R}.
The eigenvalues of M + (λ, ε) are When λ crosses the imaginary axis from right to left, both eigenvalues with positive real parts, κ − 2 , κ − 4 , cross the imaginary axis from right to left, and coincide at λ = 0. The boundaries of the essential spectrum due to the behavior at +∞ are given by the curves The set of curves (8)-(7) divides the complex plane into regions which either are covered by spectrum or, otherwise, contain only discrete eigenvalues. There is a component of it which contains the open right half-plane of the complex plane. From the estimates described in Sect. 4.2, we know that there can be no eigenvalues with positive real parts far away on the real axis. Therefore the region to the right of the rightmost parabola from (7) and (8) (9) {λ = −εν 2 + ciν; ν ∈ R} contains only discrete spectrum, i.e. isolated eigenvalues of finite multiplicity. The essential spectrum is bounded on the right by (9) and includes that curve (see Fig. 1).
Case ε = 0. We rewrite the eigenvalue problem (5) as a first order ODE: The right-hand side of the last system is an action of the matrix The eigenvalues of M − (λ) are For fixed finite β > 0, if Re λ > 0, then Re κ − 1 , κ 3 > 0, Re κ − 2 < 0. When λ crosses the imaginary axis from right to left κ − 3 crosses the imaginary axis from right to left. The eigenvalues of M + (λ) are If Re λ > 0, then Re κ + 1 , κ + 3 > 0, Re κ + 2 < 0. When λ crosses the imaginary axis from right to left both eigenvalues with positive real parts, κ − 1 , κ − 3 , cross the imaginary axis from right to left. The boundaries of the essential spectrum due to the behavior at −∞ are curves The boundaries of the essential spectrum due to the behavior at +∞ are given by the curves The rightmost curve {λ = ciν; ν ∈ R} in (10) and (11) is the imaginary axis. The same argument as in the case of ε = 0 shows that the essential spectrum belongs to the region to the left of and on the imaginary axis (see Fig. 1). The open right half plane can contain isolated eigenvalues of finite multiplicity only. The direction in which spatial eigenvalues cross the imaginary axis is related to the group velocity and determines where the system transports initial perturbations: here, to −∞.
Im λ λ λ Im λ Re Re Figure 1. The essential spectrum belongs to the region to the left of the rightmost curve and contains that curve.

Evans function and its analytic continuation.
Evans function is defined, and analytic, for λ's to the right of the essential spectrum. The discrete isolated eigenvalues of the linearized operator are zeroes of the Evans function [1].
In certain situations the Evans function can be analytically continued across the boundary of the essential spectrum as well. In particular, it is possible, if the spacial eigenvalues cross the imaginary axis on the λ plane moving in the same direction when λ crosses the imaginary axis from right to left. In the case when a unique eigenvalue with positive real part moves into the essential spectrum as λ crossed the imaginary axis, the analytic continuation of the Evans function has been constructed in [12]. In our situation the construction is similar.
The Evans function for the case ε = 0 is defined in a similar fashion.

Stability Index Bundle
In this key section we construct the Stability Index Bundle: a bundle topological properties of which can be used to locate eigenvalues of the linearized problem. We start by using the slowfast structure of the eigenvalue problem (5) to obtain a description for the flow that depends on ε (0 ≤ ε 1) continuously. Such a description allows us to define the fibers of the Stability Index Bundle exactly the way it is introduced in [1]. Moreover, it will follow from the construction that the fibers of the Stability Index are determined only by the slow dynamics of the flow. We use this information combined with estimates on the moduli of the unstable eigenvalues to choose an appropriate base for the bundle.
3.1. Fibers. The eigenvalue problem for the linearization about the front reads Again, as in the definition of the Evans function, following [1], we transform (13) into an autonomous system by introducing a new dependent variable τ defined by the relation where κ > 0 is a constant. The extended system is where derivatives are taken with respect to ξ, but now with an abuse of notation y f and u f are functions of τ .
The front (y f , u f ) is converging to its rest states at rates independent of ε on the slow scale. It has been shown in [1] that there exists κ such that (13) is C 1 .
The flow reduced to this set is together with the equation for τ : τ = κ(1 − τ 2 ).
The equationṖ = A(τ, ε, λ)P induces a flow on the space Λ k (C 4 ) Our goal now is to construct an invariant manifold for (18) which depends on ε continuously. To do so, we choose to work not with (18) directly, but with its conjugate. Indeed, for Y ∈ Λ k (C n ) one can consider its Hodge star * Y ∈ Λ n−k (C n ) (see [6,Sect.1.7]  where I n−k is the identity on Λ n−k (C n ) and (A n−k (τ, ε, λ)) * is defined as in [5].
At this point the construction [1] of the Stability Index Bundle is applicable. In our case, we call the bundle slow, since, as it is shown above, it is constructed as a perturbation of the bundle corresponding to ε = 0 case.
Assume we have a bounded simply connected domain K in C such that its boundary, contour K, does not contain any of the eigenvalues of (5). For any λ ∈ K, the fibers of the bundle are defined by means of the global unstable manifold of the point (p, q, r, τ ) = (0, 0, 0, −1).
The standard capping-off procedure [1] provides fibers at K × {τ = ±1}, where K is K with its boundary K. There exists a small γ > 0 such that each of the limiting systemṡ with Re λ > −γ, has exactly one eigenvalue with positive real part, which is given by the sum of 2κ (from the flow in τ direction) and two of the largest eigenvalues. Thus it has a unique unstable eigenvector η ε (±1, ε, λ, ξ) at τ = ±1 which provides the caps at K × {τ = ±1}.
We summarize the result.

Index Contour
To make a conclusion on existence of unstable eigenvalues based on the properties of the Stability Index Bundle, we want to construct the base of the bundle using a contour that encloses all of the unstable eigenvalues, but does not contain any of the zeroes of the extended Evans function. We also want this contour to be independent of ε. We prove the existence of such contour in Subsect. 4.3. The proof is based on two facts: all of the zeroes of the extended Evans function are isolated; the unstable discrete spectrum belongs to a bounded region. The latter and the dependence of the boundary of this region on parameter β is described in Subsect. 4.2. We start this section by obtaining properties of the fronts (Subsect. 4.1) that we use to study the location of the unstable discrete spectrum.
Remark. Monotonicity of the front for ε > 0 case has been proved in [17] under the assumption y ε > 0 and u ε > 0.
Here, for the case ε > 0, we do not discuss lim β→∞ C 2 (β) because the upper bound on the size of ε, for which the existence and uniqueness of the front has been proved, generally speaking, depends on β. The nature of this dependence has not yet been investigated.

4.3.
Construction of a contour. We want to show that it is possible to construct a closed contour which would enclose all of the unstable eigenvalues λ of the linearization of the system about the front with Re λ ≥ 0 and does not go through any of eigenvalues of either (u 0 , y 0 ) or (u ε , y ε ). We base the construction of such contour on two facts. On the right of the imaginary axis we use the estimates we obtained in Section 4.2 according to which there exists a constant C, independent of ε, such that |λ| ≤ C for Re λ ≥ 0. The situation on the left of the imaginary axis is not that simple. To describe the contour for Re λ < 0 we use Evans function.
For any contour not crossing any of the zeroes of the Evans function, the first Chern number of the Stability Index Bundle over that contour coincides with the number of the zeros of the Evans function inside of the contour [1].
Over the regions covered with the essential spectrum, the relation between the first Chern number of the Stability Index Bundle and the number of zeroes of the extended Evans function is still valid, but zeroes of the extended Evans function are not necessarily eigenvalues.
On the other hand, eigenvalues λ (embedded in the essential spectrum or not) are zeroes of the extended Evans function. As zeroes of an analytic function they are isolated, therefore it is possible to draw a contour which does not go through any of them. Moreover, from the construction of the Stability Index Bundle in Sect. 3 we see that any zero of the extended Evans function, if ε > 0 is sufficiently small, is located in a small neighborhood of a zero of the extended Evans function for a problem with ε = 0. Therefore, if we assume that the Evans function does not have any non-trivial, purely imaginary zeroes when ε = 0, then there exists ε 0 > 0 such that there is a contour independent of ε that encloses the unstable eigenvalues of the eigenvalue problems with all 0 ≤ ε ≤ ε 0 , and, at the same time, does not enclose or go through any of the zeroes λ = 0, Re λ ≤ 0 of the Evans functions corresponding to 0 ≤ ε < ε 0 .