A fully nonlinear equation for the flame front in a quasi-steady combustion model

We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasi-steady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null solution. Second, introducing a parameter $\epsilon$, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as $\epsilon\to 0$.


1.
Introduction. Flames constitute a complex physical system involving fluid dynamics, multistep chemical kinetics, as well as molecular and radiative transfer. The laminar flames of low-Lewis-number premixtures are known to display diffusivethermal instability responsible for the formation of a non-steady cellular structure (see [18]). However, the cellular instability is quite robust against these aero-thermochemical complexities and may be successfully captured by a model involving only two equations: the heat equation for the system's temperature and the diffusion equation for the deficient reactant's concentration. In suitably chosen units, the so-called thermal-diffusional model reads, see e.g., [6]: Here, Θ = (T − T u )/(T ad − T u ) is the scaled temperature, where T u and T ad correspond to the temperature of the unburned gas, and the adiabatic temperature of combustion products, respectively; Y = C/C u is the scaled concentration of the deficient reactant with C u being its value in the unburned gas; x, y, t are the scaled spatiotemporal coordinates referred to D th /U , and D th /U 2 , respectively, where D th is the thermal diffusivity of the mixture and U is the velocity of the undisturbed planar flame; Le is the Lewis number (the ratio of thermal and molecular diffusivities); σ = T u /T ad ; β = T a (1 − σ)/T ad is the Zeldovich number, assumed to be large, where T a is the activation temperature; Ω is the scaled reaction rate, where the normalizing factor 1 2 Le −1 β 2 ensures that at β 1 the planar flame propagates at the velocity close to unity.
Due to the distributed nature of the reaction rate Ω, Equations (1.1) and (1.2) are still difficult for a theoretical exploration. One therefore turns to the conventional high activation energy limit (β 1) which converts the reaction rate term into a localized source distributed over a certain interface x = ξ(t, y), the flame front. Intensity of the source varies along the front as exp 1 2 (Θ f − 1) (see [16]). Here, Θ f is the scaled temperature at the curved front, which may differ from unity (T = T ad ) by a quantity of the order of β −1 . Due to the strong temperature dependence of the reaction rate (β 1), even slight changes of Θ f may markedly affect its intensity, and thereby also local flame speed. The study of flame propagation is thus reduced to a free-interface problem. To ensure that the emerging free-interface model does not involve large parameters one should combine the limit of large activation energy (β 1) with the requirement that the product α = 1 2 β(1 − Le) remains finite, i.e., the ratio of thermal and molecular diffusivities (Le) should be closed to unity. This is the Near Equidiffusive Flames model, in short NEF, introduced in [15]. As a result, instead of the reaction diffusion problem for Θ and Y , one ends up with a free-interface problem for the new scaled temperature θ = lim β→+∞ Θ and the reduced enthalpy S = lim β→+∞ β −1 (Θ + Y − 1). More precisely, the system for the temperature θ, the enthalpy S and the moving flame front, defined by x = ξ(t, y), reads ∂θ ∂t = ∆θ, x < ξ(t, y), (1.4) θ = 1, x ≥ ξ(t, y), (1.5) ∂S ∂t = ∆S − α∆θ, x = ξ(t, y). (1.6) For some mathematical results about this problem, see [5,9,10,11,12,7]. Here, we consider only the case when α is positive, i.e., Le < 1. It will be convenient to assume periodicity in y with period , and restrict attention to y ∈ [− /2, /2]. At the front, θ and S are continuous, the following jump conditions occur for the normal derivatives: As usual one fixes the free boundary. We set ξ(t, y) In this new framework: The front is now fixed at x = 0. The first condition (1.7) reads: A very challenging problem is the derivation of a single equation for the interface or moving front ϕ, which may capture most of the dynamics and, as a consequence, yields a reduction of the effective dimensionality of the system. In this spirit, one of the authors in [17] derived asymptotically from the System (1.4)-(1.8) the Kuramoto-Sivashinsky (K-S) equation in rescaled dependent and independent variables Since then, this equation has received considerable attention from the mathematical community. We refer to the book [20] and its extensive bibliography. This paper is devoted to a quasi-steady version of the NEF model. As a matter of fact, it has been observed in similar problems (see [2]) that not far from the instability threshold the time derivatives in the temperature and enthalpy equations have a relatively small effect on the solution. The dynamics appears to be essentially driven by the front. Based on this observation one can define a quasi-steady NEF model replacing (1.9)-(1.11) by Next we consider the perturbations of temperature u and enthalpy v: Writing for simplicity x instead of x , the problem for the triplet (u, v, ϕ) reads: As in [3,4], we introduce further simplifications: we keep only linear and secondorder terms for the perturbation of the front ϕ, and first-order terms for the perturbations of temperature u and enthalpy v. This leads to the equations: At x = 0 there are several conditions. First hence up to the second-order: and keeping only the first-order for v yields: Therefore, the final system reads: (1. 13) We remark that the equation for u associated with the boundary condition u(0) = 0 entirely determines u when ϕ is given. Therefore, it can be viewed as a kind of pseudo-differential Stefan condition. We will take advantage of this remark in Section 3.
The goal of this paper is to show that this simplified NEF model still contains the dynamics of the system. It is simple enough to be integrated explicitly via a discrete Fourier transform in the variable y and therefore it allows a separation of the dependent variables. We get to a self-consistent pseudo-differential equation for the front ϕ which reads: (1.14) where the −λ k 's are the non-positive eigenvalues of the operator D yy with periodic boundary conditions at y = ± /2 (that we denote below by A) and is the symbol of operator 1 − 4D yy . Equation (1.14) can be written in the more abstract form: where L is a pseudodifferential operator whose leading part is D yy and G is a nonlinear operator whose leading term is 1 4 1 − 4D yy . This makes (1.15) a strongly nonlinear equation, more precisely it is a fully nonlinear parabolic equation: in the L 2 -setting the nonlinear part is exactly of the same order as the linear operator. This is one of the main issues of this paper. Note that the realization of the operator 1 − 4D yy in the space of continuous and -periodic functions (say C ) is defined only in a proper subspace of C 1 (the space of all the -periodic C 1 -functions). Hence, in the C -setting, the nonlinear term G ((ϕ y ) 2 ) represents the leading part of the right-hand side of (1.15). This would make the study of (1.15) more difficult than in the L 2 -setting, where we confine our analysis.
In the case when ϕ is smoother, we can rewrite Equation (1.15) as a fourth-order equation as follows: where S is nothing but the usual fourth-order differential operator Operators B and F are pseudo-differential ones with symbols, respectively, Therefore, The main feature of Equation (1.16) is that the nonlinear part is rather unusual. Actually, it has a fourth-order leading term, as S has. Therefore (1.16) is also a fully nonlinear problem, in contrast to (1.12) which is semilinear. The paper is organized as follows: In Section 3 we derive the front equation via an explicit computation of (1.13) in the strip R × [− /2, /2]. Then, in Section 4, we prove the following result  An important question, that we address in Section 5, is the link between (1.16) and K-S. Following [17], we introduce a small parameter ε > 0, setting and define the rescaled dependent and independent variables accordingly: We see that ψ solves the equation Then, we anticipate, in the limit ε → 0, that ψ ∼ Φ, where Φ solves (1.12). More precisely, we take for : which blows up as ε → 0; hence α c = 1 + 16π 2 2 0 ε. Thus, 0 becomes the new bifurcation parameter. We shall assume that 0 > 4π in order to have α c ∈ (1, 1+ε), i.e., α > α c , otherwise the trivial solution is stable and the dynamics is trivial.
The second main result of the paper is the following.
Moreover, there exists a positive constant C, independent of ε ∈ (0, ε 0 ], such that In other words, starting from the same configuration, the solution of (1.16) remains on a fixed time interval close to the solution of K-S up to some renormalization, uniformly in ε sufficiently small. Note that the initial condition for ϕ is of special type, compatible with Φ 0 and (1.12) at τ = 0. Initial conditions of this type have been already considered in [1,2,4].
Although energy methods are known to be usually inefficient in fully nonlinear problems, here we may take advantage of the special structure of F . It allows us to establish sharp a priori estimates on the remainder (more precisely on its derivative) when ε is small enough. A key point is an extension of a lemma that we already successfully used in [1,4].
Finally, in the Appendix, for the reader's convenience we provide a quite detailed proof of the existence, uniqueness and regularity of the solution to K-S which vanishes at τ = 0.
In a forthcoming paper we will incorporate the time derivatives of the temperature and enthalpy in the model; the front equation will be more involved and of higher order in time, as in [1]. Another issue we intend to address is the derivation of the front equation as a solvability condition in the spirit of [3,4].
2. Some mathematical setting. In this section we introduce some notation, the functional spaces and operators we will use below. We will mainly use the discrete Fourier transform with respect to the variable y. For this purpose, given a function f : where {w k } is a complete set of (complex valued) eigenfunctions of the operator with -periodic boundary conditions, corresponding to the non-positive eigenvalues We shall find it convenient to label this sequence as When there is no damage of confusion, we simply write λ k instead of λ k ( ). When f depends also on t and/or x, by f (·, k) we denote the k-th Fourier coefficient of f with respect to y. For instance, for fixed t and x, f (t, x, k) will denote the k-th Fourier coefficient of the function f (t, x, ·).
For integer or arbitrary real s, we denote by H s the usual Sobolev spaces of order s consisting of -periodic (generalized) functions, which we will conveniently represent as 3. The derivation of a self-consistent equation for the front. The aim of this section is the derivation of a self-consistent equation (in the Fourier variables) for the front ϕ. For this purpose, we rewrite Problem (1.13), making θ and S explicit. We get In what follows, we assume that (u, v, ϕ) is a sufficiently smooth solution to Problem (3.1) such that the function As it has been stressed in the Introduction, we use the first equation in (3.1) and the boundary condition u(·, 0, ·) = 0 as a pseudo-differential Stefan condition. We solve the problem for u via discrete Fourier transform. This leads us to the infinitely many equations A straightforward computation reveals that the solution to (3.2) which vanishes at x = 0 and tends to 0 as x → −∞ not slower than e −x/2 is given by Let us now consider the problem for v, where we disregard (for the moment) the condition v(·, 0, ·) − u x (·, 0, ·) = 1 2 (ϕ y ) 2 . Taking the Fourier transform (with respect to the variable y), we get the Cauchy problems Now, we are in a position to determine the equation for the front. Indeed, rewriting the boundary condition in Fourier variables, and using the above results, we get to the following equations for the front (in the Fourier coordinates): Let us set X k = √ 1 + 4λ k . Then, the equation for ϕ reads (in terms of X k ) as follows: for any k = 0, 1, 2, . . ., or, equivalently, or, even, and for any k = 0, 1, . . .

4.
Stability of the front. In this section we are interested in studying the stability and instability properties of the null solution to the Equations (3.4) and (3.5). In this respect we need to study the symbols appearing in (3.3).

4.1.
Study of the symbols. In this subsection, we study the main properties of the operators B, G , S , and L , F , whose symbols are respectively defined by (3.6)-(3.8) and by , for any k = 0, 1, . . . Even if all these operators depend on α, we prefer not to stress explicitly the dependence on α to avoid cumbersome notations. (v) The realization of the operator S in L 2 is the operator with H 4 as domain.
Proof. (i). To begin with, we observe that Hence, we can split Since l k → −∞ as k → +∞, 0 is an isolated point of the spectrum of L and the corresponding eigenspace is one-dimensional. Let us prove that Π is the spectral projection associated with such an eigenvalue. For this purpose, we prove that 0 is a simple pole of the function λ → R(λ, L) and compute the residual at 0. Note that for any λ ∈ σ(L) and any ψ ∈ H 2 it holds that Hence, Hence, for |λ| ≤ 1 2 min k=1,2,...
where l min = min n=1,2,... |l n | > 0. This shows that R(λI − L) has a simple pole at λ = 0 and its residual is the operator Π, which turns out to be spectral projection associated with the eigenvalue 0, which is simple. For more details, we refer the reader to e.g., [ To conclude the proof of point (i), we observe that l k < 0, for k ≥ 1, if and only if 1 + 4λ k − α > 0. Since (λ k ) is a nondecreasing sequence, l k < 0 for any k = 1, 2, . . ., if and only if 4λ 1 + 1 − α > 0, i.e., if and only if α < α c .
(ii), (iii) & (iv). It is enough to observe that b k ∼ 4λ k , f k ∼ 2λ Step 1. Using classical arguments based on a fixed point argument, one can show that for any α ∈ R and any T > 0, there exists r 0 > 0 such that, if ϕ 0 2 ≤ r 0 , the Cauchy problem for any 0 < s < t ≤ T , some positive constant C 1 and any ψ ∈ X θ (T ) (θ ∈ (0, 1)).
To prove this estimate it suffices to observe that, by Proposition 4.1(iv) for any 0 < s < T , where the last side of the previous chain of inequalities follows from Poincaré-Wirtinger inequality, and C denotes a positive constant, independent of s, t and ψ, which may vary from line to line. Estimate (4.2) now follows at once. Let us now prove properties (a) and (b). It is convenient to split the solution ϕ to Equation (3.5) along Π(L 2 ) and (I−Π)(L 2 ). We get ϕ(t, y) = p(t)w 0 +ψ(t, y) for any t > 0 and any y ∈ [− /2, /2]. Since Π commutes with both the time and the spatial derivatives, Π(D t ϕ) = D t Π(ϕ) = p and (I − Π)(D y ϕ) = D y (I − Π)(ϕ) = D y w. Moreover, for any ψ ∈ H 1 , G(ψ) = +∞ k=0 g k ψ(k)w k , so that Hence, projecting the Cauchy problem (4.1) along Π(L 2 ) and (I − Π)(L 2 ), we get the two self-consistent equations for p and ψ: and Coming back to Problem (4.1), the above results show that, if α < α c , this problem admits a unique solution, defined for all the positive times. Moreover, for any t > 0, any ω as above and some positive constant P ω independent of s, ϕ 0 and ϕ, i.e., the null solution to Equation (3.5) is (orbitally) stable with asymptotic phase.
In the case when α > α c the spectrum of L |(I−Π)(L 2 ) contains (a finite number of) eigenvalues with positive real part. Hence, the equation ψ t = Lψ+(I −Π)(G((ψ y ) 2 )) admits a backward solution, exponentially decreasing to 0 at −∞ and this implies that the null solution to Problem (4.4) and, consequently, the null solution to Problem (4.1) are unstable. For further details, we refer the reader to e.g., [8]  Step 2. We focus on the case when α < α c , the other case being simpler. Of course, we just need to deal with the function ψ = (I − Π)ϕ. We assume that ϕ 0 ∈ H 4 . We are going to show that for any ω ∈ (0, max k=1,2,... l k ), it holds that sup t>0 e −ωt ϕ(t, ·) 4 + sup t>0 e −ωt ϕ t (t, ·) 2 < +∞.

Hence, the equation for the function ψ (in Fourier coordinates) reads
for any k = 0, 1, . . . Note that the leading terms (at order 0 in ε) of b ε,k and f ε,k are 1 and −1/2, respectively. Hence, at the zero-order, we recover the K-S equation As we remind it in the Introduction, this equation has been thoroughly studied by many authors. For our purposes, we need the following classical result. For the reader's convenience we provide a rather detailed proof in Appendix A.
The above (heuristical) arguments suggest to split ψ as follows: To avoid cumbersome notation, we simply write ρ for ρ ε , when there is no damage of confusion. By assumptions (see Theorem 1.2), the initial condition for ρ is Replacing into (5.1) we get, after simplifying by ε, for any k = 0, 1, . . ., where the symbols of the operators H ε and M ε are for any ε ∈ (0, 1] and any s as above. Finally, the operator B ε is invertible both from H s to H s−2 for any s = 2, 3, . . .; (b) for any s ≥ 3, the operators F ε and M ε admit bounded realizations F ε and M ε , respectively, mapping H s into H s−3 . Moreover, for any ε ∈ (0, 1] and any s = 3, 4, . . .

Proof. (a). A straightforward computation shows that
for any k = 0, 1, . . . and any ε ∈ (0, 1]. This shows that H ε admits a bounded realization mapping H s into H s−2 for any s ≥ 2 and its norm can be bounded by a constant, independent of ε ∈ (0, 1]. Since b ε,k = εh ε,k + 1 for any k = 0, 1, . . ., the boundedness of the operator B ε from H s to H s−2 follows at once.
Showing that the operator B ε is invertible from H s into H s−2 is an easy task. It suffices to observe that b ε,k ≥ 4ελ k + 1 for any k = 0, 1, . . ..
(b). Since f k = εm ε,k − 1/2 for any k = 0, 1, . . ., we can limit ourselves to considering the operator M ε . A simple computation shows that k + 25λ k , for any k = 0, 1, 2, . . . Hence, M ε is well defined (and bounded) in H s with values in H s−3 for any s ≥ 3. Since its symbol can be estimated from above uniformly with respect to ε ∈ (0, 1], the assertion follows immediately. Since all the operators appearing in (5.4) commute with D η , the differentiated problem for ζ := ρ η reads as follows: where we have set Ψ = Φ η . Obviously, it has a null initial condition at time τ = 0. For simplicity, we denote D η by D. For an integer n ≥ 1, D n is the differentiation operator of order n. We also set D 0 = Id. Then, there exist ε 1 = ε 1 (n, T ) ∈ (0, 1) and K n = K n (n, T ) > 0 such that, if ζ ∈ Y n (T 1 ) is a solution on the time interval [0, T 1 ] of Equation (5.7) for some To prove (5.8), we multiply both sides of the equation (5.7) by (−1) n D 2n ζ and integrate by parts over (− 0 /2, 0 /2). We thus get In the following lemmata we estimate all the terms appearing in the previous equation. We first deal with the left-hand side of (5.9) which consists of the "benign" terms.
The following lemma allows us to estimate the function A ε .

5.3.
Existence and uniqueness of a solution to Equation (5.4) vanishing at τ = 0. In this subsection we are devoted to prove the following theorem. for all ξ ∈ Ξ N . The problem is subject to the initial condition ζ N (0, ·) ≡ 0. In terms of Fourier series, the variational formulation (5.9) reads as follows: f ε,k (Ψζ N ) η (·, k) ξ(k).

(5.22)
Taking ξ = w j (j = 1, . . . , N ) in (5.22), we see that the ζ N (·, k)'s verify a system of N ordinary differential equations with zero initial data. Hence, there exists a unique solution to System (5.22), defined on some maximal time interval [0, T N ), where T N may also depend on ε.