Claudia Lederman

Jean-Michel Roquejoffre

Noemi Wolanski

Lederman, Claudia

Roquejoffre, Jean-Michel

Wolanski, Noemi

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574Équations aux dérivées partielles/Partial Differential Equations(Problèmes mathématiques de la mécanique/Mathematical Problems in Mechanics)Mathematical justification of a nonlinearintegro-differential equation for the propagationof spherical flamesClaudia Lederman a, Jean-Michel Roquejoffreb, Noemi Wolanskiaa Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428Buenos Aires, Argentinab UFR-MIG, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex,FranceReceived 21 December 2001; accepted 14 January 2002Note presented by Philippe Ciarlet.Abstract This Note is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [7]. The paper [7] derives, by meansof a three-scale matched asymptotic expansion, starting from the classical thermo-diffusivemodel with high activation energies, an integro-differential equation for the flame radius.In the derivation, it is essential for the Lewis number – i.e., the ratio between thermal andmolecular diffusion – to be strictly less than unity. In this Note, we give the main ideas of arigorous proof of the validity of this model, under the additional restriction that the Lewisnumber is close to 1. To cite this article: C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I334 (2002) 569–574.  2002 Académie des sciences/Éditions scientifiques et médicalesElsevier SASJustification mathématique d’une équation intégro-différentielle nonlinéaire pour un modèle de flamme sphériqueRésumé Nous donnons dans cette Note les grandes lignes de la justification mathématiquementrigoureuse d’un modèle intégro-différentiel non linéaire d’évolution du rayon d’une flammesphérique initialement proposé par G. Joulin dans [7]. Cette équation est obtenue dans lecadre du modèle thermo-diffusif tridimensionnel aux hautes énergies d’activation, avecnombre de Lewis strictement plus petit que 1. Nous montrons dans cette note la validitédu modèle sous la restriction supplémentaire que le nombre de Lewis est assez proche de 1.Pour citer cet article : C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASVersion française abrégéeConsidérons le modèle thermo-diffusif tridimensionnel pour la propagation des flammes prémélangées.Dans l’hypothèse de la chimie simple A→ B et des hautes énergies d’activation, celui-ci s’écrit [2,4] :E-mail addresses:clederma@dm.uba.ar (C. Lederman); roque@mip.ups-tlse.fr (J.-M. Roquejoffre);Wolanski@dm.uba.ar (N. Wolanski). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-073X(02)02299-9/FLA 569C. Lederman et al. / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574{Tt −T = Yfε(T ),Yt − YLe= −Yfε(T ),((t, x) ∈ R+ × R3), (0.1)auquel nous adjoignons les conditions aux limites suivantes :T (t, r = +∞)= 0, Y (t, r = +∞)= 1. (0.2)Les notations sont classiques : T (t, x) est la température, Y (t, x) la fraction massique du réactant. Le réelLe > 0 est le nombre de Lewis, i.e. le rapport des diffusions thermiques et moléculaires. La fonction fε(T )est le terme d’Arrhénius classique fε(T ) = (1/ε2) exp((T − Le−1)/ε) pour T proche de Le−1. Le réelε > 0 représente l’inverse de l’énergie d’activation normalisée. On suppose de plus que fε s’annule pourT  θ , avec 0 < θ < Le−1.On fait dans toute cette Note l’hypothèse de la symétrie sphérique.Dans une importante contributionà la compréhension de (0.1), (0.2), Joulin [7] identifie une échelle de temps caractéristique de l’ordre deε−2 lorsque Le < 1. Il obtient de façon formelle, via un développement asymptotique à trois échelles, uneéquation asymptotique pour ρε(τ ) := Rε(τ/ε2), où Rε(t) est le rayon de la flamme à l’instant t – i.e. leplus petit rayon pour lequel T (t, ·)− Le−1 est d’ordre εLog ε :(1 − √Le)∂1/2ρ = 2Le Log(√Le3ρ)+ φ(τ), (0.3)où φ(τ) est un terme de forçage dépendant de la donnée initiale. L’opérateur ∂1/2 est la dérivée fractionnaireclassique : pour une fonction ρ(τ) de classe C1, on note∂1/2ρ(τ)= 1√π∫ τ0ρ̇(s)√τ − s ds. (0.4)Nous décrivons dans cette Note le résultat suivant, dont les preuves détaillées feront l’objet d’un articleultérieur.RÉSULTAT PRINCIPAL. – On choisitLe = 1 − δ avec0 < δ < 1, et ε > 0 petit, indépendamment deδ.Notonsτ = ε2t . Il existe: δ0 > 0, ε0 > 0, une classe de données initiales(T ε0 , Y ε0 )0<εε0 , et une fonctionφε(τ ) proche de la fonctionφ(τ) de(0.3), une solutionρ(τ) de(0.3) de temps de vieτmax > 0 tels que:– si 0< δ < δ0, 0 < ε < ε0, et0 < τ0 < τmax,– et si(T ε(t, r), Y ε(t, r)) est la solution de(0.1) issue de(T ε0 , Yε0 ),on a |Rε(τ/ε2)− ρ(τ)| = O(ε) pour τ ∈ [0, τ0].La démonstration de ce résultat utilise les trois ingrédients suivants :– l’existence et l’analyse de stabilité d’une classe particulière de solutions stationnaires de (0.1)(flammes en boules de Zeldovich) ;– la construction, en suivant les idées développées dans [7], d’une solution approchée (T εJ , YεJ ) de (0.1),(0.2) dont le rayon vérifie (0.3) à ε près ;– une analyse précise de la distance entre la solution (T εJ , YεJ ) et la solution (Tε, Y ε) de (0.1) de donnéeinitiale (T ε0 , Yε0 ), à l’aide du premier point et d’estimations sur l’opérateur d’évolution engendré par laversion linéarisée de (0.1) autour de (T εJ , YεJ ).1. IntroductionIn an important contribution [7], Joulin derives, in a formal way, a nonlinear integro-differential equationdescribing the evolution of the radius of a spherical flame, in the context of the thermo-diffusive modelfor flame propagation with high activation energies. The goal of this Note is to give the main steps of amathematically rigorous proof of this derivation. One of the motivations for doing this – besides the factthat the model has an interest of its own – is that Joulin’s derivation can be carried out for a large classof models; see, for instance, [3] or [8]. Hence it can be expected that the methods devised here have somedegree of generality.570To cite this article: C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574We consider the evolution equations for a gaseous mixture with simple chemistry A → B in the wholespace R3 [2,4]: {Tt −T = Yfε(T ),Yt − YLe= −Yfε(T ),((t, x) ∈ R+ × R3), (1.1)to which we append the following conditions at |x| = +∞:T (t, r = +∞)= 0, Y (t, r = +∞)= 1. (1.2)The notations are classical: T (t, x) is the temperature, Y (t, x) is the mass fraction of the reactant A. Thenumber Le > 0 is the Lewis number, i.e., the ratio between thermal and molecular diffusion. The functionfε(T ) is the classical Arrhenius termfε(T )= 1ε2expT − Le−1ε(1.3)for T close to Le−1. To avoid the cold boundary difficulty, the function fε is assumed to vanish for T  θ ,with 0 < θ < Le−1. The only slight difference with the most classical expressions is that, here, the flametemperature has been normalized to Le−1.In the whole work, the assumption of spherical symmetry will be made.This allows us to define, for alltime t > 0, the radius of the flame Rε(t); namely: the smallest r > 0 such that T (t, x)= Le−1 + 100εLogεif |x| = r . The paper [7] derives, by means of a three-scale formal matched asymptotic expansion, anasymptotic equation for ρε(τ ) :=Rε(τ/ε2); namely:(1 − √Le)∂1/2ρ = 2Le Log(√Le3ρ)+ φ(τ), (1.4)where φ(τ) is a forcing term depending on the initial datum. The operator ∂1/2 is the classical fractionalderivative of order 1/2; for a C1 function ρ(τ) it is defined by (0.4).It appears that Eq. (1.4) can be locally ill-posed for Le > 1, whereas it is always locally well-posedwhen Le < 1 – a deeper analysis [1] reveals that the only obstacle for global well-posedness is finite timequenching, which is indeed shown to be possible. We shall explain why it is essential to have Le < 1 in thecourse of the note. It is to be noted that the situation differs drastically from the case Le = 1. See [5] for ageneral multi-dimensional study, and the references therein.The plan of the note is therefore to state the main result – Section 2, then to give a stability resultexplaining why one must have Le < 1 – Section 3. Subsequently, following Joulin’s idea, we construct anapproximate solution to (1.1), (1.2) – Section 4, whose radius satisfies (1.4) up to O(ε) errors. Finally weexplain why the true and approximate solutions are ε2-close to each other.2. The Zeldovich flame ball and main resultThe Zeldovich flame ball is a particular nondegenerate steady solution of (1.1). Namely: if w is aparameter, we look for a solution (T εw,Yεw) of{−T = Yfε(T ),−YLe= −Yfε(T ),(R3), (2.1)T (r = +∞)= εw, Y (r = +∞)= 1. (2.2)PROPOSITION 1. – For any givenM > 0 large enough, there isε0 > 0 such that, for allε  ε0,problem (2.1), (2.2) has a unique solution(T εw,Yεw), which is C2 with respect tow – in the topologyof C2(R3), and whose radius is between1/M and M. If Rε,w is the radius of this solution we havelimε→0 Rε,w =√Le−3 exp(−w/2).MAIN RESULT. – SetLe = 1 − δ with 0 < δ < 1. Let (T ε(t, r), Y ε(t, r)) be a solution of(1.1), withinitial datum(T ε0 , Yε0 ) such that571C. Lederman et al. / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–574(i) there existw0 ∈ R, ν ∈ (0,1) and a constantC > 0, such that:∣∣(T ε0 , Y ε0 )(x)− (T εw0, Y εw0)(x)∣∣∞  Cε3 for |x|  ε−ν;(ii) (T ε0 , Yε0 ) converges to(0,1) at least as fast as|x|−1 as |x| → +∞.Setτ = ε2t . There existδ0 > 0, ε0 > 0, and a smooth functionφ(τ), determined by the limiting behaviourof the rescaled initial data at infinity, such that:– if 0 < δ < δ0 and0 < ε < ε0;– and ifρ(τ) is the solution of(1.4) with the initial datumρ(0)=√Le−3 exp(−w0/2) and lifetimeτmax,then we have∀τ0 ∈ (0, τmax), sup0ττ0∣∣Rε(τ/ε2)− ρ(τ)∣∣= O(ε).3. Linear stability analysis for the Zeldovich flame ballThe method that is devised in [7] to derive (1.4) is to notice that, when Le < 1, one can identify a slowtime scale of the order 1/ε2. This allows an explicit computation of the temperature and mass fraction atfinite distance; the slow motion of the flame is then controlled by the temperature field at |x| = +∞: if theflame were really steady, then the temperature – resp. the mass fraction – would not, in general, be 0 – resp.1 – at |x| = +∞, thus violating the conditions (1.2). This imposes the introduction of a large spatial scale,and the computation of the variations of the temperature and mass fraction on this large scale yields, to thefirst order in ε, an evolution equation for the radius of the flame.One way to interpret this 1/ε2 time scale is to relate it to the growth rate of a spherically symmetricdisturbance, and our task is to prove this growth rate is at most of order ε2.The basic problem to investigate is therefore the linear (in)stability of the Zeldovich flame ball. Namely,we look at the problemLε(u, v)=(−u− f ′ε(T0)Y0u− fε(T0)vvLe+ f ′ε(T0)Y0u+ fε(T0)v)= λ(u, v), (u, v) ∈ L2(R3). (3.1)THEOREM 1. –(i) AssumeLe = 1 − δ, δ > 0. There existsδ0 > 0 such that, for allδ ∈ (0, δ0), the following is true:there exist three constantsm> 0, M > 0 andθ ∈ [0,π/2), possibly depending onδ, such that the onlyeigenvalue of(3.1) outside the set{λ ∈C : arg(λ+M) θ, Reλm}is a complex numberε2λε such thatlimε→0λε = −(1 + √Leδ)2R20Le6e2w0 .(ii) AssumeLe = 1+δ, δ > 0. There existsδ0 > 0 and two smooth positive functionskε(δ) andk(δ) definedon (0, δ0] such thatkε(δ) k(δ) for 0 < δ < δ0, and such that−kε(δ)/ε2 is an eigenvalue to(3.1).The proof is a bootstrap process inspired from [6].Remark. – Part (i) is a first step to the proof of Joulin’s equation: the growth rate of the disturbance of theZeldovich flame ball is of order ε2. Joulin’s construction consists in devising, at each time step, solutionsthat are quasisteady, i.e., Zeldovich flame balls up to terms of order ε2. These quasisteady solutions canbe expected to have the same stability properties as the Zeldovich flame balls, as we will see in Section 5below.Part (ii) of this theorem explains why Joulin’s derivation cannot work in the case Le > 1.572Pour citer cet article : C. Lederman et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–5744. Construction of an approximate solutionConsider ν > 0 as small as needed. Joulin’s idea is to assume that there is a solution (TJ ,YJ ) of1{|x|ε−ν}Tt −T = Yfε(T ),1{|x|ε−ν}Yt − YLe = −Yfε(T ),(4.1)which indeed expresses the fact that there is a slow time scale. Eq. (1.4) comes out as an asymptoticcompatibility condition for the derivatives of T and Y at |x| = ε−ν . Here we do not attempt to prove thatthis problem has an exact solution, all the more as what we are interested in is the full system (1.1). Instead,we construct an approximate solution to (4.1), which is given in the following theorem:THEOREM 2. – Letw0 ∈ R, ε > 0 andR0 = Rε,w0 . Let (T0, Y0) be the Zeldovich flame ball(T εw0 , Y εw0)in B(0, ε−ν) and letT0, Y0 be radially symmetric and continuous inR3 with |x|T0(x), |x|Y0(x) boundedand satisfying, forT 0(x)= 1ε T0(xε), Y 0(x)= 1ε(Y0(xε)− 1), that(i)∥∥|x|T 0(x)∥∥L∞(|x|>ε1−ν), ∥∥|x|Y 0(x)∥∥L∞(|x|>ε1−ν) C independent ofε,(ii)∥∥|x|T 0(x)∥∥C3+α(ε1−ν|x|ε1−ν+r0), ∥∥|x|Y 0(x)∥∥C3+α(ε1−ν|x|ε1−ν+r0) CwithC, r0 > 0, 0 < α < 1 independent ofε,(iii)∂2∂r2(|x|T 0(x))∣∣|x|=ε1−ν+0)= ∂2∂r2(|x|Y 0(x))∣∣|x|=ε1−ν+0)= 0,(iv)(|x|(T 0)r)∣∣x=ε1−ν+0 =w0, (|x|(Y 0)r)∣∣x=ε1−ν+0 = Le(w0 + 2 Log(√Le3R0)),(v) |x|T 0(x), |x|Y 0(x) converge uniformly on compact subsets of|x|> 0, asε → 0.(4.2)There existτmax > 0 independent ofε and a solution(T εJ , YεJ ) to{Tt −T = Yfε(T )+ F1(t, x)+µ1,Yt − YLe= −Yfε(T )+ F2(t, x)+µ2,((0, τmax/ε2)× R3), (4.3)where the functionsF1, F2 are supported on[0, τmax/ε2)×B(0, ε−ν ) and, for allτ0 ∈ (0, τmax), we have:|Fi(t, x)|  Cε2 for 0  t  τ0/ε2, i = 1,2; moreoverµi = hi ds whereds is the surface measure on[0, τmax/ε2)× ∂B(0, ε−ν) and|hi |  Cε2 for 0  t  τ0/ε2, i = 1,2. The constantC may depend onτ0.Moreover, for everyτ0 ∈ (0, τmax), the couple of functions(T εJ , Y εJ ) is continuous in[0, τ0/ε2] × R3,C12((0, τ0/ε2]×{|x|  ε−ν}), C1,2((0, τ0/ε2]×{|x|  ε−ν}). The rescaled radiusρεJ (τ ) of (T εJ , Y εJ ) satisfiesEq. (1.4) with a forcing termφε, depending on the rescaled initial datum at|x| = +∞, that convergesuniformly asε → 0.Remarks. – 1. The function (T εJ , YεJ ) constructed in Theorem 2 is a solution to (4.1) in |x|< ε−ν and in|x|> ε−ν , with a jump of the gradient at |x| = ε−ν of order ε2+ν | Log2 ε|. The time derivatives Tt and Ytin |x|< ε−ν are of order ε2.2. The real number τmax is the lifetime of the solution ρ(τ) to (1.4) with forcing termφ(τ)= limε→0φε(τ )and initial datum ρ(0)=√Le−3 exp(−w0/2).3. The ε closedness to the Zeldovich solution and conditions (iii) and (iv) in Theorem 2 are compatibilityconditions. Indeed, problem (4.1) leads to a singularly perturbed fractional derivative system, whosestructure will be studied somewhere else.573C. Lederman et al. / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 569–5745. ConclusionSet (u, v)= (T ε, Y ε)− (T εJ , Y εJ ) where (T ε, Y ε) is the solution of the full system (1.1), (1.2) with initialdatum satisfying the assumptions of Section 2. Define – as in Section 3 – LεJ (ε2t) the linear operator aroundthe approximate solution (T εJ , YεJ ). Then (u, v) verifies(u̇, v̇)+LεJ(ε2t)(u, v)= Fε(u, v)+ (µ1,µ2)+ (F1,F2),where Fε has quadratic behaviour around (u, v) = (0,0); the measures µi and the functions Fi beingdefined in Theorem 2 and of order ε2−2ν at most. A detailed analysis of the properties of the evolutionoperator generated by LεJ (ε2t) and a classical stability argument yields that (u, v) is of size ε2−2ν at most.The detailed proofs and discussion will appear in a forthcoming paper.Acknowledgement. C.L. and N.W were supported by Grants: ANPCYT-PICT 03-05009 and CONICET-PIP0660/98. J.-M.R. was supported by the CNRS “Young researcher Grant” titled “Combustion diphasique”. Bothteams are grateful to their home institutions for several fruitful meetings.References[1] J. Audounet, V. Giovangigli, J.-M. Roquejoffre, A threshold phenomenon in the propagation of a point sourceinitiated flame, Phys. D 121 (1998) 295–316.[2] H. Berestycki, B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, in:H. Brezis, Lions (Eds.), Collège de France Seminar, Vol. 10, Pitman–Longman, Harlow, UK, 1991.[3] J.D. Buckmaster, G. Joulin, P. Ronney, The effects of radiation on flame balls at zero gravity, Combustion andFlame 79 (1990) 381–392.[4] J.D. Buckmaster, G.S.S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982.[5] J. Fernandez Bonder, N. Wolanski, A free-boundary problem in combustion theory, Interfaces Free Bound 2 (2000)381–411.[6] L. Glangetas, J.-M. Roquejoffre, Bifurcations of travelling waves in the thermo-diffusive model for flamepropagation, Arch. Rational Mech. Anal. 134 (1996) 341–402.[7] G. Joulin, Point-source initiation of lean spherical flames of light reactants: an asymptotic theory, Comb. Sci. andTech. 43 (1985) 99–113.[8] G. Joulin, Preferential diffusion and the initiation of lean flames of light fuels, SIAM J. Appl. Math. 47 (1987)998–1016.574

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

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Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

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