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Nonlinear stability analysis of singular travelling waves in combustion— a one-phase problem
Nonhnear Analysis, Theory, Printed in Great Britam.
Methods
& Applicarions.
Vol.
16, No.
10, pp. X81-892,
1991 8
0362-546X/91 $3.00+ .CU 1991 Pergamon Press plc
NONLINEAR STABILITY ANALYSIS OF SINGULAR TRAVELLING WAVES IN COMBUSTIONA ONE-PHASE PROBLEM M. BRAUNER
C. Centre
de Recherche
en Mathematiques
de Bordeaux, Universite Bordeaux 33405 Talence Cedex, France S. NOOR
Department
M.I.S.,
Ecole Centrale
I, 351 Cours
de la Liberation,
EBAD
de Lyon,
B.P.
163, 69131 Ecully Cedex,
France
and CL. CNRWEcole
Normale
Superieure
SCHMIDT-LAINB
de Lyon, Laboratoire de Mathematiques, 69364 Lyon Cedex 07, France
46 AllCe d’Italie,
(Received 30 November 1989; received for publication 20 September 1990) Key words and phrases: Nonlinear combustion
stability
analysis,
travelling
waves,
moving
fronts,
weighted
norms,
models.
1. INTRODUCTION
in combustion, a gamut of simpler equations DUE TO the complexity of the full governing energy asymptotics is an models have been derived. First, it is well known that activation effective tool for dealing with the highly nonlinear reaction terms. Second, a complete chemicalkinetic description is often not available. For this reason simplified kinetic schemes are normally adopted, the simplest being the classical one-step irreversible scheme, represented by [Y] + products. Another attractive simplification is the equidiffusion hypothesis, d: (corresponding to the ratio of thermal/molecular diffusivities) scalar problem. The near equidiffusional flames (N.E.F.) theory is characterized
(9 where I = O(1) is the reduced (ii)
in which the Lewis number is taken to 1, leading to a by the requirements
[5]
2-l = 1 - 1/e, Lewis number
and 13the activation
energy,
and
H = Hf + O(K’)
where H is the enthalpy Y + T (the subscript f (for fresh) stands for the value at --oo while b (for burnt) corresponds to the value at +co). The second requirement corresponds to using the following expansions for the temperature T and the reactant mass fraction Y T = To + K’T,
+ ...
Y = (Hf - T,,) + K’(H1 881
- TJ + a...
(1.1)
C. M. BRAUNERet al.
882
Within this framework, flame sheet < = r(t) [5]
the basic equations
for a planar
flame become
on either side of the
azr,
ar,
at=ax2 aH, -=
at
In the burnt
gas the assumption
of equilibrium
At the flame sheet, r(t), the jump
- f(t(t)-).
leads to a2H,
aH, -=
at
conditions
(1.3)
ax2’
are
W,l = 0
IT,1 = 0,
where [f] = f(<(t)‘)
+ I$.
ax2
T,=T,,
(1.2)
2
a2H,
The boundary
conditions
TO(-co, t) = T, = 0,
(1.4)
are
H,(koo,
t) = 0.
(1.6)
System (1.2)-( 1.6) defines a two-phase free boundary problem with the flame sheet as the moving boundary. We point out that N.E.F.s are a restricted class of solutions, therefore, initial conditions must be consistent with the assumption that H is constant to leading order. As far as the stability question is concerned, the initial conditions are chosen as small disturbances of the travelling wave solutions of the above problem, which may be easily derived. These disturbances should not affect the conditions at the front, otherwise the equilibrium assumption would be violated. In this paper we are interested in the nonlinear stability analysis of travelling waves in the equidiffusional case I = 0, under a special class of disturbances, namely those which perturb the temperature only, so the (perturbation of) enthalpy H, is identically zero. Therefore, we consider the following one-phase problem, subject to an adequate initial condition: find a temperature To and a front 5 obeying
ar, a2G -=at ax2 ’
r,=
x < t-(f) x 2 at)
T-t,
a&
ax
= x=
(1.7)
r,.
t-(t)
Let us mention that (1.7) is also related to a one-phase and Ludford [17], describing the deflagration-detonation deflagration, under an appropriate stimulus, accelerates
version of a model derived by Stewart transition (D.D.T.) by which a and evolves into a detonation. As the
Nonlinear stability analysis
detonation corresponds to the emergence wave propagating downstream, namely
of a progressive
883
wave, the model is built to keep the
(1.8) on either side of the flame sheet l(t), F and [F,] being prescribed at the front [17-191. An extensive numerical investigation of this two-phase problem [l l] has finally shown the behaviour of the solution. Another two-phase system stemming from the N.E.F. theory describes the propagation of a primary reaction premixed flame, including density variation effects [8]. Again this formulation can be reduced to (1.7) for special parameter values. In general, two-phase problems encounter exchange of stability (Hopf bifurcations for the N.E.F.s [l], sharp transition due to the emergence of an eigenvalue from infinity in the D.D.T. model [4]), however, they admit a large domain of stability. Therefore, although it is always stable, the one-phase problem (1.7) can be considered as a paradigm reflecting many situations, which makes its mathematical study more valuable (see [lo] for an I.V.P. approach). We are going to extend to this singular case, a method developed by Sattinger [14, 151, for smooth travelling waves (see also [7, 13]), which consists of: (i) linear stability analysis in suitable Banach spaces with weighted norm; (ii) appeal to the implicit function theorem for the nonlinear stability. Here the framework is rather different since the speed of the front is involved in the formulation (see Section 2), and eventually generates a nonlocal term. In the linear analysis (Section 3) the front is eliminated due to a Lyapunov-Schmidt-like procedure. A sharp regularity result in Holder spaces ([12, 161) is invoked in Section 4. The reader is referred to [ 1 l] for some tedious technical details which are omitted here from the proofs. 2. MATHEMATICAL
FORMULATION
We are concerned with the nonlinear stability of the following one-phase problem, a front r = c(t) and a function (temperature) u = u(q, t) are to be determined u, = u,,, u(<(t), 0 = u* 9 u(-00, where U, is a fixed positive number. It is straightforward to check that (2.1) u(q, t) = U(v -k ct) with velocity -c, c > 0 y = rj + ct,
in which
--co < D < T(f) u,(r(t)-,
t) = 1
(2.1)
t) = 0,
admits
a travelling
wave
(T.W.)
solution
c = l/u*
(2.2) U(y) = u*ecy. The method does not require the explicit knowledge of (I, but only its asymptotic decay as y approaches -co. Because of the translation invariance, @(y) = U(y - h) is also a T.W. solution whenever h E R. We refer the reader to the end of Section 3 for some geometrical insights.
C. M. BRAUNER et al.
884
In the moving
coordinate
frame y = q + ct, problem 24, + cur = uvv,
--Q, < y < h(t)
u(h(O, 0 = u*,
u,(h(t)-,
U(-co, As already data
(2.1) now reads (h(t) = r(t) - ct)
stated in the Introduction,
t> = 1
(2.3)
t) = 0.
the stability
question
consists
h(0) = 0,
U(Y>0) = U(Y) + E%(Y),
in assigning
an initial (2.4)
where E > 0 and u0 is a given smooth function with compact support in (-00, 0). The latter assumption on u0 is to meet the requirements of N.E.F. and for the sake of simplicity, but it can be weakened. As usual in autonomous T.W. problems, we do not think of classical stability, but rather of “orbital stability” [14]: as t + +a, h(t) converges to a phase shift h, which can be calculated by formal integration of (2.3) 0
h, = --EC
I’--co
uo dr.
(2.5)
Therefore, we do not expect u to converge to CT,but rather to approach Uhm, in an exponential way. Finally it is convenient to reformulate (2.3) in a fixed coordinate frame x = y - h(t)
u, + (c - ht)u, = uxx, 40, t) = u* >
u,(o-,
-w
t) = 1,
U(-co,
t) = 0
(2.6)
U(0, x) = U(x) + &z&)(X). Our goal is to establish the existence of a function u = U&(X,t) and a front h = h”(t), solutions of (2.6), such that z.P -+ U and h” + h, when t --) sco (as e-“’ for some o > 0). Definitions of the functional spaces are deferred to the next section. More precisely we will look for U’ and h” in the form U&(X,f) = U(x) + &U&(X,t),
h”(t) = EZ’(~).
Clearly it is equivalent to prove the existence and exponential some value z,. Substituting (2.7) into (2.6) leads to v: + cv; - v;, - z:u,
(2.7)
decay of U’ to 0 and of zE to
= &Z$,E
u&(0, t) = u,E(O,t) = 0
(2.8)
U&(X,0) = Q(X).
Remark 2.1. Taking second-order Stefan equation [4].
formally condition.
x = 0 into (2.8), it turns out that z:(t) = -u&(0, t) which is a Replacing z: by this value leads to a fully nonlinear parabolic
Nonlinear stability analysis 3. LINEAR
STABILITY
885
ANALYSIS
The linear stability analysis corresponds to the resolution of (2.8) with E = 0 up + cv,” - l& - zpu, = 0 vO(O,t) = v,“(O,t) = 0
(3.1)
vO(x, 0) = u,(x). For this purpose we start by defining a class of weighted Banach spaces introduced by Sattinger [14, 151and an operator which will play a crucial role in the analysis. The space of real continuous, bounded, functions with the sup norm]] * )I0 will be denoted by Cj. As far as the space variable is concerned, x always belongs to the half-line (-a~, 01, and uniform continuity is implicit. For j an integer 2 1, the notation Ci is straightforward. Definition 3.1. For q(x) = e-cx’2, C,,, = (v, qv E Cjj endowed with the norm ]Iv&,~ = I/q&,. Definition 3.2. For j an integer natural norm 11 VJJq, j.
2 1, C,,j = (v, vdo) E CQ,O,p = 0, . . . , j), endowed with the
In the sequel we will denote C,,, by X as the reference space. On the space X, we define the unbounded operator L by Ly, = vxx - wx (3.2) D(L) = (u, E Cg,2, PAO) = CCJ@)~THEOREM 3.1. L is a sectorial operator in the space X. It has an isolated (simple) eigenvalue at the origin, while the essential spectrum is the half-line (-00, -c2/4],
Remark 3.1. A straightforward computation shows that, in the space Ci, 0 is not an isolated eigenvalue, which compromises the stability. The introduction of the weight q shifts the essential spectrum to the left and leaves the eigenvalue 0 (due to the translation invariance) isolated. Proof of theorem 3.1. To determine the spectrum a(L) and the resolvent set p(L), it is convenient to introduce the unbounded operator M in Cj Mu = v,,
C2 --V
4 (3.3)
D(M) =
v E c;,
v,(O) = iv(O)
1 Clearly M is derived from operator L by the straightforward LEMMA 3.1.
p(M)
.
1 transformation
L = q_lMq.
= p(L).
Proof. If A E p(L), then, for any f E C4,0, the equation (L - A)u = f has a unique solution But v = qu belongs to D(M), (A4 - A)v = g, g = qf, and u E D(L), llu&,,, 5 W)l]fl],,,. llvl10I K(/z)llgllO, therefore v E p(M). The converse is true.
886
C.
M. BRAUNER et
For g E Cj , A E @, we consider
the resolvent
LEMMA 3.2. p(M) Proof.
= C*\(-co,
al.
-c2/4].
Vxx -
equation
v,(O)
=
; v(0).
(3.4)
Clearly r(A) = is analytic in C\( -00, -c2/4], and the solution of the homogeneous equation Aeraox + Be-‘(X)X. It can be checked that the solution of (3.4) is given by
is v(x) =
(3.5) in which A has to be nonzero,
in order to have 2r(A) - c # 0. Furthermore
we have the estimate (3.6)
where r,(n) = Re r(A) > 0, and the lemma
3.2 is proved.
The next lemma will demonstrate that L is a sectorial that L is a closed densely defined operator).
operator
(cf. [9]. See [l l] for a proof
LEMMA 3.3. Whenever 6 E (71/4,7r/2), the sector Ss = (A, larg A( < 7c - 6, A # Oj is in the resolvent set p(L) and there exists C(6) > 0 such that, for any f E X (3.7)
Proof.
Clearly
S6 C p(L). After lemma
3.1, we have
IIW - Wfll,,o 5 mMl,,o~
vf
where K(I) is the constant in (3.6), and r(k) = d[(c2/4) + A]. For I@)( L d(IAI sin 6) and r,(A) = Re A 2 c,(S)J& where c,(6)
= min(J(sin
(3.8)
E x9 any
A E S,,
one
has
6), sin 6 - cos 6).
On the other hand, the ratio 12r(A) + c(/(2r(h) - c( converges to 1 as I@)( -+ to, while it is of order c2/)Aj when r(A) approaches c/2. Therefore, K(1) I C(s)/]J.j. (See [ll] for details.) Lemma 3.3 finishes the proof of theorem 3.1. However, u, solution of (3.4), by u, in the proof of lemma 3.2.
(3.7) may be improved
by replacing
Nonlinear stability analysis COROLLARY
3.1. Whenever
887
6 E (7r/4, n/2), f E X
IIW - WII,,,
-$$ Ilfllq,o,
5
v?L ES*.
(3.9)
As 0 is an isolated eigenvalue, o1 = (0) and o2 = (- co, -c2/4] U f-co] are spectral sets. Let Ej be the projections associated with these spectral sets and Xj = E’(X), j = 1,2. Then X = Xi @ X,, the Xj are invariant under L, and, if Lj is the restriction of L to Xj, then L,: Xl --f X, is bounded, D(L,) LEMMA
3.5. X, is spanned
Proof.
results
and
01
= 02.
on X by E,(v)
= (c j!_ a, dx)U,.
onto Xi such that L 0 E, = E, 0 L = 0.
projection about
=
a(L,)
by U, = ecx and E, is defined
E, is the unique
The following
= D(L) II x,
a(L,)
L, follow
from standard
analytic
semi-group
3.2. (i) L, is also a sectorial operator, which generates (ii) V 6 E (0, c2/4), [(ezLZ(Iz(XZ) 5 Cst e-“, V t > 0; (iii) erL2 has the representation
an analytic
theory
(cf. e.g.
]6, 91). COROLLARY
etL2 = &
semi-group
erL2;
(AZ - L,)-‘e”dl, Ir
where I is any contour We now return
which lies strictly
arg P, -+ fB, B E (7r/2, n), as II( + 00.
to (3. l), to which we apply the operators v” = E,v”
yields,
in p(L,),
+ E2vo =p’U,
E, and E2. The splitting + w”
of (3.11)
since E, v, = E, v,, = 0 pp = zp (3.12) wp + cwp - w,9; = 0.
In order to derive the boundary
conditions
associated
to w,, we calculate
wO(0, t) = vO(0, t) - pO(t)U,(O)
= -PO(i)
w,“(O, t) = v,o(O,t) - pO(t)U,,(O)
= -cpO(t)
(3.13) hence p” may be eliminated
through
(3.13) w,“(O, t) = cwO(0, t).
Summing
up, we find that w” verifies
the following
wp = L2 w”, whose solution namely
problem
w’(O) = E,u,
is w’(t) = efLZE2uo from corollary
(3.14) in the space X2 E X2
3.2. Next, p” is explicitly
pO(t) = -wO(O, t).
(3.15) given by (3.13), (3.16)
888
After
C.
(3.12),
p” and
z” differ
M.
et al.
BRAUNER
up to a constant
only.
For p’(O)U, = EluO and z’(O) = 0
(see (2.4)) 0 zO(t> =
p(t) - c
ug dx.
(3.17)
s --m
Remark 3.1. The technique
used in (3.11) enables us to fully decouple (3. l), and reduces the problem to a P.D.E. (3.15) in a space of co-dimension 1, plus a trivial relation (3.16) in one dimension. Therefore, we refer to it as a Lyapunov-Schmidt-like procedure. We will use it again in the nonlinear analysis (Section 4). The next step is to examine LEMMA 3.6. Whenever
the asymptotic
behaviour
0 < w < c2/4, (IwO(t)JI,,, 5 cst e-“‘,
Proof. After corollary I,U’ = wp, which solves
3.2(ii),
one has
IjL, ~‘(t)jJ,,~
It follows
from
IIvOwIl,, 1. From
3.6 and
5 cste-U’.
(3.18) The same
result
holds
for
(3.19)
l//O(O) = LUO.
I cst e-“’ which obviously
lemma (3.17)
vt>o.
)(w’(t)\l,,,
wt” = &Jo, Therefore,
of w” for f large.
yields (3.18).
(3.16) that p’(t) -+ 0 as e-“’
when
t -+ co, and so does
”0 p(t)
-+ Zm = -c
u() dx.
(3.20)
! --m To take into account this exponential decay as time approaches introduce a class of suitable Banach spaces [ 141.
Definition 3.1. Let Y be a Banach e”(]O, +a); In the definition,
infinity,
it is convenient
to
space, o > 0
Y) = ]U E Ci([O, +m); Y), w; e”‘IIIp(t)ll. < ~1.
case Y = C4,j, the weighted sup-norm will be denoted by II~[14,j,o. With lemma 3.6 simply reads: w” E c“([O, +co); C,,,), 0 < w < c2/4.
this
To conclude this section devoted to linear analysis, we attempt to present some geometrical interpretation of the problem, going back to formula (2.2), (2.3) in the moving coordinate frame y = q + ct. The wave U gives rise to a one-parameter family of waves (U’], Uh(y) = U(y - h), which can be regarded as a one-dimensional manifold. The derivative
is equal to -U,
.
889
Nonlinear stability analysis
h
Fig. 1
We look for a formal expansion u(y, t) = U(y) + e&y, t) + . . . . h(t) = s&(t) < 0 (say), --co < y < 0. Note that we are linearizing about U not a shift of it. It turns out that (0,e) verifies 0, = Lir,
cl,=,
= &
(3.21)
t(t) = -O(O, t). In (3.21) L is defined as in (3.2) and has 0 as a simple eigenvalue with eigenvector U,. The critical (or centre) manifold is precisely the curve Uh and the flow in this manifold is given by dh/dt = 0 [9, p. 1741. The solution ir of (3.21) writes (3.22)
o(t) = A0 ir + G(t),
where ti, verifies I?( = L,ti
(3.23)
G(O) = u. - A00
which is exactly (3.15) obtained in a direct but formal way. As t + a, C(t) -+ A, 0 hence the final position of 2(t) is 0 .i&
=
-A,
ir(O>
=
A,
q,(O)
=
A0
=
UO
-c
-co 4. NONLINEAR
dx.
(3.24)
ANALYSIS
We now turn to the full nonlinear problem (2.8) with E > 0. We apply the same Lyapunov-Schmidt-like procedure as in Section 3. However, the speed of the front may not be completely eliminated. We see that vE=E1vE+E2vE=pEUX+
wE
(4.1)
ug d_x
(4.2)
0
P: = zf,
p&(O) = -c 1 -ca
w,” = L, WE + Epfv,,
p”(t) = -W&(0, t) (4.3)
w”(O) = E2uo.
890
BRAUNER et al.
C. M.
In order to deal with (4.3), we need some insight on abstract Cauchy problems spaces and Holder regularity of solutions. First we recall the following definition. Definition
4.1. Let Y be a Banach Ckq[o,
+a);
in Banach
space, k E N, 0 < 6’ < 1.
Y) = (rp E Ck([O, $00); Y), (y?‘ky)
< 031,
where
(P) (0)
IIv(t + h) - cm0Y
sup
=
IV
t,t+h>o
Ihi iho
This space is equipped
*
with the norm lllVlllk,e = lIG&k,~(0,+m),Y) + (V)@).
We also need the space C$‘([O, +m); Y) of functions of C’,‘([O, +m); Y) such that p(O) = 0. Next, we define the operator K: f r--) u where u is the solution of the auxiliary Cauchy problem u, = L,u Operator
K has two kinds of important
f
u(0)= 0.
f,
properties
(4.4)
which are collected
into two lemmas.
LEMMA 4.1. For any o, 0 < o < c2/4, the operator K is a bounded transformation from ?([O, +a); X,) into ?([O, +a~); E,C,,,). This result follows from corollary 3.1 and the representation (3.10) of the semi-group efL2. Since the proof corresponds to those of lemmas 3.4 and 3.5 of Sattinger [14] (see also [l l]), we shall omit it. LEMMA 4.2. The operator K is a bounded transformation from C$‘([O, C’,‘([O, too); X,) n C”,‘([O, +a); D(L,)). Moreover, u,(O) = 0. This lemma is based on corollary 3.2 and the following abstract result. THEOREM 4.1. Let Y be a Banach e ‘A such that
space, A the infinitesimal
ItetAIIs(y) s Me-“‘,
Let u be the solution
of the Cauchy
generator
fco); X,)
of an analytic
into
semi-group
M and 6 > 0.
(4.5)
problem
24, = Au +
f,
u(0)= 0.
(4.6)
Then the mapping f ++u is a bounded transformation from C$‘([O, 400); Y) into C’,O([O, +oo); Y). This theorem is a by-product of abstract results for evolution equations in Banach spaces (cf. Sinestrari [16, p. 531, Pazy [12, p. 478]), extended to the case T = +o3 under the additional hypothesis (4.5)-see [ 111. We are now in position
to reformulate
(4.2), (4.3) with the help of the operator
F(w”; E) = wE - ~K[p;(cp”U,
K
+ w;)] - w” = 0 (4.7)
p”(t) = -W&(0, t).
Nonlinear
stability
analysis
891
For 0 < o < cz/4 and 0 < B < 1, we define the functional space w = (W E P([o,
+oo); x,)
n c”~o([o, +w); D(L,)), ~~(0, 0) = 0)
n mo, +4; w,,,)
(4.8)
equipped with the natural norm. THEOREM 4.2. w” E W and the mapping 5 defined by (4.7) is a Frtchet differentiable from W x R+ into W.
mapping
Proof. (i) w” is smooth because of our assumptions on uo, and we already know w” E P’([O, +m); E,C,,,), therefore w” E W. (ii) Let us check that if w E W, S(w; E) E W. Following lemmas 4.1 and 4.2, it is sufficient to show that the quantity in brackets belongs to P([O, +w); X,) fl Ci([O, +oo); X,) and is zero at t = 0. Let f = p,(cpU, + w,), p(t) = - ~(0, t) IIf(t)ll,,o
llw*wll,,o) 5 Ill411, &II Noll,,o + IIwxwll,,A,
5 IPrl * (clp(t)l
+
where the triple-bar is the norm of C’,‘([O, +oo); X). Therefore
Ilf IIp,o,o 5 4llwlll1,e * l14q,l,w~ sof belongs to sW([O,+a); X,). On the other hand, w, E PO([O, +co); X,)
f E Ci([O, +a); X2) for p E c’sB([o, +a)).
and
Finally f(0) = 0 since p,(O) = - w,(O, 0) = 0. (iii) Let us show that 5 is C’. The only point to be checked is that 5w is continuous. calculate, for a, E W S,(w; 8) * v = 9 - &Kb, VI = -%a 472 = PA-d09
We
+ vl21
*)(cPUx + wx)
(4.9)
*)v, + v,).
It is not difficult to verify that v, c p, and o, u IJI~are continuous
from W into
em([O,+a); x2) n C,B([o, +m); x2), the arguments being similar to the ones of (ii); thus lemmas 4.1 and 4.2 apply. We may thus apply the implicit function theorem to the mapping 5. To this end we compute the Frechet derivative of 5 at (w’, 0), which clearly is the identity. Therefore, by the implicit function theorem we have the following result, where o is any fixed number in the interval (0, c2/4). THEOREM 4.3. There exists a C’ mapping E c~ wE from some interval [0, so) into W such that S(w”; E) = 0. Therefore problem (4.3) admits a unique solution (wE,pE) for E E [0, so) such that, as t -+ +a, 1)w”(t))),, 1 5 cst e-“‘.
For p”(t) = -w&(0, t), Ip”(t)l I cst e-“’ also.
(4.10)
C. M. BRAUNER et al
892
Going back to U’ and z&--See (4.1), COROLLARY4.1. Whenever asf-++oo
(4.2)-we
E E [0, ao), problem
have the following
corollary.
(2.8) admits a unique solution
[z”(t) - z-1 5 csfepw’,
z, = -c
(u’, z”) such that
(4.11)
Ou. dx.
I
--m
Finally we have the following question
raised in Section
theorem
which provides
a complete
answer to the stability
1.
THEOREM 4.4. Define h, = h5 = --EC j!_ u. dx. Whenever unique solution (u, h) = (u’, h”) such that, as t -+ +co
E E [0, Q), problem
(2.6) admits a
I(u”(t) - U((,,, 5 cstepO’
(4.12)
IF(t) - h:l 5 cste-w’. Acknowledgement-The collaboration between C.M.B. and C.S.L. was made MPB-SPI Instabilitts des flammes laminaires de premelange 8720 N5/25).
possible
by a CNRS
grant
(ATP
REFERENCES
ALABAU F. (to appear).
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
BERTSCHM., HILHORSTD. & SCHMIDT-LAIN~CL., The well-posedness of a free boundary problem arising in combustion theory (to appear). BRAUNERC. M., NOOREBAD S. & SCHMIDT-LAIN~~ CL., Sur la stabilite d’ondes singulieres en combustion, C. r. Acad. Sci. Paris 308, 159-162 (1989). LUNARDIA. & SCHMIDT-LAIN~ CL., Une nouvelle formulation de modeles de fronts en problemes totalement non lineaires, C.r. Acad. SC;. Paris 311, 597-602 (1990). Stabilite et instabilite des ondes stationnaires d’un probleme de detonation a deux phases, C.r. Acad. Sci. Paris 311, 697-700 (1990). BUCKMASTER J.D. & LUDFORDG. S. S., Lectures on mathematical combustion, CBMS-NSF Conference series Applied Mathematics, (1983), p. 43. SIAM. DUNFORDN. & SCHWARTZJ. T., Linear Operators I. Interscience, New York (1957). FIFE P. & MCLEOD J. B.. The approach of solutions of nonlinear diffusion equations to traveling front solutions, Archs ration. Mech. Analysis 65, 335-361 (1977). VAN HARTENA. & MATKOWSKY B. J.. A new model in flames theory. SIAM J. apol. Math. 42, 850-867 (1982). HENRY D., Geometric theory of semilinear parabolic equations, L&tare Notes-b Mathematics 840. Springer, New York (1981). HILHORSTD. & HULSHOFJ., An elliptic-parabolic problem in combustion theory: convergence to traveling waves (to appear). NOOR EBAD S., Aspects theoriques et numeriques de stabilite d’ondes dans des modeles de combustion, Ph.D. thesis, Lyon (1989). PAZY A., Semi-groups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44. Springer, Berlin (1983). ROYTBURDV., A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics, J. difs. Eqns 56, 40-62 (1985). SATTINCERD. H., On the stability of waves of nonlinear parabolic systems, Adv. Math. 22, 312-355 (1976). SATTINGERD. H.. Weiehted norms for the stability of traveling waves, J. diff. Eans 25, 130-144 (1977). SINESTRARIE., On the-abstract Cauchy problem of paraboliciype in spaces of continuous functions, J. math. Analysis Applic. 107, 16-66 (1985). STEWARTD. S. & LUDFORDG. S. S., Evolution of near Chapman-Jouguet deflagrations, Trans. 28th Conf of Army Math., AR0 Report 83-1, pp. 133-142 (1983). STEWARTD. S., KAPILAA. K. & LUDFORDG. S. S., Deflagrations and detonations for small heat release, J. Me%. ThPor. Appl. 3. 105-115 (1984). STEWARTD. S., Transition to detonation on a model problem, J. M&. ThPor. Appl. 4, 103-137 (1985).
Methods
& Applicarions.
Vol.
16, No.
10, pp. X81-892,
1991 8
0362-546X/91 $3.00+ .CU 1991 Pergamon Press plc
NONLINEAR STABILITY ANALYSIS OF SINGULAR TRAVELLING WAVES IN COMBUSTIONA ONE-PHASE PROBLEM M. BRAUNER
C. Centre
de Recherche
en Mathematiques
de Bordeaux, Universite Bordeaux 33405 Talence Cedex, France S. NOOR
Department
M.I.S.,
Ecole Centrale
I, 351 Cours
de la Liberation,
EBAD
de Lyon,
B.P.
163, 69131 Ecully Cedex,
France
and CL. CNRWEcole
Normale
Superieure
SCHMIDT-LAINB
de Lyon, Laboratoire de Mathematiques, 69364 Lyon Cedex 07, France
46 AllCe d’Italie,
(Received 30 November 1989; received for publication 20 September 1990) Key words and phrases: Nonlinear combustion
stability
analysis,
travelling
waves,
moving
fronts,
weighted
norms,
models.
1. INTRODUCTION
in combustion, a gamut of simpler equations DUE TO the complexity of the full governing energy asymptotics is an models have been derived. First, it is well known that activation effective tool for dealing with the highly nonlinear reaction terms. Second, a complete chemicalkinetic description is often not available. For this reason simplified kinetic schemes are normally adopted, the simplest being the classical one-step irreversible scheme, represented by [Y] + products. Another attractive simplification is the equidiffusion hypothesis, d: (corresponding to the ratio of thermal/molecular diffusivities) scalar problem. The near equidiffusional flames (N.E.F.) theory is characterized
(9 where I = O(1) is the reduced (ii)
in which the Lewis number is taken to 1, leading to a by the requirements
[5]
2-l = 1 - 1/e, Lewis number
and 13the activation
energy,
and
H = Hf + O(K’)
where H is the enthalpy Y + T (the subscript f (for fresh) stands for the value at --oo while b (for burnt) corresponds to the value at +co). The second requirement corresponds to using the following expansions for the temperature T and the reactant mass fraction Y T = To + K’T,
+ ...
Y = (Hf - T,,) + K’(H1 881
- TJ + a...
(1.1)
C. M. BRAUNERet al.
882
Within this framework, flame sheet < = r(t) [5]
the basic equations
for a planar
flame become
on either side of the
azr,
ar,
at=ax2 aH, -=
at
In the burnt
gas the assumption
of equilibrium
At the flame sheet, r(t), the jump
- f(t(t)-).
leads to a2H,
aH, -=
at
conditions
(1.3)
ax2’
are
W,l = 0
IT,1 = 0,
where [f] = f(<(t)‘)
+ I$.
ax2
T,=T,,
(1.2)
2
a2H,
The boundary
conditions
TO(-co, t) = T, = 0,
(1.4)
are
H,(koo,
t) = 0.
(1.6)
System (1.2)-( 1.6) defines a two-phase free boundary problem with the flame sheet as the moving boundary. We point out that N.E.F.s are a restricted class of solutions, therefore, initial conditions must be consistent with the assumption that H is constant to leading order. As far as the stability question is concerned, the initial conditions are chosen as small disturbances of the travelling wave solutions of the above problem, which may be easily derived. These disturbances should not affect the conditions at the front, otherwise the equilibrium assumption would be violated. In this paper we are interested in the nonlinear stability analysis of travelling waves in the equidiffusional case I = 0, under a special class of disturbances, namely those which perturb the temperature only, so the (perturbation of) enthalpy H, is identically zero. Therefore, we consider the following one-phase problem, subject to an adequate initial condition: find a temperature To and a front 5 obeying
ar, a2G -=at ax2 ’
r,=
x < t-(f) x 2 at)
T-t,
a&
ax
= x=
(1.7)
r,.
t-(t)
Let us mention that (1.7) is also related to a one-phase and Ludford [17], describing the deflagration-detonation deflagration, under an appropriate stimulus, accelerates
version of a model derived by Stewart transition (D.D.T.) by which a and evolves into a detonation. As the
Nonlinear stability analysis
detonation corresponds to the emergence wave propagating downstream, namely
of a progressive
883
wave, the model is built to keep the
(1.8) on either side of the flame sheet l(t), F and [F,] being prescribed at the front [17-191. An extensive numerical investigation of this two-phase problem [l l] has finally shown the behaviour of the solution. Another two-phase system stemming from the N.E.F. theory describes the propagation of a primary reaction premixed flame, including density variation effects [8]. Again this formulation can be reduced to (1.7) for special parameter values. In general, two-phase problems encounter exchange of stability (Hopf bifurcations for the N.E.F.s [l], sharp transition due to the emergence of an eigenvalue from infinity in the D.D.T. model [4]), however, they admit a large domain of stability. Therefore, although it is always stable, the one-phase problem (1.7) can be considered as a paradigm reflecting many situations, which makes its mathematical study more valuable (see [lo] for an I.V.P. approach). We are going to extend to this singular case, a method developed by Sattinger [14, 151, for smooth travelling waves (see also [7, 13]), which consists of: (i) linear stability analysis in suitable Banach spaces with weighted norm; (ii) appeal to the implicit function theorem for the nonlinear stability. Here the framework is rather different since the speed of the front is involved in the formulation (see Section 2), and eventually generates a nonlocal term. In the linear analysis (Section 3) the front is eliminated due to a Lyapunov-Schmidt-like procedure. A sharp regularity result in Holder spaces ([12, 161) is invoked in Section 4. The reader is referred to [ 1 l] for some tedious technical details which are omitted here from the proofs. 2. MATHEMATICAL
FORMULATION
We are concerned with the nonlinear stability of the following one-phase problem, a front r = c(t) and a function (temperature) u = u(q, t) are to be determined u, = u,,, u(<(t), 0 = u* 9 u(-00, where U, is a fixed positive number. It is straightforward to check that (2.1) u(q, t) = U(v -k ct) with velocity -c, c > 0 y = rj + ct,
in which
--co < D < T(f) u,(r(t)-,
t) = 1
(2.1)
t) = 0,
admits
a travelling
wave
(T.W.)
solution
c = l/u*
(2.2) U(y) = u*ecy. The method does not require the explicit knowledge of (I, but only its asymptotic decay as y approaches -co. Because of the translation invariance, @(y) = U(y - h) is also a T.W. solution whenever h E R. We refer the reader to the end of Section 3 for some geometrical insights.
C. M. BRAUNER et al.
884
In the moving
coordinate
frame y = q + ct, problem 24, + cur = uvv,
--Q, < y < h(t)
u(h(O, 0 = u*,
u,(h(t)-,
U(-co, As already data
(2.1) now reads (h(t) = r(t) - ct)
stated in the Introduction,
t> = 1
(2.3)
t) = 0.
the stability
question
consists
h(0) = 0,
U(Y>0) = U(Y) + E%(Y),
in assigning
an initial (2.4)
where E > 0 and u0 is a given smooth function with compact support in (-00, 0). The latter assumption on u0 is to meet the requirements of N.E.F. and for the sake of simplicity, but it can be weakened. As usual in autonomous T.W. problems, we do not think of classical stability, but rather of “orbital stability” [14]: as t + +a, h(t) converges to a phase shift h, which can be calculated by formal integration of (2.3) 0
h, = --EC
I’--co
uo dr.
(2.5)
Therefore, we do not expect u to converge to CT,but rather to approach Uhm, in an exponential way. Finally it is convenient to reformulate (2.3) in a fixed coordinate frame x = y - h(t)
u, + (c - ht)u, = uxx, 40, t) = u* >
u,(o-,
-w
t) = 1,
U(-co,
t) = 0
(2.6)
U(0, x) = U(x) + &z&)(X). Our goal is to establish the existence of a function u = U&(X,t) and a front h = h”(t), solutions of (2.6), such that z.P -+ U and h” + h, when t --) sco (as e-“’ for some o > 0). Definitions of the functional spaces are deferred to the next section. More precisely we will look for U’ and h” in the form U&(X,f) = U(x) + &U&(X,t),
h”(t) = EZ’(~).
Clearly it is equivalent to prove the existence and exponential some value z,. Substituting (2.7) into (2.6) leads to v: + cv; - v;, - z:u,
(2.7)
decay of U’ to 0 and of zE to
= &Z$,E
u&(0, t) = u,E(O,t) = 0
(2.8)
U&(X,0) = Q(X).
Remark 2.1. Taking second-order Stefan equation [4].
formally condition.
x = 0 into (2.8), it turns out that z:(t) = -u&(0, t) which is a Replacing z: by this value leads to a fully nonlinear parabolic
Nonlinear stability analysis 3. LINEAR
STABILITY
885
ANALYSIS
The linear stability analysis corresponds to the resolution of (2.8) with E = 0 up + cv,” - l& - zpu, = 0 vO(O,t) = v,“(O,t) = 0
(3.1)
vO(x, 0) = u,(x). For this purpose we start by defining a class of weighted Banach spaces introduced by Sattinger [14, 151and an operator which will play a crucial role in the analysis. The space of real continuous, bounded, functions with the sup norm]] * )I0 will be denoted by Cj. As far as the space variable is concerned, x always belongs to the half-line (-a~, 01, and uniform continuity is implicit. For j an integer 2 1, the notation Ci is straightforward. Definition 3.1. For q(x) = e-cx’2, C,,, = (v, qv E Cjj endowed with the norm ]Iv&,~ = I/q&,. Definition 3.2. For j an integer natural norm 11 VJJq, j.
2 1, C,,j = (v, vdo) E CQ,O,p = 0, . . . , j), endowed with the
In the sequel we will denote C,,, by X as the reference space. On the space X, we define the unbounded operator L by Ly, = vxx - wx (3.2) D(L) = (u, E Cg,2, PAO) = CCJ@)~THEOREM 3.1. L is a sectorial operator in the space X. It has an isolated (simple) eigenvalue at the origin, while the essential spectrum is the half-line (-00, -c2/4],
Remark 3.1. A straightforward computation shows that, in the space Ci, 0 is not an isolated eigenvalue, which compromises the stability. The introduction of the weight q shifts the essential spectrum to the left and leaves the eigenvalue 0 (due to the translation invariance) isolated. Proof of theorem 3.1. To determine the spectrum a(L) and the resolvent set p(L), it is convenient to introduce the unbounded operator M in Cj Mu = v,,
C2 --V
4 (3.3)
D(M) =
v E c;,
v,(O) = iv(O)
1 Clearly M is derived from operator L by the straightforward LEMMA 3.1.
p(M)
.
1 transformation
L = q_lMq.
= p(L).
Proof. If A E p(L), then, for any f E C4,0, the equation (L - A)u = f has a unique solution But v = qu belongs to D(M), (A4 - A)v = g, g = qf, and u E D(L), llu&,,, 5 W)l]fl],,,. llvl10I K(/z)llgllO, therefore v E p(M). The converse is true.
886
C.
M. BRAUNER et
For g E Cj , A E @, we consider
the resolvent
LEMMA 3.2. p(M) Proof.
= C*\(-co,
al.
-c2/4].
Vxx -
equation
v,(O)
=
; v(0).
(3.4)
Clearly r(A) = is analytic in C\( -00, -c2/4], and the solution of the homogeneous equation Aeraox + Be-‘(X)X. It can be checked that the solution of (3.4) is given by
is v(x) =
(3.5) in which A has to be nonzero,
in order to have 2r(A) - c # 0. Furthermore
we have the estimate (3.6)
where r,(n) = Re r(A) > 0, and the lemma
3.2 is proved.
The next lemma will demonstrate that L is a sectorial that L is a closed densely defined operator).
operator
(cf. [9]. See [l l] for a proof
LEMMA 3.3. Whenever 6 E (71/4,7r/2), the sector Ss = (A, larg A( < 7c - 6, A # Oj is in the resolvent set p(L) and there exists C(6) > 0 such that, for any f E X (3.7)
Proof.
Clearly
S6 C p(L). After lemma
3.1, we have
IIW - Wfll,,o 5 mMl,,o~
vf
where K(I) is the constant in (3.6), and r(k) = d[(c2/4) + A]. For I@)( L d(IAI sin 6) and r,(A) = Re A 2 c,(S)J& where c,(6)
= min(J(sin
(3.8)
E x9 any
A E S,,
one
has
6), sin 6 - cos 6).
On the other hand, the ratio 12r(A) + c(/(2r(h) - c( converges to 1 as I@)( -+ to, while it is of order c2/)Aj when r(A) approaches c/2. Therefore, K(1) I C(s)/]J.j. (See [ll] for details.) Lemma 3.3 finishes the proof of theorem 3.1. However, u, solution of (3.4), by u, in the proof of lemma 3.2.
(3.7) may be improved
by replacing
Nonlinear stability analysis COROLLARY
3.1. Whenever
887
6 E (7r/4, n/2), f E X
IIW - WII,,,
-$$ Ilfllq,o,
5
v?L ES*.
(3.9)
As 0 is an isolated eigenvalue, o1 = (0) and o2 = (- co, -c2/4] U f-co] are spectral sets. Let Ej be the projections associated with these spectral sets and Xj = E’(X), j = 1,2. Then X = Xi @ X,, the Xj are invariant under L, and, if Lj is the restriction of L to Xj, then L,: Xl --f X, is bounded, D(L,) LEMMA
3.5. X, is spanned
Proof.
results
and
01
= 02.
on X by E,(v)
= (c j!_ a, dx)U,.
onto Xi such that L 0 E, = E, 0 L = 0.
projection about
=
a(L,)
by U, = ecx and E, is defined
E, is the unique
The following
= D(L) II x,
a(L,)
L, follow
from standard
analytic
semi-group
3.2. (i) L, is also a sectorial operator, which generates (ii) V 6 E (0, c2/4), [(ezLZ(Iz(XZ) 5 Cst e-“, V t > 0; (iii) erL2 has the representation
an analytic
theory
(cf. e.g.
]6, 91). COROLLARY
etL2 = &
semi-group
erL2;
(AZ - L,)-‘e”dl, Ir
where I is any contour We now return
which lies strictly
arg P, -+ fB, B E (7r/2, n), as II( + 00.
to (3. l), to which we apply the operators v” = E,v”
yields,
in p(L,),
+ E2vo =p’U,
E, and E2. The splitting + w”
of (3.11)
since E, v, = E, v,, = 0 pp = zp (3.12) wp + cwp - w,9; = 0.
In order to derive the boundary
conditions
associated
to w,, we calculate
wO(0, t) = vO(0, t) - pO(t)U,(O)
= -PO(i)
w,“(O, t) = v,o(O,t) - pO(t)U,,(O)
= -cpO(t)
(3.13) hence p” may be eliminated
through
(3.13) w,“(O, t) = cwO(0, t).
Summing
up, we find that w” verifies
the following
wp = L2 w”, whose solution namely
problem
w’(O) = E,u,
is w’(t) = efLZE2uo from corollary
(3.14) in the space X2 E X2
3.2. Next, p” is explicitly
pO(t) = -wO(O, t).
(3.15) given by (3.13), (3.16)
888
After
C.
(3.12),
p” and
z” differ
M.
et al.
BRAUNER
up to a constant
only.
For p’(O)U, = EluO and z’(O) = 0
(see (2.4)) 0 zO(t> =
p(t) - c
ug dx.
(3.17)
s --m
Remark 3.1. The technique
used in (3.11) enables us to fully decouple (3. l), and reduces the problem to a P.D.E. (3.15) in a space of co-dimension 1, plus a trivial relation (3.16) in one dimension. Therefore, we refer to it as a Lyapunov-Schmidt-like procedure. We will use it again in the nonlinear analysis (Section 4). The next step is to examine LEMMA 3.6. Whenever
the asymptotic
behaviour
0 < w < c2/4, (IwO(t)JI,,, 5 cst e-“‘,
Proof. After corollary I,U’ = wp, which solves
3.2(ii),
one has
IjL, ~‘(t)jJ,,~
It follows
from
IIvOwIl,, 1. From
3.6 and
5 cste-U’.
(3.18) The same
result
holds
for
(3.19)
l//O(O) = LUO.
I cst e-“’ which obviously
lemma (3.17)
vt>o.
)(w’(t)\l,,,
wt” = &Jo, Therefore,
of w” for f large.
yields (3.18).
(3.16) that p’(t) -+ 0 as e-“’
when
t -+ co, and so does
”0 p(t)
-+ Zm = -c
u() dx.
(3.20)
! --m To take into account this exponential decay as time approaches introduce a class of suitable Banach spaces [ 141.
Definition 3.1. Let Y be a Banach e”(]O, +a); In the definition,
infinity,
it is convenient
to
space, o > 0
Y) = ]U E Ci([O, +m); Y), w; e”‘IIIp(t)ll. < ~1.
case Y = C4,j, the weighted sup-norm will be denoted by II~[14,j,o. With lemma 3.6 simply reads: w” E c“([O, +co); C,,,), 0 < w < c2/4.
this
To conclude this section devoted to linear analysis, we attempt to present some geometrical interpretation of the problem, going back to formula (2.2), (2.3) in the moving coordinate frame y = q + ct. The wave U gives rise to a one-parameter family of waves (U’], Uh(y) = U(y - h), which can be regarded as a one-dimensional manifold. The derivative
is equal to -U,
.
889
Nonlinear stability analysis
h
Fig. 1
We look for a formal expansion u(y, t) = U(y) + e&y, t) + . . . . h(t) = s&(t) < 0 (say), --co < y < 0. Note that we are linearizing about U not a shift of it. It turns out that (0,e) verifies 0, = Lir,
cl,=,
= &
(3.21)
t(t) = -O(O, t). In (3.21) L is defined as in (3.2) and has 0 as a simple eigenvalue with eigenvector U,. The critical (or centre) manifold is precisely the curve Uh and the flow in this manifold is given by dh/dt = 0 [9, p. 1741. The solution ir of (3.21) writes (3.22)
o(t) = A0 ir + G(t),
where ti, verifies I?( = L,ti
(3.23)
G(O) = u. - A00
which is exactly (3.15) obtained in a direct but formal way. As t + a, C(t) -+ A, 0 hence the final position of 2(t) is 0 .i&
=
-A,
ir(O>
=
A,
q,(O)
=
A0
=
UO
-c
-co 4. NONLINEAR
dx.
(3.24)
ANALYSIS
We now turn to the full nonlinear problem (2.8) with E > 0. We apply the same Lyapunov-Schmidt-like procedure as in Section 3. However, the speed of the front may not be completely eliminated. We see that vE=E1vE+E2vE=pEUX+
wE
(4.1)
ug d_x
(4.2)
0
P: = zf,
p&(O) = -c 1 -ca
w,” = L, WE + Epfv,,
p”(t) = -W&(0, t) (4.3)
w”(O) = E2uo.
890
BRAUNER et al.
C. M.
In order to deal with (4.3), we need some insight on abstract Cauchy problems spaces and Holder regularity of solutions. First we recall the following definition. Definition
4.1. Let Y be a Banach Ckq[o,
+a);
in Banach
space, k E N, 0 < 6’ < 1.
Y) = (rp E Ck([O, $00); Y), (y?‘ky)
< 031,
where
(P) (0)
IIv(t + h) - cm0Y
sup
=
IV
t,t+h>o
Ihi iho
This space is equipped
*
with the norm lllVlllk,e = lIG&k,~(0,+m),Y) + (V)@).
We also need the space C$‘([O, +m); Y) of functions of C’,‘([O, +m); Y) such that p(O) = 0. Next, we define the operator K: f r--) u where u is the solution of the auxiliary Cauchy problem u, = L,u Operator
K has two kinds of important
f
u(0)= 0.
f,
properties
(4.4)
which are collected
into two lemmas.
LEMMA 4.1. For any o, 0 < o < c2/4, the operator K is a bounded transformation from ?([O, +a); X,) into ?([O, +a~); E,C,,,). This result follows from corollary 3.1 and the representation (3.10) of the semi-group efL2. Since the proof corresponds to those of lemmas 3.4 and 3.5 of Sattinger [14] (see also [l l]), we shall omit it. LEMMA 4.2. The operator K is a bounded transformation from C$‘([O, C’,‘([O, too); X,) n C”,‘([O, +a); D(L,)). Moreover, u,(O) = 0. This lemma is based on corollary 3.2 and the following abstract result. THEOREM 4.1. Let Y be a Banach e ‘A such that
space, A the infinitesimal
ItetAIIs(y) s Me-“‘,
Let u be the solution
of the Cauchy
generator
fco); X,)
of an analytic
into
semi-group
M and 6 > 0.
(4.5)
problem
24, = Au +
f,
u(0)= 0.
(4.6)
Then the mapping f ++u is a bounded transformation from C$‘([O, 400); Y) into C’,O([O, +oo); Y). This theorem is a by-product of abstract results for evolution equations in Banach spaces (cf. Sinestrari [16, p. 531, Pazy [12, p. 478]), extended to the case T = +o3 under the additional hypothesis (4.5)-see [ 111. We are now in position
to reformulate
(4.2), (4.3) with the help of the operator
F(w”; E) = wE - ~K[p;(cp”U,
K
+ w;)] - w” = 0 (4.7)
p”(t) = -W&(0, t).
Nonlinear
stability
analysis
891
For 0 < o < cz/4 and 0 < B < 1, we define the functional space w = (W E P([o,
+oo); x,)
n c”~o([o, +w); D(L,)), ~~(0, 0) = 0)
n mo, +4; w,,,)
(4.8)
equipped with the natural norm. THEOREM 4.2. w” E W and the mapping 5 defined by (4.7) is a Frtchet differentiable from W x R+ into W.
mapping
Proof. (i) w” is smooth because of our assumptions on uo, and we already know w” E P’([O, +m); E,C,,,), therefore w” E W. (ii) Let us check that if w E W, S(w; E) E W. Following lemmas 4.1 and 4.2, it is sufficient to show that the quantity in brackets belongs to P([O, +w); X,) fl Ci([O, +oo); X,) and is zero at t = 0. Let f = p,(cpU, + w,), p(t) = - ~(0, t) IIf(t)ll,,o
llw*wll,,o) 5 Ill411, &II Noll,,o + IIwxwll,,A,
5 IPrl * (clp(t)l
+
where the triple-bar is the norm of C’,‘([O, +oo); X). Therefore
Ilf IIp,o,o 5 4llwlll1,e * l14q,l,w~ sof belongs to sW([O,+a); X,). On the other hand, w, E PO([O, +co); X,)
f E Ci([O, +a); X2) for p E c’sB([o, +a)).
and
Finally f(0) = 0 since p,(O) = - w,(O, 0) = 0. (iii) Let us show that 5 is C’. The only point to be checked is that 5w is continuous. calculate, for a, E W S,(w; 8) * v = 9 - &Kb, VI = -%a 472 = PA-d09
We
+ vl21
*)(cPUx + wx)
(4.9)
*)v, + v,).
It is not difficult to verify that v, c p, and o, u IJI~are continuous
from W into
em([O,+a); x2) n C,B([o, +m); x2), the arguments being similar to the ones of (ii); thus lemmas 4.1 and 4.2 apply. We may thus apply the implicit function theorem to the mapping 5. To this end we compute the Frechet derivative of 5 at (w’, 0), which clearly is the identity. Therefore, by the implicit function theorem we have the following result, where o is any fixed number in the interval (0, c2/4). THEOREM 4.3. There exists a C’ mapping E c~ wE from some interval [0, so) into W such that S(w”; E) = 0. Therefore problem (4.3) admits a unique solution (wE,pE) for E E [0, so) such that, as t -+ +a, 1)w”(t))),, 1 5 cst e-“‘.
For p”(t) = -w&(0, t), Ip”(t)l I cst e-“’ also.
(4.10)
C. M. BRAUNER et al
892
Going back to U’ and z&--See (4.1), COROLLARY4.1. Whenever asf-++oo
(4.2)-we
E E [0, ao), problem
have the following
corollary.
(2.8) admits a unique solution
[z”(t) - z-1 5 csfepw’,
z, = -c
(u’, z”) such that
(4.11)
Ou. dx.
I
--m
Finally we have the following question
raised in Section
theorem
which provides
a complete
answer to the stability
1.
THEOREM 4.4. Define h, = h5 = --EC j!_ u. dx. Whenever unique solution (u, h) = (u’, h”) such that, as t -+ +co
E E [0, Q), problem
(2.6) admits a
I(u”(t) - U((,,, 5 cstepO’
(4.12)
IF(t) - h:l 5 cste-w’. Acknowledgement-The collaboration between C.M.B. and C.S.L. was made MPB-SPI Instabilitts des flammes laminaires de premelange 8720 N5/25).
possible
by a CNRS
grant
(ATP
REFERENCES
ALABAU F. (to appear).
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
BERTSCHM., HILHORSTD. & SCHMIDT-LAIN~CL., The well-posedness of a free boundary problem arising in combustion theory (to appear). BRAUNERC. M., NOOREBAD S. & SCHMIDT-LAIN~~ CL., Sur la stabilite d’ondes singulieres en combustion, C. r. Acad. Sci. Paris 308, 159-162 (1989). LUNARDIA. & SCHMIDT-LAIN~ CL., Une nouvelle formulation de modeles de fronts en problemes totalement non lineaires, C.r. Acad. SC;. Paris 311, 597-602 (1990). Stabilite et instabilite des ondes stationnaires d’un probleme de detonation a deux phases, C.r. Acad. Sci. Paris 311, 697-700 (1990). BUCKMASTER J.D. & LUDFORDG. S. S., Lectures on mathematical combustion, CBMS-NSF Conference series Applied Mathematics, (1983), p. 43. SIAM. DUNFORDN. & SCHWARTZJ. T., Linear Operators I. Interscience, New York (1957). FIFE P. & MCLEOD J. B.. The approach of solutions of nonlinear diffusion equations to traveling front solutions, Archs ration. Mech. Analysis 65, 335-361 (1977). VAN HARTENA. & MATKOWSKY B. J.. A new model in flames theory. SIAM J. apol. Math. 42, 850-867 (1982). HENRY D., Geometric theory of semilinear parabolic equations, L&tare Notes-b Mathematics 840. Springer, New York (1981). HILHORSTD. & HULSHOFJ., An elliptic-parabolic problem in combustion theory: convergence to traveling waves (to appear). NOOR EBAD S., Aspects theoriques et numeriques de stabilite d’ondes dans des modeles de combustion, Ph.D. thesis, Lyon (1989). PAZY A., Semi-groups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44. Springer, Berlin (1983). ROYTBURDV., A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics, J. difs. Eqns 56, 40-62 (1985). SATTINCERD. H., On the stability of waves of nonlinear parabolic systems, Adv. Math. 22, 312-355 (1976). SATTINGERD. H.. Weiehted norms for the stability of traveling waves, J. diff. Eans 25, 130-144 (1977). SINESTRARIE., On the-abstract Cauchy problem of paraboliciype in spaces of continuous functions, J. math. Analysis Applic. 107, 16-66 (1985). STEWARTD. S. & LUDFORDG. S. S., Evolution of near Chapman-Jouguet deflagrations, Trans. 28th Conf of Army Math., AR0 Report 83-1, pp. 133-142 (1983). STEWARTD. S., KAPILAA. K. & LUDFORDG. S. S., Deflagrations and detonations for small heat release, J. Me%. ThPor. Appl. 3. 105-115 (1984). STEWARTD. S., Transition to detonation on a model problem, J. M&. ThPor. Appl. 4, 103-137 (1985).