C. R. Acad. Sci. Paris, t. 330, Série I, p. 77–81, 2000Analyse mathématique/Mathematical Analysis(Problèmes mathématiques de la mécanique/Mathematical Problems in Mechanics)

Instability of the free boundary in a two-dimensionalcombustion modelClaude-Michel BRAUNER a, Alessandra LUNARDI b

a Mathématiques appliquées de Bordeaux, Université Bordeaux I, 33405 Talence cedex, FranceE-mail: brauner@math.u-bordeaux.fr

b Dipartimento di Matematica, Via D’Azeglio 85/A, 43100 Parma, ItaliaE-mail: lunardi@prmat.math.unipr.it

(Reçu le 4 novembre 1998, accepté après révision le 23 novembre 1999)

Abstract. We prove instability of the front in a free boundary problem inR2 modelling premixedflame propagation, transforming it to a fully nonlinear problem with fixed boundary. Thedifficulty due to the spectrum of the linear part, which contains an interval[0, ωc],ωc > 0, isovercome by an argument from the theory of dynamical systems in Banach spaces. 2000Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Instabilité de la frontière libre dans un modèle de combustionbidimensionnel

Résumé. Nous montrons l’instabilité du front dans un problème à frontière libre dans le plan,modélisant la propagation de flammes prémélangées, en le transformant en un problèmetotalement non linéaire dans un domaine fixe. La difficulté due au spectre de l’opérateurlinéarisé, qui contient un intervalle[0, ωc], ωc > 0, est résolue en faisant appel à lathéorie des systèmes dynamiques dans les espaces de Banach. 2000 Académie dessciences/Éditions scientifiques et médicales Elsevier SAS

Version française abrégée

Dans le cadre des flammes de prémélange proches de l’équidiffusion, le modèle thermo-diffusif donnelieu à un problème à frontière libre dans le plan ([8,9]) pour les inconnuesθ et S, correspondantrespectivement à la température et à l’enthalpie,

∂θ

∂t+∂θ

∂x= ∆θ, x < s(t, y), θ= 1, x> s(t, y),

∂S

∂t+∂S

∂x= ∆S − λ∆θ, x 6= s(t, y),

θ(t,−∞, y) = S(t,−∞, y) = S(t,+∞, y) = 0.

Note présentée par Gérard IOOSS.

0764-4442/00/03300077 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.Tous droits réservés.

77

C.-M. Brauner, A. Lunardi

Au front s, θ etS sont continus, tandis que les dérivées normales satisfont des conditions de saut[∂θ

∂n

]=− exp(S),

[∂S

∂n

]= λ

[∂θ

∂n

].

Ce système admet une solution «onde progressive », unique à une translation près,

s= 0, θ0 = expx, S0 = λx expx si x < 0, θ0 = 1, S0 = 0 si x> 0.

Dans cette Note, nous démontrons rigoureusement que l’onde progressive et le front sont instables. Laméthode est la suivante ([3–5]) : après avoir fixé le front enx= 0, nous décomposonsθ etS en

θ= θ0 + sθ0x + v, S = S0 + sS0

x +w,

et nous éliminons le fronts grâce aux conditions de saut sur les dérivées normales. On arrive au problèmetotalement non linéaire

ut = Lu +F(u), B(u) = G(u).

En plus de l’existence et de l’unicité locale en temps, notre résultat principal concerne l’instabilitéponctuelle du front. On s’appuie sur un théorème de Henry [6] pour certains systèmes dynamiques dans lesespaces de Banach, que nous adaptons à notre situation. La remarque principale est que l’élément maximalωc du spectre de l’opérateur linéariséL avec des conditions aux limites homogènes est une valeur propre.Ceci nous permet de construire, pour des conditions initiales arbitrairement proches de l’onde progressive,une solution telle que le front soit minoré, en valeur absolue, par une constante, ceci au moins pour untemps grand.

1. Introduction

In the context of near-equidiffusion flames the diffusional-thermal model gives rise to a free boundaryproblem inR2 ([8,9]) for the unknownsθ andS, corresponding to the temperature and the enthalpy,

∂θ

∂t+∂θ

∂x= ∆θ, x < s(t, y), (1)

θ= 1, x> s(t, y), (2)

∂S

∂t+∂S

∂x= ∆S − λ∆θ, x 6= s(t, y), (3)

θ(t,−∞, y) = S(t,−∞, y) = S(t,+∞, y) = 0. (4)

At the fronts, θ andS are continuous, while their normal derivatives satisfy the jump conditions[∂θ

∂n

]=− exp(S),

[∂S

∂n

]= λ

[∂θ

∂n

]. (5)

The system (1)–(5) admits a planar travelling wave solution, unique up to translations:

s0 = 0, θ0 = expx, S0 = λx expx if x< 0, θ0 = 1, S0 = 0 if x> 0. (6)

In this Note we prove rigorously that forλ > 1 the travelling wave and the front are unstable.

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Free boundary instability

2. Reformulation as a fully nonlinear problem

We use the method of our previous works [3–5,1], splitting (after fixing the front atx= 0):

θ = θ0 + sθ0x + v, S = S0 + sS0

x +w, (7)

and eliminating the fronts thanks to conditions (5),

s(t, y) = [v] =−v(t,0−, y). (8)

We arrive at afully nonlinearevolution equation foru = (v,w),

ut = Lu +F(u), B(u) = G(u), (9)

where the linear operatorL is given by

Lu = L(v,w) = (∆v − vx,∆w−wx − λ∆v), (10)

and the linear boundary operatorB is the tripletBu = (B0u,B1u,B2u) with

B0(v,w) = [w− λv], B1(v,w) = [(w− λv)x]− λ[v], B2(v,w) =w(0+)− vx(0−) + v(0−). (11)

The functionsF0 andG are defined by:

F(u) = F̃(u)(x, y)− ∆v(0−, y)− vx(0−, y) + f1(u)(0−, y)

1− v(0−, y) + vx(0−, y)

([v]θ0

xx + vx, [v]S0xx +wx

), (12)

F̃(u) =(f1(u), f2(u)

), (13)

f1(u) = [vy ]2(θ0xx + [v]θ0

xxx + vxx)− 2[vy]

([vy]θ0

xx + vxy)− [vyy]

([v]θ0

xx + vx),

f2(u) = [vy ]2(S0xx + [v]S0

xxx +wxx)− 2[vy]

([vy]S0

xx +wxy)− [vyy]

([v]S0

xx +wx)

−λ[vy]2(θ0xx + [v]θ0

xxx + vxx)

+ 2λ[vy]([vy]θ0

xx + vxy)

+ λ[vyy ]([v]θ0

xx + vx), (14)

G(u) =(0,0, g(u)

), g(u) = 1 +w(0+)−

(1 + [vy]2

)−1/2expw(0+). (15)

With the aid of the general theory of fully nonlinear parabolic problems [7] we can show, as in [2], localexistence and uniqueness of a solution of (9) belonging to the usual parabolic Hölder space, for every smallinitial datumu0 belonging to

I ={u0 ∈X2+α : v(0−, y)− vx(0−, y) 6=−1, B0u0 =B1u0 = 0,

B2u0 = G(u0), B0(Lu0 +Fu0) = 0},

where fork ∈N, 0<α< 1

Xk+α ={u = (v,w) : v ∈Ck+α

((−∞,0)×R

), v = 0 in (0,+∞)×R,

w ∈Ck+α((−∞,0)×R

)∩Ck+α

((0,+∞)×R

),

limx→±∞

∣∣v(x, y)∣∣+ ∣∣w(x, y)

∣∣= 0, ∀y}.

(See[1] for the one-dimensional case.)

79

C.-M. Brauner, A. Lunardi

3. Instability

Our main result will be an instability theorem. In order to state it we need some spectral properties of therealizationLα of L with homogeneous boundary conditions inXα.

For everyk, ω ∈R we set

r1 = r1(ω,k) =1 +√

1 + 4ω+ 4k2

2, r2 = r2(ω,k) = 1− r1,

and we consider thedispersion relation

D(k,ω,λ) =λ(r2

2 − k2)

1− 2r1− r2(1− 2r1).

It is known (see[9], Figure 3) that fork in an interval[0, kc], kc being the characteristic wavenumber, thereexistsω > 0 such that the dispersion relation is satisfied. The range of the increasing functionk 7→ ω(k) isan interval[0, ωc].

THEOREM 1. –Let α ∈ (0,1). Then the whole interval[0, ωc] consists of eigenvalues ofLα, while thehalf plane{ω : Reω > ωc} is contained in the resolvent set ofLα. For ω = ωc there are eigenfunctionsof the typeuc(x, y) = (vc(x)g(y),wc(x)g(y)), g(y) being any function such thatg′′(y) = −k2

cg(y) andvc(x) = expr1(ωc, kc)x.

COROLLARY 2. –Let λ > 1. Fix α ∈ [0,1), and setM = eLα : D(Lα) 7→ D(Lα). Denoting byρ thespectral radius ofM we have

ρ= eωc > 1.

Note that the subset ofσ(Lα) consisting of elements with positive real part is not a spectral set. We proveinstability by adapting a result of Henry ([6, p. 105]) about dynamical systems in Banach spaces. Indeed,in our case the set of the admissible initial dataI is not a linear space but rather a manifold, due to thenonlinear compatibility conditions atx= 0. This enables us to prove the instability of the null solution intheC2+α norm. However, a sharper result is needed to prove the instability of the front.

THEOREM 3. –LetX be a real Banach space. For everyn ∈N let Tn be a map from a neighborhood ofthe origin inX with Tn(0) = 0, letM be a continuous linear operator onX with spectral radiusρ greaterthan1, and

Tn(x) =Mx+ O(‖x‖2

), asx→ 0, uniformly in n ∈N.

Assume in addition thatρ is an eigenvalue, and that there exist an eigenfunctionu ∈X and an elementx′ ∈X ′ (the space of all linear continuous functions fromX toR) such thatx′(u) 6= 0. Then there isC′ > 0such that for everyδ > 0 there existsx0 ∈X with norm less or equal toδ, such that ifxn+1 = Tn(xn) thenfor someN (depending onδ), the sequencex1, x2, . . . , xN is well defined and|x′(xN )|>C′.

The theorem is applied to the spaceX = P (X2+α), whereP is a suitable projection on the subspace ofX2+α consisting of the functionsu0 which satisfy the homogeneous boundary conditionsB0u0 =B1u0 =B2u0 = 0, B0Lu0 = 0, and

Tn :B(0, r)⊂ P (X2+α) 7−→ P (X2+α), Tn(u0) = P(u(n,u0, n− 1)

), n ∈N,

whose derivative at0 isM = P eL|P (X2+α) = eL|P (X2+α). Here we denote byu(n,u0, n− 1) the solutionof (9) with initial conditionu(n− 1) = u0, evaluated at timen. Moreover, for any fixedy0 ∈R we choose

x′(u) = v(0−, y0).

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Free boundary instability

The eigenfunctionuc of M with eigenvalueρ= eωc , given by Theorem 1, hasx′(u) = 1, providedg ischosen in such a way thatg(y0) = 1.

Applying Theorem 3 we get that there existsC′ > 0 such that for everyδ > 0 there isu0 ∈ I, withPu0 ∈ P (X2+α) with norm not exceedingδ, such that the corresponding solutionu(t, · ) is defined (atleast) in an interval[0,N ] and|x′(P (u(N, · )))|>C′. P is defined in such a way that the first componentof P (v,w) atx= 0 equalsv(0, · ). Recalling (8), this implies that∣∣s(N,y0)

∣∣= ∣∣x′(P (u(N, · )))∣∣>C′,

so that the front is pointwise unstable.

References

[1] Alabau F., Lunardi A., Behaviour near the travelling wave solution of a free boundary system in combustion theory,Dynam. Syst. and Appl. 1 (1992) 391–418.

[2] Brauner C.-M., Hulshof J., Lunardi A., A general approach to stability in free boundary problems, J. Differ. Eq.(à paraître).

[3] Brauner C.-M., Lunardi A., Schmidt-Lainé Cl., Stability of travelling waves with interface conditions, Nonlin. Anal.TMA 19 (1992) 465–484.

[4] Brauner C.-M., Lunardi A., Schmidt-Lainé Cl., Multidimensional stability analysis of planar travelling waves, Appl.Math. Lett. 7 (1994) 1–4.

[5] Brauner C.-M., Lunardi A., Schmidt-Lainé Cl., Stability of travelling waves in a multidimensional free boundaryproblem, Nonlin. Anal. TMA (à paraître).

[6] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer-Verlag, Berlin,1980.

[7] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.[8] Matkowski B.J., Sivashinsky G.I., An asymptotic derivation of two models in flame theory associated with the

constant density approximation, SIAM J. Appl. Math. 37 (1979) 686–699.[9] Sivashinsky G.I., On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math. 39 (1980) 67–82.

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