STABILITY OF ZND DETONATION WAVES IN THE

MAJDA COMBUSTION MODEL

Jean-Michel ROQUEJOFFRE^{(1)} Jean-Paul VILA^{(2)}

ABSTRACT. This paper is the first of a series aimed at deriving uniform stability results for strong ZND - Zeldovich-Neumann-Döring - detonation waves, in the context of singular perturbations. In this work, the Majda and Majda-Rosales models with small viscosity are examined, and a dispersion relation accounting for the linear stability of the wave is derived, by means of matched asymptotics. A rigorous justification follows, in which the main difficulty is a uniform spectral estimate.

(1) UFR MIG, UMR C.N.R.S. 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 TOULOUSE CEDEX, FRANCE.

(2) Département GMM, UMR C.N.R.S. 5640, INSA, Complexe scientifique de Rangueil, 31077 TOULOUSE CEDEX, FRANCE.

INTRODUCTION

This paper is a preliminary study in a program aimed at a mathematically rigorous account of stability - and, more interestingly, instability - properties of detonation waves in fluids, in the context of singular perturbations. We start in this paper with the Majda [14] and Majda-Rosales [15] models for combustion [15]. Both are reduced models, the second one being derived from the full reactive gas dynamics system in which the velocity is near-sonic. In both models, two variables come into play: the mass fraction of the reactant, denoted by Z(t, x), and a lumped variable u(t, x) accounting for the density, velocity, and temperature of the fluid. The evolution equations write [14]

(1)  \{ \begin{matrix}(u + q_{0}Z)_{t} − εu_{xx} + (f(u))_{x} & = 0 \\ Z_{t} & = −ϕ(u)Z \end{matrix}

or

(2)  \{ \begin{matrix}(u + q_{0}Z)_{t} − εu_{xx} + (f(u))_{x} & = 0 \\ Z_{x} & = ϕ(u)Z \end{matrix}

The coefficient q_{0} describes the specific heat release, and the viscosity ε is assumed to be small. The function f is strictly convex, and the function ϕ is the usual Arrhenius reaction term, to which an ignition temperature has been added. The assumptions on ϕ will be made precise in the next section. In the sequel, the computations will be carried out on system (1): the same properties are available in system (2), with exactly similar proofs.

Strong ZND - Zeldovich-Neumann-Döring - detonation wave solutions of (1) are travelling waves of the form (u_{ε}(x − st), Z_{ε}(x − st)), looking like a hydrodynamic shock, followed by a chemical reaction. They satisfy

(3)  \{ \begin{matrix}−s(u_{ε} + q_{0}Z_{ε})′ − εu′′_{ε} + (f(u_{ε}))′ & = 0 \\ −sZ′_{ε} & = −ϕ(u_{ε})Z_{ε} \end{matrix}

Their existence is established in [14], for arbitrary diffusion coefficients; the singular perurbation study is also set up. See also Larrouturou [11] for a related system with diffusion in both equations. A singular perturbation analysis for the full system including (Navier-Stokes) gas dynamics may be found in Gasser-Szmolyan [6]; for completeness its analogue for system (3) will be performed in Section 1 below. An existence theorem is given by Gardner in [5] for the same Navier-Stokes system - via the Conley index, but with O(1) diffusion coefficients.

As for the stability of the travelling waves for the Majda-Rosales model, there are two papers we are aware of. The first one, due to Levy [12], proves the convergence towards the strong detonation of the solution to the Riemann Problem corresponding to the nondiffusive version of (1), i.e. with ε = 0. The second one, due to Liu and Ying [13], does not assume ε to be small, and proves the nonlinear stability of the strong detonations under the assumption that q_{0} is small.

The aim of this paper is to study the linear stability of the strong detonation waves, under the assumption that ε is small, but with no restriction on the size of q_{0}. Here, the main point in the analysis will be the linear stability of the waves, i.e. the resolution the differential eigenvalue problem, with unknown (λ, v, Z):

(4)  \{ \begin{matrix}−εv′′ + ((f′(u_{ε}) − s)v)′ − sq_{0}Y′ & = λ(v + q_{0}Y) \\ −sY′ & = −(ϕ(u_{ε})Y + ϕ′(u_{ε})Z_{ε}v)+λY \end{matrix}

In this approach, and in order to make things clearer, we will first give a formal derivation of a dispersion relation, which will be obtained as the result of a matched asymptotics procedure. In order to make this rigorous, two main steps will be required: first, one will have to make sure that, as ε → 0, possible destabilizing eigenvalues will remain bounded. Then we will compute an approximate dispersion relation, and prove that it actually converges to the formal one. Let us mention that, at this stage, it would be easy to conclude to nonlinear stability and derive estimates - see, for example, [8] - which are not uniform with respect to ε. Such is not our goal, since we are aiming at uniform results with respect to ε.

The second step of the linear analysis is classical, and will be quickly sketched in the last section. The first step is represents the main difficulty, and the strategy that we will follow is a bootstrap method inspired from [7]: one first computes a crude upper bound for |λ|, and this yields expansions for the eigenfunctions. This in turn implies less crude bounds for λ, and so on.

The organisation of the paper reflects the above steps. In Section 1, the singular perturbation setting for the wave will be briefly sketched, the eigenvalue problem set up and the two main results stated. Section 2 will be devoted to the formal dispersion relation when a simplifying assumption on ϕ is made. Sections 3 to 5, which constitute the main body of the paper, will present uniform spectral estimates:

O(\frac{1}{ε}) estimates will be derived - Section 3; this will lead us to o(\frac{1}{ε}) - Section 4. In Section 5 we will deduce O(1) estimates. Finally, the formal computation of Section 2 will be briefly justified in Section 6, and eigenvalue properties will be derived.

1. FACTS ABOUT THE ZND WAVE AND MAIN RESULTS

Although the singular perturbation analysis is already performed in [6] for the full Navier-Stokes equations, the analysis for system (3) is so short that recalling it will not make this paper swell beyond a reasonable limit.

Assumptions on ϕ

1. ϕ is nonnegative and smooth enough;

2. there exists u_{I} ∈ IR such that ϕ(u) = 0 if u ≤ u_{I} and ϕ(u) > 0 if u > u_{I}; moreover

(1.1)  lim inf_{u→+∞} ϕ(u) > 0.

Assumption 2 - ignition temperature - is essential. A widely made commodity assumption is Assumption 3 below, which allows explicit computations.

3. There exists u_{II} > u_{I} such that ϕ(u) = 1 if u ≥ u_{II}.

The problem under consideration is the following: given a left state ( \begin{matrix}u_{L} \\ 0 \end{matrix} ) and

a right state ( \begin{matrix}u_{R} \\ 1 \end{matrix} ) one looks for a solution (u, Z) with speed s connecting these

two states. Integration of the equation for u in (3) between +∞ and x yields the first-order system

(1.2)  \{ \begin{matrix}εu′ & = ψ(u, Z) \\ sZ′ & = ϕ(u)Z \end{matrix}

with

(1.3)  ψ(u, Z) = f(u) − f(u_{R}) − s(u + q_{0}Z) + s(u_{R} + q_{0}).

This in particular implies

(1.4)  s = \frac{f(u_{L}) − f(u_{R})}{u_{L} − u_{R} + q_{0}},

which in turn implies that q_{0} must not be larger than some critical value q^{CJ}, the critical quantity q^{CJ} being the Chapman-Jouguet value. The strict convexity of f implies

(1.5)  u_{L} > u_{R} + q_{0},

and this inequality will be crucial in assessing the stability of the wave.

A strong detonation wave is made, in the limit ε = 0, of a hydrodynamic shock

connecting ( \begin{matrix}u_{R} \\ 1 \end{matrix} ) to ( \begin{matrix}u_{∗} \\ 1 \end{matrix} ) followed by a smooth chemical reaction, i.e u and Z

decrease along the curve {ψ(u, Z) = 0} - the slow manifold. The intermediate state u_{∗} is determined by the classical Rankine-Hugoniot relations

(1.6)  s = \frac{f(u_{∗}) − f(u_{R})}{u_{∗} − u_{R}}.

and there holds u_{R} < u_{L} < u_{∗}, f′(u_{R}) < s < f′(u_{L}) < f′(u_{∗}).

The above considerations may be made rigorous as follows: set M = {(u, Z) :

u > u_{L}, ψ(u, Z) = 0\}. One notices that M is normally hyperbolic - i.e. \frac{∂ψ}{∂u} =

f′(u) − s > 0. Hence - Fenichel [4], see also a proof in [10] - for ε > 0 small enough, there exists a smooth curve M_{ε} which is ε-C^{1} close to M, and invariant under the flow (1.2). Even more interesting is that the stable manifold of the latter curve is ε-C^{1} close to the stable manifold of M. This implies the following: an orbit of (1.2)

starting - at x = +∞ - from ( \begin{matrix}u_{R} \\ 1 \end{matrix} ) will be attracted to \{ψ(u, Z) = 0\}, and hence

will tend to ( \begin{matrix}u_{L} \\ 0 \end{matrix} ) as x → −∞. The ε-C^{1} closedness of the stable manifolds implies

the desired shape for the wave.

Remark. We have implicitly assumed u_{L} > u_{I}; this means that the parameters have been suitably designed so that this inequality holds. If such is not the case, there is incomplete combustion: see [15].

Notations and normalizations.

1. From now on, the ZND wave will be denoted by (u_{ε}, Z_{ε}). Moreover we will set

(1.7)  f(u) = g(u) + su.

2. Let u_{B}(ξ) be the travelling wave of the viscous steady Burgers equation:

(1.8)  −u′′_{B} + g(u_{B})′ = 0,  u_{B}(−∞) = u_{∗}, u(+∞) = u_{R}.

In the sequel the following normalization will be adopted:

(1.9)  u_{ε}(0) = u_{B}(0) = \frac{u_{∗} + u_{R}}{2} .

3. Let us define x_{1}^{ε} and x_{2}^{ε} by

(1.10)  \begin{aligned}x_{1}^{ε} & = max\{x ∈ IR : u_{B}(\frac{x}{ε}) = u_{∗} − ε^{2M}\} \\ x_{2}^{ε} & = min\{x ∈ IR : u_{B}(\frac{x}{ε}) = u_{R} + ε^{2M}\} \end{aligned}

the constant M will be as large as will be suitable to us in the sequel. Let us also introduce the auxiliary point

(1.11)  x_{3}^{ε} = min\{x ∈ IR : u_{B}(\frac{x}{ε}) = u_{∗} − ε^{M}\}

There holds x_{1}^{ε} < x_{3}^{ε} < x_{2}^{ε}; moreover there exists a constant C > 0 independent of ε such that

(1.12)  −\frac{1}{C} εLogε ≤ −x_{1}^{ε} ≤ −CεLogε; −\frac{1}{C} εLogε ≤ x_{2}^{ε} ≤ −CεLogε.

This can easily be seen on the explicit expression of u_{B}.

4. The left half complex plane will be denoted by P_{−}.

The linearisation of (1) around (u_{ε}, Z_{ε}) leads to the eigenvalue problem

(1.14)  \{ \begin{matrix}−εv′′ + ((g′(u_{ε}))v)′ − sq_{0}Y′ & = λ(v + q_{0}Y) \\ −sY′ & = −(ϕ(u_{ε})Y + ϕ′(u_{ε})Z_{ε}v)+λY \end{matrix}

The main result of this paper is in fact the following

Theorem 1.1. Let Assumptions 1 and 2 hold. There exists C > 0, independent of ε, such that, if (λ, v, y) is a nontrivial solution of (1.14) with λ ∈ P_{−}, then |λ| ≤ C.

Linear stability may be obtained from the above result at the expense of the additional Assumption 3.

Theorem 1.2. Let Assumptions 1 to 3 hold. Assume u_{II} < u_{L}. If ε > 0 is small enough, then for all nontrivial solution (λ, v, Y) ∈ C^{|} ^{∗}×L^{2}(IR) × L^{2}(IR), there holds: λ ∉ P_{−}.

When the function ϕ does not satisfy Assumption 3 anymore, we do not know whether Theorem 1.2 holds - and in fact it may well not hold. However we know that destabilizing eigenvalues have nonzero imaginary parts.

Theorem 1.3. If ε > 0 is small enough then, for any nontrivial solution (λ, v, Y) ∈ P_{−}\{0}×L^{2}(IR) × L^{2}(IR), there holds: λ ∉ IR.

The rest of the paper is devoted to the proof of these three results. Let us end this section with the following fact, which will prove to be important in the sequel. It comes from the comparison between the first inequality of (1.12) and the ε-closedness of the perturbed slow invariant manifold to the unperturbed one for the flow (1.2).

Lemma 1.4. (i). (Behaviour in a neighbourhood of the shock layer) There holds

(1.15)  |Z_{ε} − 1| + |u_{ε} − u_{B}| = O(εLogε) on [x_{1}^{ε}, x_{2}^{ε}].

(ii) (Behaviour outside the shock layer). The function u_{ε} is strictly increasing on ]−∞, x_{3}^{ε}]. Moreover, the sequence (u_{ε}, Z_{ε})_{ε} is bounded, in the C^{1}-norm, on ]−∞, x_{3}^{ε}].

The proof is straightforward in view of the above considerations.

2. FORMAL ARGUMENT FOR LINEAR STABILITY

In this section, Assumption 3 will be understood to hold.

Let us assume that (λ, v, Y) is a nontrivial solution of (1.14), with Reλ ≤ 0, and (v, Y) ∈ L^{2}(IR). In this paragraph the strong detonation will be denoted by (u, Z), and the viscosity ε will be deleted whenever it does not play a significant role.

Three cases are to be studied, depending on whether the coordinate x is before, after, or inside the shock which, in the limit ε → 0, will be assumed to be located at x = 0. The two first cases will yield outer expansions for v and Y, whereas the last one will yield an inner expansion. A dispersion relation will be obtained by matching the values of v and Y resulting from the outer and inner expansions at x = 0^{−}, or equivalently, ξ := \frac{x}{ε} = −∞.

Case 1. x > 0. Then u is almost equal to u_{R} and Z is almost equal to 1. Moreover the viscosity may be taken equal to 0. Then ϕ(u) = ϕ′(u) = 0. The

equation for Y is then Y′ = −\frac{λ}{s}Y, which implies Y = 0. An equation for v is then g′(u_{R})v′ = λv; because g′(u_{R}) < 0 there holds v = 0.

Case 2. x < 0. Here, once again, the viscosity may be neglected. Let us recall that, in this zone, there holds u_{II} < u_{L} ≤ u ≤ u_{∗}, which implies ϕ(u) ≡ 1 and

ϕ′(u) ≡ 0. The equation for Y is then Y′ = \frac{1 − λ}{s}Y, which implies Y = Y_{0}e^{\frac{1−λ}{s} x}. The equation for v now reads

(g′(u)v)′= λv + q_{0}Y;

recalling that in this zone there holds g′(u) > 0 we have:

(2.1)  v(x) = \frac{q_{0}Y_{0}}{g′(u)} ∫_{−∞}^{x} exp(λ ∫_{y}^{x} \frac{dz}{g′(u(z))} + \frac{1 − λ}{s}y) dy

Case 3. Inner expansion. Setting ξ = \frac{x}{ε}, we look for an ansatz for v and Y of the type

(2.2)  \begin{aligned}v(ξ) & = \frac{u′(ξ)}{ε} + v_{1}(ξ) \\ Y (ξ) & = O(1) \end{aligned}

Here, the superscript ′ denotes the differentiation with respect to ξ. Let us explain two point of the above ansatz: first, we chose this form because the x-derivative of the wave is a solution to (1.14) with λ = 0, and this is really the form that it has inside the shock. Next, one may notice that an undetermined coefficient should be added to the first term in the expansion of u; we have decided to put it equal to 1, as a normalisation.

Let us not forget that, in the shock, the variable Z is almost equal to 1. System (1.14), in the scaled variable, reads

\{ \begin{matrix}−v′′ + (g′(u)v)′ & = sq_{0}Y′ + ελ(v + q_{0}Y) \\ sY′ & = ε(ϕ(u)Y + ϕ′(u)v)−ελY \end{matrix}

Therefore the equation for Y reads sY′ = ϕ′(u)u′ + O(ε); as a consequence there holds

Y (ξ) = \frac{ϕ(u)}{s}

and in particular

(2.3)  Y_{0} = Y (−∞) = \frac{1}{s}

Turn to the equation for v. The \frac{1}{ε}-terms only say that u′ is indeed, in the limit ε → 0, the derivative of the Burgers wave. More interesting is the equation for the zero-order terms, which asserts that

−v′′ + (g′(u)v)′= sq_{0}Y′ + λu′.

Integrating this equation between −∞ and +∞, taking into account, according to Case 1, that v(+∞) = Y (+∞) = 0, and remembering (2.3) we obtain

g′(u_{∗})v_{1}(−∞) = λ(u_{∗} − u_{R}) + q_{0}.

Matching v_{1}(−∞) with the value at 0 of the outer expression for v yields the dispersion relation:

(2.4)  λ(u_{∗} − u_{R}) + q_{0} = \frac{q_{0}}{s} ∫_{−∞}^{0} exp(λ ∫_{y}^{0} \frac{dz}{g′(u(z))} + \frac{1 − λ}{s}y) dy

One easily sees that λ = 0 is a solution of (2.4). It is also possible to prove the following

Proposition 2.1. Equation (2.4) has no nonzero solution λ such that Reλ ≤ 0. Moreover 0 is a simple root of equation (2.4).

PROOF. Integrating by parts the right handside of (2.4), we obtain that (2.4) is equivalent to a relation of the form λΓ(λ) = 0, where

(2.5)  \begin{matrix}Γ(λ) = & u_{∗} − u_{R} \\ − q_{0} ∫_{−∞}^{0}(\frac{1}{g′(u)} + \frac{1}{s})exp[−λ(∫_{y}^{0} \frac{dz}{g′(u)} + \frac{y}{s} )]exp(\frac{y}{s} ) dy \end{matrix}

Let us assume the existence of a root λ of Γ with nonpositive real part; this implies

(2.6)  u_{∗} − u_{R} ≤ q_{0} ∫_{−∞}^{0}(\frac{1}{g′(u)} + \frac{1}{s})exp(\frac{y}{s} ) dy.

Let us now recall that, in the after-shock zone, there holds

sZ′ = Z, and u′ = \frac{sq_{0}Z′}{g′(u)} = \frac{q_{0}Z}{g′(u)}.

Therefore (2.6) implies

u_{∗} − u_{R} ≤ u_{∗} − u_{L} + q_{0},

an obvious contradiction to (1.5). •

The dispersion relation for system (2) can be obtained in a totally similar fashion, and reads

sq_{0} + λ(u_{∗} − u_{R}) = q_{0}(λ + s) ∫_{−∞}^{0} exp(λ ∫_{y}^{0} \frac{dz}{g′(u)} + y) dy

It is also possible to prove that no eigenvalue of nonpositive real part can satisfy such a relation. The proof is left to the reader.

3. UNSTABLE EIGENVALUES ARE O(\frac{1}{ε})

In the next paragraphs 3 to 5, Assumption 3 will be suppressed.

The strategy followed in this section is inspired from the well-known method used to prove that a second-order elliptic operator is sectorial in L^{2}; see Henry [8], Chapter 1. Some care is, however, needed in estimating the coefficients, especially in the shock layer. All in all, we have

Lemma 3.1. There exists C > 0, independent of ε, such that, for every nontrivial solution of (1.14) with λ ∈ P_{−}, there holds

(3.1)  |λ| ≤ \frac{C}{ε} .

PROOF. Let us first notice that the function Y (x) has the expression

(3.2)  Y (x) = \{ \begin{matrix}& 0 & if x ≥ x_{2}^{ε} \\ ∫_{x_{2}^{ε}}^{x} exp(∫_{y}^{x} \frac{ϕ(u_{ε}) − λ}{s})ϕ′(u_{ε})Z_{ε}v dy & & if x ≤ x_{2}^{ε} \end{matrix}

We shall need the following quantity

\begin{matrix}∫_{−∞}^{+∞} ϕ(u_{ε})Y v̄ dy = & ∫_{−∞}^{x_{1}^{ε}} ϕ(u_{ε})Y v̄ dy + ∫_{x_{1}^{ε}}^{x_{2}^{ε}} ϕ(u_{ε})Y v̄ dy \\ := & I_{1} + I_{2} \end{matrix}

We notice that, if u_{II} < u_{∗}, then I_{2} = 0. We will not, however, make this assumption here. There holds

\begin{aligned}|I_{2}| & ≤ (∫_{x_{1}^{ε}}^{x_{2}^{ε}} |ϕ(u_{ε})v| dy)^{2} \\ & ≤ CεLogε∥v∥_{L^{2}}^{2} \end{aligned}

On the other hand, we have

|I_{1}| ≤ ∫_{−∞}^{x_{1}^{ε}} |v(x)||∫_{x_{2}^{ε}}^{x} exp(∫_{y}^{x} \frac{ϕ(u_{ε}) − λ}{s} dz)|ϕ′(u_{ε})v(y)| dy| dx

≤ C ∫_{−∞}^{x_{1}^{ε}} |v(x)|exp(\frac{1 − Reλ}{s}(x − x_{1}^{ε}))(∫_{x_{1}^{ε}}^{x_{2}^{ε}} |v(y)| dy) dx

≤ \frac{C\sqrt{ε}|Logε|}{1 − Reλ}∥v∥_{L^{2}}

As a consequence, and because Reλ ≤ 0,

(3.3)  | ∫_{−∞}^{+∞} ϕ(u_{ε})Y v̄ dy| ≤ C\sqrt{ε}|Logε|∥v∥_{L^{2}}.

Multiplication of the equation for v by v̄ yields

\frac{ε}{2} ∫_{−∞}^{+∞}(|v′|^{2} + (g′(u_{ε})v)′v̄) dx = λ∥v∥_{L^{2}} + ∫_{−∞}^{+∞}(q_{0}ϕ′(u_{ε})Z_{ε}|v|^{2} + q_{0}ϕ(u_{ε})Y v̄) dx

Taking the real and imaginary parts and remembering (3.3) we have

(3.4)

ε∥v′∥_{L^{2}}^{2} ≤ − ∫_{−∞}^{+∞} g′′(u_{ε})u′_{ε}|v|^{2} dx + (Reλ + C)∥v∥_{L^{2}}
|Imλ| ∥v∥_{L^{2}}^{2} ≤ ∫_{−∞}^{+∞} |g′(u_{ε})(v′_{1}v_{2} − v_{1}v′_{2})| dx
+ C(εLog^{2}ε + C\sqrt{ε}|Logε|)∥v∥_{L^{2}}

The term ∫_{−∞}^{+∞} g′′(u_{ε})u′_{ε}|v|^{2} dx is dangerous, because it may contain terms of the form \frac{1}{ε}∥v∥_{L^{2}}^{2}. We may treat it as follows.

\begin{aligned}∫_{−∞}^{+∞} g′′(u_{ε})u′_{ε}|v|^{2} dx & ≥ O(1)∥v∥_{L^{2}}^{2} + ∫_{x_{1}^{ε}}^{x_{2}^{ε}} g′′(u_{ε})u′_{ε}|v|^{2} dx \\ & ≥ (g(u_{∗}) − g(u_{R}))∥v∥_{∞}^{2} + O(1)∥v∥_{L^{2}}^{2} \end{aligned}

Then we use the classical inequality

∥v∥_{∞}^{2} ≤ ηε∥v′∥_{L^{2}}^{2} + \frac{C_{η}}{ε} ∥v∥_{L^{2}}^{2},

valid for all η > 0; inequalities (3.4) then become, for some constant C > 0 independent of ε:

(3.5)  \begin{aligned}ε∥v′∥_{L^{2}}^{2} & ≤ (Reλ + \frac{C}{ε} )∥v∥_{L^{2}}^{2} \\ |Imλ| ∥v∥_{L^{2}}^{2} & ≤ O(1)(1 + ∥v∥_{L^{2}}∥v′∥_{L^{2}}) \\ & ≤ \frac{ε}{2}∥v′∥_{L^{2}}^{2} + \frac{C}{ε} ∥v∥_{L^{2}}^{2} \end{aligned}

This implies

|Imλ| ≤ C(Reλ + \frac{1}{ε});

hence the lemma. •

4. UNSTABLE EIGENVALUES ARE o(\frac{1}{ε})

The strategy that we are going to use to prove the above statement is to stretch the x-coordinate - both in the shock zone and in the outer zone -, and to let ε to 0; this will contradict a suitably chosen normalisation of v. An intermediate result about the linearized Burgers operator in the stretched variables will be needed. Our objective is thefefore the

Lemma 4.1. For every eigenvalue λ of (1.14) in P_{−}, there holds ελ = o(1).

Let us set once again ξ = \frac{x}{ε}, and introduce the obvious notations ξ_{1}^{ε} and ξ_{2}^{ε} -

recall that ξ_{1}^{ε} < 0 and ξ_{2}^{ε} > 0. For every η ∈]0, \frac{1}{2}[, let α_{η}^{ε} > 0 be chosen so that

(4.1)  ε^{2η} ≤ min(e^{α_{η}^{ε}ξ_{1}^{ε}}, e^{−α_{η}^{ε}ξ_{2}^{ε}}) ≤ max(e^{α_{η}^{ε}ξ_{1}^{ε}}, e^{−α_{η}^{ε}ξ_{2}^{ε}}) ≤ ε^{η};

from the definition of x_{1}^{ε} and x_{2}^{ε} the sequence (α_{η}^{ε})_{ε} is bounded from above and bounded away from 0; therefore we will now call these real numbers α_{η}. On the other hand, we notice that α_{η} tends to 0 as η → 0. In the sequel, the real number η will be taken as small as is convenient to us.

Let w_{η}(ξ) be equal to e^{−α_{η}|ξ|} outside [−1, 1] and suitably modified on [−1, 1] so that it is positive and C^{∞}. If I is an interval of IR, let B_{w_{η},k}(I) be the weighted space with norm

(4.2)  ∥u∥_{w_{η},k} = ∥w_{η}u∥_{C^{k}(I)}.

Let now T_{ε} be defined on

X_{η}^{ε} := {u ∈ C^{2}([x_{1}^{ε}, +∞[), u(x_{1}^{ε}) = 0, −u′′ + u ∈ B_{w_{η},0}([x_{1}^{ε}, +∞[)},

by the expression

(4.3)  T_{ε}v = −v′′ + (g′(u_{B})v)′

The superscript′ here denotes the differentiation with respect to the variable ξ. When ε = 0, i.e. the interval of definition is IR, it will simply be called T .

Lemma 4.2. For every λ ∈ P_{−}^{∗}, the operator T_{ε} − λI is an isomorphism from X_{η}^{ε} to B_{w_{η},0}([x_{1}^{ε}, +∞[). Moreover there exists a function C_{η}(λ), bounded on every compact subset of P_{−}^{∗}, such that

(4.4)  ∀λ ∈ P_{−}^{∗}, ∥(T_{ε} − λI)^{−1}∥_{w_{η},0} ≤ C_{η}(λ).

PROOF. That T_{ε} − λI is an isomorphism from X_{ε} to C([ξ_{1}^{ε}, +∞[) comes from the fact that T_{ε}u′_{B} = 0 and u′_{B} < 0; one concludes with the maximum principle. See [8], Chap. 5 to see how the argument works. Let us now assume the estimate of the lemma to be false; if this is so then there exists a sequence (f_{ε})_{ε} going to 0 and a sequence (u_{e})_{ε}; with ∥u_{ε}∥ = 1, such that ∥T_{ε}u_{ε} − λI∥_{w,0} → 0. Let ζ_{ε} be a maximum point for |w_{η}u_{ε}|; still calling u_{ε} the function ξ ↦ u_{ε}(ξ − ζ_{ε}), we end up, by sending ε to 0, to the existence of

- an unbounded interval I,

- a linear operator T̃, defined on I,

- a bounded nonzero function u, such that

- T̃ is either T + λI, or an second order operator with zero-order coefficient of the form μ + λ, μ ∈ IR_{+}^{∗},

- there holds T̃u = 0. This is a contradiction in all cases. •

PROOF OF LEMMA 4.1. Let us normalize v so that

(4.5)  ∥v∥_{L^{1}(IR)} = 1.

Let us fix η > 0 so that the above considerations work. Let us assume the existence of a sequence (λ_{ε})_{ε} such that (ελ_{ε})_{ε} tends to a constant μ ∈ P_{−}^{∗}.

We have

(4.6)  ∥Y ∥_{∞} ≤ C(εLogε∥v∥_{L^{∞}([x_{1}^{ε},x_{2}^{ε}])} + ∥v∥_{L^{∞}(]−∞,x_{1}^{ε}])}).

Indeed, the expression of Y (3.2) implies that, for x ≤ x_{2}^{ε} there holds

|Y (x)| = | ∫_{x_{2}^{ε}}^{x} exp(∫_{y}^{x} \frac{ϕ(u) − λ}{s})ϕ′(u_{ε})Z_{ε}v dy|

= ∫_{−∞}^{x_{1}^{ε}} ... + ∫_{x_{1}^{ε}}^{+∞} ...
≤ C((x_{2}^{ε} − x_{1}^{ε})∥v∥_{L^{∞}([x_{1}^{ε},x_{2}^{ε}])} + ∥v∥_{L^{∞}(]−∞,x_{1}^{ε}])})

hence the result.

1. Outer expansion. Let us see what happens when x ≤ x_{1}^{ε}. Let us set

μ_{ε} = ελ, ξ = \frac{x − x_{3}^{ε}}{ε}; let us also set, for i ∈ \{1, 2, 3\}: ξ_{i}^{ε} = \frac{x_{i}^{ε} − x_{1}^{ε}}{ε}. There holds

(4.7)  L_{μ}v := −v′′ + (g′(u)v)′ − μv = εq_{0}Y.

We claim that, if μ ∈ P_{−}^{∗}, then L_{μ}, with Dirichlet condition at ξ = 0, is an isomorphism between the space of bounded uniformly continuous functions on ] − ∞, 0] and its domain; moreover the norm of its inverse is bounded independently of ε. To see this, one may notice that, for x ≤ x_{3}^{ε}, there holds u′_{ε} > 0 because of Lemma 1.4; then the same analysis as in Lemma 4.2 applies.

All the solutions of L_{μ}v = 0, v(0) = v_{0} satisfy

(4.8)  |v(ξ)| ≤ C|v_{0}|e^{−kξ}

for some constants k > 0, C > 0 independent of ε. From (4.8) and the uniform boundedness of L_{μ}^{−1}, there exists C > 0 such that

|v(x)| ≤ Ce^{k^{\frac{x−x^{ε}_{3}}{ε}}} |v(x_{3}^{ε})| + O(ε)∥Y ∥_{∞}.

Hence there exists k > 0 such that there holds, for all x ≤ x_{1}^{ε} - because of (1.15):

\begin{aligned}|v(x)| & ≤ C(∥Y ∥_{∞} + ε^{k}|v(x_{3}^{ε})|) \\ & ≤ Cε^{k}(1 + ∥v∥_{L^{∞}(]−∞,x_{3}^{ε}])}) \end{aligned}

The real number k may indeed take any value in ]0, +∞[; in the above inequality we have renamed k the minimum of the two numbers k and 1. As a consequence we have

(4.9)  ||v(x)||_{L^{∞}(]−∞,x_{1}^{ε}])} ≤ Cε^{k}∥v∥_{L^{∞}([x_{1}^{ε},x_{3}^{ε}])}.

2. Inner expansion and conclusion. Let us now apply Lemma 4.2; one of its consequences is that

∥v∥_{w_{η},0} ≤ C(ε^{−2η}|v(x_{1}^{ε})| + ε) ≤ C(ε^{−2η}∥v∥_{L^{∞}(]−∞,x_{1}^{ε}])} + ε);

using (4.9) we find out that

(4.10)  ∥v∥_{w_{η},0} ≤ Cε^{k−2η}

However, this inequality in turn implies that the L^{1}-norm of v is of order ε^{k−2η}.

Choosing η < \frac{k}{2} contradicts (4.5).

5. UNSTABLE EIGENVALUES ARE O(1)

We may now prove Theorem 1.1. Now that we know ελ to be negligible, the quantity ελI is a perturbation of the Burgers operator in the stretched coordinates. In the external zone, the linear problem contains two large parameters of a priori

different scales: \frac{1}{ε} and λ, whose locations in the equation forbid a classical smallparameter analysis - see [9], Section 3. An extra lemma is therefore required.

The bound on λ will be obtained as follows: we will perform an inner expansion of v and Y just as in the formal derivation of the dispersion relation; if, on the other hand, λ is too large, we will contradict the matching relations at x = x_{1}^{ε}.

Let us therefore assume the existence of an unbounded family (λ_{ε})_{ε} of eigenvalues in P_{−}; it can be assumed that lim_{ε→0} |λ_{ε}| = +∞. In the course of the proof we will

simply set λ_{ε} := λ; what is given to us by the preceding section is ελ = o(1), and this will be used in the next

Lemma 5.1. Let (a_{ε}(x))_{ε} be a sequence of functions, bounded in C^{1}(] − ∞, x_{3}^{ε}]). The problem

(5.1)  −εv′′ + (g′(u_{ε})v)′ − (a_{ε}(x) + λ)v = 0, x < x_{3}^{ε}

has two solutions u_{1}^{−}(λ, x) and u_{2}^{−}(λ, x) such that • there holds

(5.2)  u_{1}^{−}(λ, x) = \frac{1}{g′(u_{ε})}exp(λ(∫_{x_{3}^{ε}}^{x} \frac{dy}{g′(u_{ε}(y))} + O(ελ)x)),

• there holds

(5.3)  u_{2}^{−}(λ, x) = exp(∫_{x_{3}^{ε}}^{x}(\frac{g′(u_{ε}(y))}{ε} − \frac{λ}{g′(u_{ε}(y))} + O(ελ^{2})) dy).

Although the functions u_{i}^{−} obviously depend of ε, we have chosen not to make their dependence with respect to ε more explicit, in order to keep the notations as light as possible.

PROOF. The result is would be obvious if g′(u_{ε}) was a constant. The idea is

that, in the coordinate ξ = \frac{x − x_{3}^{ε}}{ε}, the function g′(u_{ε}) is almost a constant. Because

x < x_{3}^{ε}, Lemma 1.4 tells us that u′_{ε}(x) = O(1); hence \frac{du_{ε}}{dξ} = O(ε).

Let us first look for u_{1}^{−} under the form u_{1}^{−}(λ, x) = e^{φ(ξ)}. The function φ is a solution of the Riccati equation

−(φ′′ + φ′^{2}) + g′(u_{ε})φ′ = ε(λ + a_{ε}(εξ) − g′′(u_{ε})u′_{ε});

we by setting ψ = φ′ we transform the above equation into

(5.4)  ψ′ − g′(u_{ε})ψ + ψ^{2} = −ε(λ + a_{ε}(εξ) − g′′(u_{ε})u′_{ε}).

We now look for ψ as

ψ(ξ) = ε\frac{λ + a_{ε} − g′′(u_{ε})u′_{ε}}{g′(u_{ε})} + ζ;

we obtain an equation of the form

ζ′ − (g′(u_{ε}) + k_{ε,λ}(ξ))ζ + ζ^{2} = h_{ε,λ}(ξ).

Because of Lemma 1.4 we have k_{ε,λ} = O(ελ) and h_{ε,λ}(ξ) = O(ε^{2}λ^{2}); these estimates hold uniformly with respect to ξ. To obtain the function ζ, it only remains to apply the Banach fixed point theorem to the formula

ζ(ξ) = ∫_{0}^{ξ} exp(∫_{ξ′}^{ξ}(g′(u_{ε}) + k_{ε,λ})) (−ζ^{2} + h_{ε,λ}(ξ′)) dξ′,

in the vicinity of 0. This fixed point will work because g′(u_{ε}) is bounded from below by a positive constant. We obtain ζ = O(ε^{2}λ^{2}), which is exactly what we require.

As for the function u_{2}^{−}, let us notice that we may also look for it under the form u_{2}^{−}(λ, x) = e^{φ(ξ)}; setting once again φ′ = ψ we end up with (5.4). This time we look for ψ as the largest root ψ_{0} of equation

h_{ε,λ}(ψ) := ψ^{2} − g′(u_{ε})ψ + ελ = 0;

we have

ψ_{0}(ξ) = g′(u_{ε}) − \frac{ελ}{g′(u_{ε})} + O(ε^{2}λ^{2}).

Setting ζ := ψ − ψ_{0} we obtain, this time, an equation for ζ under the form

ζ′ + h′_{ε,λ}(ψ_{0})ζ = k_{ε,λ}(ξ, ζ) = O(ζ^{2}) + O(ε);

moreover h′_{ε,λ}(ψ_{0}) = g′(u_{ε}) + O(ελ). The function ζ may thus be obtained as the unique O(ε) fixed point of the integral equation

ζ(ξ) = ∫_{−∞}^{ξ} exp(− ∫_{ξ′}^{ξ} h′_{ε,λ})) (k_{ε,λ}(ξ′, ζ)) dξ′,

This gives the function u_{2}^{−}, as required. •

PROOF OF THEOREM 1.1. Assume, without loss of generality, that

(5.5)  g(\frac{u_{∗} + u_{R}}{2} ) = 0.

This time the normalisation for v will be

(5.6)  v(0) = \frac{u′_{B}(0)}{ε}.

1. Inner expansions. We look for an expansion of v of the form

(5.7)  \begin{aligned}v & = \frac{u′_{B}(ξ)}{ε} + v_{1}^{+}(λ, ξ) \\ Y & = Y_{1}^{+} + εY_{2}^{+} \end{aligned}

with

sY_{1}^{+}′ = −ελY_{1}^{+} + ϕ′(u_{ε})u′_{ε},

and it will be our task to prove that v_{1}^{+} and Y_{2}^{+} are of order 1. Let us recall that we have, in the scaled variables

(5.8)  \{ \begin{matrix}−v′′ + (g′(u_{ε})v)′ & = sq_{0}Y′ + ελ(v + q_{0}Y) \\ sY′ & = ε(ϕ(u_{ε})Y + ϕ′(u_{ε})Z_{ε}v)−ελY \end{matrix}

1.a Estimate for Y_{1}^{+}.

The equation for Y_{1}^{+} in (5.8) implies

\begin{aligned}Y_{1}^{+}(ξ) & = − ∫_{ξ}^{+∞} e^{− \frac{ελ}{s} (ξ−ζ)}ϕ′(u_{ε})u′_{ε} dζ \\ & = \frac{ϕ(u_{ε})}{s} + ελ ∫_{ξ}^{ξ_{2}^{ε}} e^{− \frac{ελ}{s} (ξ−ζ)}ϕ(u_{ε}) dζ \end{aligned}

Hence there holds, because Reλ ≤ 0:

(5.9)  Y_{1}^{+}(ξ) = \frac{ϕ(u_{ε})}{s} + O(ελLogε).

2.b Estimate for Y_{2}^{+}. The equation for Y_{2}^{+} reads

sY_{2}^{+}′ = −ελY_{2}^{+} + ϕ(u_{ε})Y_{1}^{+} + ϕ′(u_{ε})Z_{ε}v_{1}^{+} + O(ε)Y_{2}^{+};

hence

Y_{2}^{+}(ξ) = − ∫_{ξ}^{+∞} e^{− \frac{ελ+O(ε)}{s} (ξ−ζ)}(ϕ(u_{ε})Y_{1}^{+} + ϕ′(u_{ε})Z_{ε}v_{1}^{+}) ds := Mv_{1}^{+}.

The operator M is an affine integral operator and we have, from the same considerations as in 1.a and from (4.1):

(5.10)  \begin{aligned}|Y_{2}^{+}(ξ)| & ≤ C_{η}ε^{−η}(1 + ∥v_{1}^{+}∥_{∞}1I_{[ξ_{1}^{ε},ξ_{2}^{ε}]}(ξ)) \\ ∥Y_{2}^{+}∥_{B_{wη} ([ξ_{1}^{ε},+∞[)} & = O(ε^{−η})(1 + ∥v_{1}^{+}∥_{B_{wη} ([ξ_{1}^{ε},+∞[)}) \end{aligned}

where 1I_{A} denotes the characteristic function of A.

1.c Estimate for v_{1}^{+}.

The equation for v_{1}^{+} writes

\begin{matrix}−v_{1}^{+}′′ & + (g′(u_{B})v_{1}^{+})′= −(ελ + εϕ′(u_{B}))v_{1}^{+} + O(ε^{2}Logε)v_{1}^{+} \\ + & (λ + q_{0}ϕ′(u_{ε})Z_{ε})u′_{ε} − (g′′(ũ)u_{1}u′_{B})′+ε^{2}Mv_{1}^{+} + O(ε)1I_{[ξ_{1}^{ε},ξ_{2}^{ε}]}(ξ) \end{matrix}

where g′′(ũ) stands for the first-order Taylor expansion of g′(u_{ε}); we have set u_{ε} = u_{B} + εu_{1}, with u_{1} = O(Logε). The conditions are

(5.11)  v_{1}^{+}(λ, 0) = 0, v_{1}^{+}(λ, +∞) = 0.

For a given η > 0, and given f ∈ B_{w_{η}}([0, +∞[), there exists a unique function v ∈ C^{2}([0, +∞[ such that

−v′′ + (g′(u_{B})v)′= f, v(0) = v(+∞) = 0.

This function v is given by

(5.12)  M_{1}v = ∫_{0}^{ξ}(∫_{ζ}^{+∞} f(η) dη)exp(∫_{ζ}^{ξ} g′(u_{B}(η)) dη) dζ.

The operator M_{1} is a linear operator in B_{w_{η}}([0, +∞[); there holds

∥M_{1}∥_{w_{η},1} = O(1).

Recall that B_{w_{η},k}(I) is given by definition (4.2).

Let us come back to the function v_{1}^{+}; let us set

(5.14)  v_{1}^{+} = M_{1}((λ + q_{0}ϕ′(u_{ε})Z_{ε})u′_{ε} − (g′′(ũ)u_{1}u′_{B})′)+ṽ_{1}^{+} := v̄ + ṽ_{1}^{+};

we have

(5.15) \begin{matrix}ṽ_{1}^{+}(λ, ξ) = & ε^{2}M_{1}Mv_{1}^{+} \\ + & M_{1}(O(ε)1I_{[ξ_{1}^{ε},ξ_{2}^{ε}]} − (ελ + εϕ′(u_{B}))(ṽ_{1}^{+} + v̄) + O(ε)(v̄ + ṽ_{1}^{+})′) \end{matrix}

which implies, because ελ = o(1):

(5.16)  |ṽ_{1}^{+}(λ, 0)| + |ṽ_{1}^{+}′(0)| = O(ελ).

Now, we need to follow the function ṽ_{1}^{+} down to ξ = ξ_{1}^{ε}. Write down (5.15) as the following second-order equation:

(5.17)  \begin{matrix}− ṽ_{1}^{+}′′ + (g′(u_{B})ṽ_{1}^{+})′−(ελ + εϕ′(u_{B}))ṽ_{1}^{+} \\ = & ε^{2}Mv_{1}^{+} + (O(ε)1I_{[ξ_{1}^{ε},ξ_{2}^{ε}]} − (ελ + εϕ′(u_{B}))v̄ + O(ε)(v̄ + ṽ_{1}^{+})′) \end{matrix}

The homogeneous equation associated to (5.17) has two independent solutions U_{1}^{ε} and U_{2}^{ε} such that there holds, as ξ → −∞:

|U_{1}^{ε}(ξ)| ≤ Cu′_{B}(ξ); |U_{2}^{ε}(ξ)| ≤ Ce^{−O(εReλ)ξ}

for some C > 0 independent of ε. This can be proved exactly as in the classical analysis of asymptotically homogeneous systems; see Coddington-Levinson [3]. The variation of constants formula and (5.16) then imply - the computation is left to the reader -

∥ṽ_{1}^{+}∥_{L^{∞}([ξ_{1}^{ε},0])} = 0(ελLogεe^{O(εLogε Reλ)}).

As a consequence of (5.14) and the definition of M_{1} (5.12), we have

(5.18)

v_{1}^{+}(λ, ξ_{3}^{ε}) =v̄(ξ_{3}^{ε}) + O(εLogελe^{O(εLogε Reλ)})
=λ(u_{∗} − u_{R}) + q_{0}ϕ(u_{∗}) + O(εLogελe^{O(εLogε Reλ)})
=(u_{∗} − u_{R} + O(ε^{1+O(εReλ)}Logε))λ + q_{0}ϕ(u_{∗})

The same will be true for v_{1}^{+}(λ, ξ_{1}^{ε}).

2. Outer expansion and conclusion. The same argument as in the proof of estimate (4.10) will be used, but this time we need more precise asymptotics. From (3.2) we have, for all x ∈] − ∞, x_{3}^{ε}]:

(5.19)  \begin{aligned}Y (x) & = ∫_{x_{3}^{ε}}^{x} Z_{ε}(y)e^{∫{x \atop y} \frac{ϕ(uε)−λ}{s}} ϕ′(u_{ε})v dy + Y (x_{3}^{ε})e^{∫{x \atop x^{ε}_{3}} \frac{ϕ(uε)−λ}{s}} \\ & = Y_{1}^{−}(x) + Y_{2}^{−}(x) \end{aligned}

We incidentally notice that we have

Y_{1}^{−}(x) = Z_{ε}(x) ∫_{x_{3}^{ε}}^{x} e^{\frac{−λ}{s} (x−y)}ϕ′(u_{ε})v dy.

On the other hand, let W_{ε}(x) be the wronskian of (5.1); we have

W_{ε}(x) = W_{ε}(x_{3}^{ε})exp(−ε^{−1} ∫_{x_{3}^{ε}}^{x} g′(u_{ε}) dz) dy

and

W_{ε}(x_{3}^{ε}) = | \begin{matrix}u_{1}^{−}(λ, x_{3}^{ε}) & u_{2}^{−}(λ, x_{3}^{ε}) \\ u_{1}^{−}(λ, x_{3}^{ε})′ & u_{2}^{−}(λ, x_{3}^{ε})′ \end{matrix} |

= | \begin{matrix}\frac{1}{g′(u_{ε}(x_{3}^{ε}))} & 1 \\ −\frac{g′′(u_{ε}(x_{3}^{ε}))}{g′(u_{ε}(x_{3}^{ε}))^{2}} + \frac{λ}{g′(u_{ε}(x_{3}^{ε}))^{2}} & \frac{g′(u_{ε}(x_{3}^{ε}))}{ε} − \frac{λ}{g′(u_{ε}(x_{3}^{ε}))} \end{matrix} |

= \frac{1 + O(ελ)}{ε}

For every f in L^{1}(] − ∞, x_{3}^{ε}]) ∩ L^{∞}(] − ∞, x_{3}^{ε}]), the only solution of

(5.20)  \begin{aligned}−εv′′ + (g′(u_{ε})v)′ − (λ + ϕ′(u_{ε})Z_{ε})v & = f(x), x < x_{3}^{ε} \\ v(−∞) = 0, v(x_{3}^{ε}) & = α \end{aligned}

is given by

(5.21)

(1 + O(ελ))v(x)

= −\frac{u_{2}^{−}(λ, x)}{u_{2}^{−}(λ, x_{3}^{ε})}u_{1}^{−}(λ, x) ∫_{−∞}^{x_{3}^{ε}} f(y)u_{2}^{−}(λ, y)exp(− ∫_{x_{3}^{ε}}^{y} ε^{−1}g′(u_{ε}) dz) dy

+ \frac{u_{2}^{−}(λ, x)}{u_{2}^{−}(λ, x_{3}^{ε})} ∫_{x}^{x_{3}^{ε}} f(y)u_{1}^{−}(λ, y)exp(− ∫_{x_{3}^{ε}}^{y} ε^{−1}g′(u_{ε}) dz) dy

+ u_{1}^{−}(λ, x) ∫_{−∞}^{x} f(y)u_{2}^{−}(λ, y)exp(− ∫_{x_{3}^{ε}}^{y} ε^{−1}g′(u_{ε}) dz) dy + α\frac{u_{2}^{−}(λ, x)}{u_{2}^{−}(λ, 0)}

:= ∑_{i=1}^{3} T_{i}(λ)(Y_{1}^{−} + Y_{2}^{−})(x) + α\frac{u_{2}^{−}(λ, x)}{u_{2}^{−}(λ, 0)}

Therefore we are led to estimating six functions T_{i}(λ)Y_{j}, for (i, j) ∈ {1, 2, 3} × {1, 2}. We claim the existence of C > 0, independent of ε and λ, such that

(5.22)  ∀i ∈ \{1, 2, 3\}, \begin{aligned}∥T_{i}(λ)Y_{1}^{−}∥_{L^{∞}(]−∞,x_{3}^{ε}])} & ≤ \frac{C}{|λ|}∥v∥_{L^{∞}(]−∞,x_{3}^{ε}])} \\ ∥T_{i}(λ)Y_{2}^{−}∥_{L^{∞}(]−∞,x_{3}^{ε}])} & ≤ C(ε + \frac{1}{|λ|}) \end{aligned}

The more complex estimates concern the T_{i}(λ)Y_{1}^{−}; as for the T_{i}(λ)Y_{2}^{−}, their computation is almost explicit as soon as estimate (5.9) is known. Therefore the relevant estimate for them can be checked by inspection.

The estimates for the T_{i}(λ)Y_{1}^{−} are all based on the same integration by parts principle. Therefore we are only going to check T_{3}(λ)Y_{1}^{−}, leaving the two remaining estimates to the reader. We have, from (5.21) and Lemma 5.1- with a_{ε} = ϕ′(u_{ε})Z_{ε} and f(x) = q_{0}ϕ(u_{ε})(Y_{1}^{−} + Y_{2}^{−}):

(1 + O(ελ))T_{3}(λ)Y_{1}^{−}

=q_{0} ∫_{−∞}^{x} ϕ(u_{ε})Y_{1}^{−}(y)exp(λ ∫_{y}^{x} \frac{1 + O(ελ)}{g′(u_{ε}(z))} dz) dy
=q_{0} ∫_{−∞}^{x} ϕ(u_{ε})Z_{ε}(y)(∫_{x_{3}^{ε}}^{y} e^{\frac{λ}{s} z}ϕ′(u_{ε})v dz)exp(−\frac{λ}{s}y − λ ∫_{y}^{x} \frac{1 + O(ελ)}{g′(u_{ε}(z))} dz) dy

= −[q_{0}ϕ(u_{ε})\frac{Y_{1}^{−}(y)}{(1 + O(ελ))λ(\frac{1}{s} + \frac{1}{g′(u_{ε})})}exp(λ ∫_{y}^{x} \frac{1 + O(ελ)}{g′(u_{ε}(z))} dz)]_{−∞}^{x_{3}^{ε}}

+ ∫_{−∞}^{x_{3}^{ε}}[\frac{(1 + O(ελ))Z_{ε}}{\frac{λ}{s} + \frac{λ}{g′(u_{ε})}} ]′exp(−\frac{λ}{s}y + λ ∫_{y}^{x} \frac{1 + O(ελ)}{g′(u_{ε}(z))} dz)ϕ(u_{ε})v(y)dy

=O(\frac{1}{|λ|}∥v∥_{L^{∞}(]−∞,x_{3}^{ε}])})

All in all, we have obtained

∥v∥_{L^{∞}(]−∞,x_{3}^{ε}])} ≤ C(\frac{∥v∥_{L^{∞}(]−∞,x_{3}^{ε}])}}{|λ|} + \frac{1}{|λ|} + ε + exp ∫_{x_{3}^{ε}}^{x} \frac{g′(u_{ε})}{ε});

hence we have, in particular:

|v(x_{1}^{ε})| = O(\frac{1}{|λ|}) + O(εLogε).

This obviously contradicts (5.18). •

6. EIGENVALUE PROPERTIES

This paragraph is devoted to the proof of Theorems 1.2 and 1.3; now that we know the potential unstable eigenvalues to be bounded, we only require classical tools of singular perturbation theory [9] or geometric singular perturbation theory [10]; and we are only going to give a brief sketch of the proof.

Let us introduce the Evans function of the problem; see [1] for its definition and main properties. We construct it as follows:

a. It is possible to improve (5.7) and (5.18), to show that the function (ε, λ)λ ↦ v_{1}^{+}(λ, .) belongs to C^{1}([0, ε_{0}] × B(0, 1), C([ξ_{1}^{ε}, +∞[), and is even analytic with respect to λ, for ε_{0} small enough. This just stems from the analyticity of the equation with respect to its parameters.

b. Section 5 also tells us that the linear problem (1.14), posed on ] − ∞, x_{3}^{ε}], has

a unique L^{2} solution (v_{ε}^{−}, Y_{ε}^{−}) such that Y_{ε}^{−}(x_{3}^{ε}) = \frac{ϕ(u_{∗})}{s}. Moreover, this solution is smooth in ε and λ, for small enough ε and bounded λ. We call v_{ε}^{−}(λ, .) the vcomponent and Y_{ε}^{−}(λ, .) the Y -component.

c. Call (v_{ε}^{+}(λ, .), Y_{ε}^{+}(λ, .) the functions given by (5.7) and computed all through

step 1 of the proof of Theorem 1.1. We have Y_{ε}^{+}(λ, x_{3}^{ε}) = \frac{ϕ(u_{∗})}{s}.

d. Finally, let us emphasize the dependence with respect to ε of the function u_{2}^{−} of Lemma 5.1 by denoting it u_{2}^{−,ε}. The Evans function of (1.14) is given by

(6.1) E_{ε}(λ) = det ( \begin{matrix}u_{2}^{−,ε}(λ, x_{1}^{ε}) & v_{ε}^{−}(λ, x_{1}^{ε}) & v_{ε}^{+}(λ, x_{1}^{ε}) \\ u_{2}^{−,ε}(λ, x_{1}^{ε})′ & v_{ε}^{−}(λ, x_{1}^{ε})′ & v_{ε}^{+}(λ, x_{1}^{ε})′ \\ 0 & \frac{ϕ(u_{∗})}{s} + O(εLogε) & \frac{ϕ(u_{∗})}{s} + O(εLogε) \end{matrix} )

Lemma 6.1. Consider the function Γ(λ) defined in Section 2.The functions E_{ε}(λ) is analytic in intP_{−} and continuous up to the imaginary axis. Moreover there holds

(6.2)  lim_{ε→0} \frac{εE_{ε}(λ)}{λg′(u_{ε}(x_{1}^{ε}))} = Γ(λ)

uniformly on every compact subset of P_{−}.

PROOF. From Lemma 5.1 we know that εu_{2}^{−,ε}′(λ, x_{1}^{ε}) ∼ g′(u_{ε}(x_{1}^{ε}), and Step 1 of the proof of Theorem 1.1 tells us that v_{ε}^{+}(λ, x_{1}^{ε}) = O(εLogε). Inspection of identity (5.21) reveals that all the terms factored with u_{2}^{−,ε} become O(ε) at x = x_{1}^{ε}. Hence the dominant term is T_{3}Y_{1}. More precisely we have

(6.3)  v^{−}(λ, x_{1}^{ε}) ∼ \frac{q_{0}Y_{0}}{g′(u)} ∫_{−∞}^{x} exp(λ ∫_{y}^{x} \frac{dz}{g′(u(z))} + \frac{1 − λ}{s}y) dy + O(εLogε)

This identity also holds for the derivatives of both quantities with respect to λ; because λ is now bounded this may be inferred from Lemma 5.1. From these considerations the limit (6.3) is true on P_{−}^{∗} However, λ = 0 is an eigenvalue; therefore

\frac{εE_{ε}(λ)}{λg′(u_{ε}(x_{1}^{ε}))} exists and converges, uniformly on bounded subsets in λ ∈ P_{−}, to the

function Γ(λ) defined in Section 2.

The continuity up to the boundary is trivial. •

From now on we set

(6.4)  Γ_{ε}(λ) = \frac{εE_{ε}(λ)}{λg′(u_{ε}(x_{1}^{ε}))}.

PROOF OF THEOREM 1.2. We assume the existence of a sequence of eigenvalues (λ_{ε})_{ε} ∈ P_{−} which may, thanks to the preceding three sections, be assumed to converge to λ ∈ P_{−}. From Proposition 2.1 we have Γ(λ) = 0: a contradiction. •

PROOF OF THEOREM 1.3. The argument that we are going to give is quite reminiscent of the one in [2].

Assume the existence of a sequence (λ_{ε})_{ε} of eigenvalues in IR_{−}. We always may assume the existence of

λ ∈ P_{−} := lim_{ε→0} λ_{ε}.

Let ϕ_{0} be any function satisfying Assumptions 1-3; Theorem 1.2is therefore applicable to ϕ_{0}. Set ϕ_{t} = tϕ_{0} + (1 − t)ϕ; we have ϕ_{1} = ϕ. We denote by Γ_{ε}^{t} the corresponding dispersion functions, and Γ^{t} their limits as ε → 0. A straightforward adaptation of Rouché’s Theorem tells us that there exists t_{0} ∈ [0, 1], a sequence (t_{n})_{n} converging to t_{0} and a sequence (λ_{n})_{n} of real negative zeroes of Γ_{ε}^{t_{n}} such that

(i) either (λ_{n})_{n} converges to 0,

(ii) or (λ_{n})_{n} converges to a real negative μ double zero of Γ_{ε}^{t_{0}}.

In both cases, there is a t_{0} ∈ [0, 1] such that Γ_{ε}(t_{0}) has a double zero; therefore [1], (1.14) has a double real nonpositive eigenvalue. This is impossible due to the fact that (u_{2}^{−,ε}, v_{ε}^{−}) form an independent system, as can be seen from their asymptotic behaviours x → −∞. •

REFERENCES

[1] - J.C. ALEXANDER, R. GARDNER, C.K.R.T. JONES, A topological invariant arising in the stability analysis of travelling waves, J. reine angew Math., 410 (1990), pp. 167-212.

[2] - J.C. ALEXANDER, M.G. GRILLAKIS, C.K.R.T. JONES, B. SANDST-EDE, Stability of pulses on optical fibers with phase-sensitive amplifiers, Z. Angew. Math. Phys., 48 (1997), pp. 175-192.

[3] - E.A. CODDINGTON, N. LEVINSON, Ordinary differential equations, Mc-Graw Hill, New York, 1955.

[4] - N. FENICHEL, Geometric singular perturbation theory, J. Diff. Eq., 31 (1979), pp. 53-98.

[5] - R.A. GARDNER, On the detonation of a combustible gas, Trans. Am. Math. Soc., 277 (1983), pp. 431-468.

[6] - I. GASSER, P. SZMOLYAN, A geometric singular perturbation analysis of detonation and deflagration waves, SIAM J. Math. Anal., 24 (1993), pp. 968-986.

[7] - L. GLANGETAS, J.M. ROQUEJOFFRE, Bifurcations of travelling waves in the thermo-diffusive model for flame propagation, Arch. Rat. Mech. Anal., 134 (1996), pp. 341-402.

[8] - D. HENRY, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer Verlag, New-York, 1981.

[9] - F. HOPPENSTEADT, Properties of solutions of ordinary differential equations with small parameters, Comm. Pure Appl. Math., 24 (1971), pp. 807-840.

[10] - C.K.R.T. JONES, Geometric singular perturbation theory, CIME Lectures on dynamical systems (1994), Springer.

[11] - B. LARROUTUROU, Remarks on a model for combustion waves, Nonlinear Analysis, TMA, 9 (1985), pp. 905-935.

[12] - A. LEVY, On Majda’s model for dynamic combustion, Comm. Partial Diff. Eq., 17 (1992), pp. 657-698.

[13] - T.P. LIU, L.A. YING, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), pp. 519-528.

[14] - A. MAJDA, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41 (1981), pp. 70-91.

[15] - A. MAJDA, R, ROSALES, Weakly nonlinear detonation waves, SIAM J. Appl. Math., 43 (1983), pp. 1186-1118.