Multi-dimensional travelling-wave solutions of a flame propagation model

set in the infinite cylindrical domain X = {(xl, y) ~ R • o~}, where o~ is a bounded and smooth open domain in R n~l. This equation arises in combustion theory: it describes the propagation of a curved premixed flame in the infinite tube X, in the framework of the classical thermo-diffusive model, under the assumption that the Lewis number is equal to unity. Referring to [3], [4], [13], [15] e.g. for more details, we simply recall here the equations of the thermo-diffusive model, which is derived in the framework of the well known constant-density approximation. We consider a curved flame propagating in the infinite cylindrical tube S = R • o) ( R N. For x E 27, we write x = (x l , y) with x l E R and y E~o. With the assumption of a single one-step chemical reaction N -+ N, the equations of the thermo-diffusive model are


Introduction
In this paper we show the existence of a travelling wave solution of the equation ~u ~u 8-7 + ~(y) ~ = ~u + g(u), (1.1) set in the infinite cylindrical domain X = {(xl, y) ~ R • o~}, where o~ is a bounded and smooth open domain in R n~l.This equation arises in combustion theory: it describes the propagation of a curved premixed flame in the infinite tube X, in the framework of the classical thermo-diffusive model, under the assumption that the Lewis number is equal to unity.
Referring to [3], [4], [13], [15] e.g. for more details, we simply recall here the equations of the thermo-diffusive model, which is derived in the framework of the well known constant-density approximation.We consider a curved flame propagating in the infinite cylindrical tube S = R • o) (R N.For x E 27, we write x = (xl, y) with xlER and y E~o.With the assumption of a single one-step chemical reaction N -+ N, the equations of the thermo-diffusive model are Here u is the normalized temperature and v is the mass fraction of the reactant.Moreover, ec(y) is the xa-component of the velocity field V = (o~(y), 0) which is given, parallel to the tube walls 827 and divergence free.Lastly, the terms Au, Av L--e and f(u) v correspond to the thermal diffusion, the molecular diffusion (the non-dimensional positive parameter Le is the Lewis number of the reactant ~), and the chemical reaction respectively.u(-= 0, v(-= 1, (1.5) u(+ cx~, y) _--1, v(+ 0% y) _--0 for y E e).
Problem (1.6)-(1.8)had been previously investigated by the first two authors in [3], under the additional assumption essentially that ~ does not differ much from a constant.More precisely, under the assumption that max ~(y) --~(y) <o~) --rain ~(y) < 2 f g(s) ds, (1.15) x yE~o y~o~ 0 it was shown in [3] that there exists a solution (u, c) to (1.6)-(1.8),and that this solution satisfies e + rain ~(y) > 0. (1.16) This inequality, which is derived from the additional assumption (1.15), is crucially used in [3] to derive some apriori estimates.In contrast with this situation, for the solutions we construct here, c + ~(y) may in general change sign in the domain to.This phenomenon may be interpreted as an "inversion of the velocity field".Indeed, (I.2)- (1.3) show that e + to(y) is the mixture velocity in the reference frame RT in which the solution is stationary (Rf is a reference frame attached to the flame and moves with the velocity --c with respect to the original reference frame Ro).Then, (1.16) means th at, at every point, the velocity in the reference frame Rf points from left to right, i.e. from the fresh mixture towards the burnt gases, a physically natural situation.But, for solutions satisfying c + min o~(y) < 0, (1.17) there are regions of the tube (where e § ~(y) < 0) where the velocity is directed in the opposite way, from the burnt gases towards the fresh mixture !It is important to realize here that this non-classical situation is by no means unphysicM: it has indeed been known for a long time (not for a flame in a tube, but in other geometrical configurations~ such as for a counterflow diffusion flame; see WILLIAMS [15, p. 418]) that the mixture velocity in the neighborhood of the flame may be pointing from the burnt gases towards the fresh mixture.This simply means that the convective effects are locally dominated by the diffusive effects.Moreover, in these conditions, (1.14) says that the average velocity in the reference frame Rf is necessarily positive.Several results in this direction are shown in Section 3. We prove there that, the function g corresponding to the reaction term being given, one can choose the function o~ (far enough from a constant) so that the corresponding solution (u, c) of (1.6)-(1.8)actually satisfies (1.17) (such travelling wave solutions have been numerically computed in [2]).Moreover, we show that condition (1.15) is, in some sense, optimal to insure that the inversion of the velocity field does not occur (i.e. that property (1.16) is fulfilled).
The existence of travelling fronts in two dimensions but with different boundary conditions has been examined in only one other work to our knowledge.In an interesting paper, R. GARDNER [8] establishes the existence Of c and u solution of problems of the type ) with 0 </3 < 89 and ~ is the maximal positive solution of 4/' + f($) = 0, 4~(0) = 4~(L) = 0 (for L sufficiently large).R. GARDNER [8] uses a method relying on the Conley index.

Proof of Existence
In this section we prove Theorem 1.1.As in [3], this proof reduces to studying an analogous problem posed on the bounded cylindrical domain R, = (--a, a) • for a E ~ and then in examining the passage to the limit as a-+ + co.Solving the problem in R~ is essentially the same here as in [3].However [because the solution here does not necessarily satisfy (!.16)], the arguments in [3] would fail to yield the limit as a-+ + ~.Hence, we need here another approach to the derivation of the a priori estimates and to the limiting procedure.
We first consider the problem with the mixed boundary conditions: 3) To this system we add the following normalization condition which we trade against the freedom to choose c: The role of this condition (and also analogous normalization conditions) is discussed in BERESTYCKI & LARROUTUROtl [3] and in BERESTYCI<I, NICOLAENKO (~ SCHEURER [5].

4). []
The proof follows the steps of the one given in [3] with a few minor modifications.For the sake of completeness, we repeat it here.It rests on the following a priori estimates (in what follows we always assume that the definition of g is extended to all of R by setting g(s) =--0 for s~ [0, 1]): Lemma 2.2.Suppose g ~ M on [0, 1] and o~ o ~ or <~ oq for all y E -~.Then there is a constant K, depending only on a, ~o, oq and M, such that any solution (u, c) of (2.1)-(2.4)satisfies lel =< K, (2.5) and:

Ilull l< a) g. [] (2.6)
Proof of Lemma 2.2.Since g(s) -~ 0 outside the interval [0, 1], it follows from the maximum principle that 0 < u < 1 in R a. Hence, by a result of BERESTYCKI & NIRENBERG [6, Theorem 4.1], we know that u~, > 0 in (--a, a)• (actually, it is assumed in [6, Theorem 4.1] that u satisfies Dirichlet data on the whole boundary of Ra; but of course the same result holds under the present conditions).Using this information, we infer some inequalities from the maximum principle.Let 0r and 0q be such that % -< 0~(y) ~ 6r 1 for all y C ~, and denote Zo, zl the solutions of the following ordinary differential equations:

Proof of Theorem 1.I, divided into several steps
Step 1: A priori estimate on e~.Lemma 2.4.There is a constant K ~ 0, independent of a, such that for any a ~ 1 the solution (u``, ca)of (2.1),(2.Step 2: Existence of a limit. From the estimate (2.27) on c a we see by the classical elliptic estimates (see AOMON, DOUGLIS & NIRENBERG [1]) that, for any p > 1, ua is bounded in the W 2,p norm in any rectangle (xl, xl + 1)• contained in Ra, and this holds independently of a and xl.In particular, there is a K > 0 independent of a such that
Moreover, we can fred a sequence an -+ -k cx~ such that ca, -+ c in 1% and u,, -+ u locally in cgl norm.Obviously we obtain a solution of the equation in 27 (c -1-offy)) Uxl = Au -k g(u), we obtain From now on in this section, (u, c) will always denote the limit of (uan, Can).
It now remains to study the limits of u(x~, y) as xl-+ • cx~.In particular, we wish to prevent u a from converging locally to some constant 9 E [0, 0] W {1}.
Step 3: Energy estimates Here and hereafter the measure dx~ dy is understood to refer to integrals taken over ! or parts of I.
a) Assume first that t3+ = 0, and integrate the equation (2.1) satisfied by ua, " on the domain R~ = (0, a,,) • o.We get gl "-~xl (an, y) dy + j ~x (~ y) dy 9 (2.45) But we know from Lemma 2.6 that ua,, converges in the cgl sense to 0 on any compact subset off.Then the third and fifth terms in (2.45) converge to (c + (or 0 I~o[ 8u~,, and 0 respectively.Moreover, using the fact that ~ > 0 and Lemma 2.7, we obtain (1 -o)[,o l(c + (o,)) > 8, (2.46) whence (2.44).b) Assuming now that /3+ = 1, we can argue as in the proof of Lemma 2.7 to show that u (and not ua) also satisfies  We can now conclude the proof of Theorem 1.1, using the following lemma.Lemma 2.9.Under the assumption (o~) + 7 > O, there are a unique 2 > 0 and a corresponding "eigenfunetion" ~ = W(y) (which is strictly positive in -~) that solve the following problem: This result is a particular case of Theorem 3.4 in BERESTYCKI • NIRENBERG [7].We refer the reader to " [7] for the complete proof.
Here we use Lemma 2.9 to define a barrier function for u.First we choose a real 7 in such a way that (o~) q-7 > 0 and 7 < ca for large a (i.e. 7 = c -for some small e > 0).Since ~ is defined up to a multiplicative constant, we may as well assume that ~(y) ~ 0 on N. Then we consider the function qi defined by ~(xl, y) = e z~' ~(y).
( shows that t3+ cannot be equal to 0. Thus /3+ : 1 and the proof of Theorem 1.1 is complete.[]

Lemma 2 . 5 .
The following integrals are bounded

rStep 4 :
If f g(u) ~ + oo when z-+ + oo, then U'(+ c~) = --c~, which is impossible Qz sirlce Uis bounded.For the second integral in (2.32), we multiply (1.6) by u before integrating on Qz, and we conclude in the same manner.[] Existence of limits as x~ -~ ~ oo Since ux~ ~ 0, we know that the limits lim u(x~, y) =/~• exist.By xl-~--k oo considering the sequence of functions