J. Math. Study
doi: 10.4208/jms.v49n2.16.01
Vol. 49, No. 2, pp. 93-110
June 2016
Kuramoto-Sivashinsky Equation and Free-interface
Models in Combustion Theory
Claude-Michel Brauner∗
School of Mathematical Sciences and Fujian Provincial Key Laboratory
on Mathematical Modeling & High Performance Scientific Computing,
Xiamen University, Xiamen 361005, Fujian, P.R. China
Institut de Mathématiques de Bordeaux, Université de Bordeaux,
33405 Talence Cedex, France.
Received 31 March, 2016; Accepted 15 May, 2016
Abstract. In combustion theory, a thin flame zone is usually replaced by a free interface. A very challenging problem is the derivation of a self-consistent equation for the
flame front which yields a reduction of the dimensionality of the system. A paradigm
is the Kuramoto-Sivashinsky (K–S) equation, which models cellular instabilities and
turbulence phenomena. In this survey paper, we browse through a series of models in
which one reaches a fully nonlinear parabolic equation for the free interface, involving pseudo-differential operators. The K–S equation appears to be asymptotically the
lowest order of approximation near the threshold of stability.
AMS subject classifications: 35K55, 35R35, 35B35, 35B40, 80A25
Key words: Free interface, combustion theory, Kuramoto-Sivashinsky equation, instability, fully
nonlinear parabolic equation.
1 Introduction
Interface phenomena are commonplace in physics, chemistry, biology, and various disciplines bridging these fields (see Fife [18]), such as combustion and flame. The latter domain constitutes an intricate physical system involving fluid dynamics, multistep chemical kinetics, and molecular and radiative heat transfer. In the middle of the 20th century,
the Russian School [23] introduced formal asymptotic methods based on large activation
energy which have allowed simpler descriptions, especially when a thin flame zone is replaced by a free interface, commonly called the flame front. A very challenging problem
is the derivation of a single equation for the free interface, which may capture most of the
∗ Corresponding author.
Email address: [email protected] (C.-M. Brauner)
http://www.global-sci.org/jms
93
c
2016
Global-Science Press
94
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
dynamics and, as a consequence, yields a reduction of the effective dimensionality of the
system.
In premixed gas combustion, thermal-diffusive instability is a result of the competition between the exothermic reaction and the heat diffusion, which in turn exhibits
chaotic dynamics. Near the instability threshold it is possible to (asymptotically) separate the spatial and temporal coordinates, and further reduce the system to a single
geometrically invariant surface dynamics equation (see Frankel and Sivashinsky [22]):
Vn = 1 +(α − 1)κ + κss ,
(1.1)
where Vn is the normal velocity of the flame sheet, s is the arc-length along the interface,
and κ is its curvature. The parameter, α, reflects the physico-chemical characteristics of
the combustible; cellular instability occurs when α exceeds unity.
The coordinate-free model (1.1), especially its weakly-nonlinear approximation, the
Kuramoto-Sivashinsky (K–S) equation:
1
2
Φτ + νΦηηηη + Φηη + (Φη ) = 0, ν > 0,
(1.2)
2
appears in a variety of domains in physics and chemistry which include free interfaces.
As it models cellular instabilities (see Sivashinsky [33]), pattern formation, turbulence
phenomena (see Kuramoto [27, 28] who independently derived K–S in a study of turbulence in the Belousov-Zhabotinsky reaction), and transition to chaos (see Hyman and
Nicolaenko [25]), the K–S equation has received considerable attention from the mathematical community (see Temam [34] and the references therein). Several authors have
restricted their attention to a differentiated version of (1.2) which includes a nonlinearity
of Burgers type.
The K–S model comprises a balance between several effects: loosely speaking, this
equation arises when the competing effects of a destabilizing linear part νΦηηηη + Φηη
and a stabilizing nonlinearity 12 Φ2η are the dominant processes. The linear instability is
2 (stabilizing) and D
itself the result of a competition between two linear operators, νDηη
ηη
(destabilizing). Comparable competition holds in other dissipative systems with similar
dynamics, such as the Burgers-Sivashinsky equation (see Berestycki et al. [1]) and the Q–S
equation (see Section 2.2).
In this survey, we browse through a series of free-interface problems in combustion
theory (also a model for supercooling). Our viewpoint is twofold:
(i) First, after a number of simplifications, derive a self-consistent equation for the interface (more precisely for the corrugated perturbation of a planar front) whose general
form on an interval (−ℓ/2, ℓ/2) with periodic boundary conditions reads:
∂
B ϕ = S ( ϕ)+ F (( ϕy )2 ),
∂t
or, inverting operator B whenever it is possible:
ϕt = L ( ϕ)+ G (( ϕy )2 ).
(1.3)
(1.4)
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
95
The most general situation is that (1.3) and (1.4) are fully nonlinear parabolic equations (see
Lunardi [31]) with pseudo-differential operators (or Fourier multipliers), in contrast to
K–S which is a semilinear equation. An issue is the stability of the one-dimensional Traveling Wave (TW) solution with respect to two-dimensional disturbances and the computation of a critical value, αc , of parameter α as in (1.1), corresponding to the threshold of
stability.
(ii) Second, in rescaled dependent and independent variables, recover asymptotically K–
S as a limit of (1.3) when a small parameter ε, frequently defined as ε = α − 1, tends to 0.
In other words, the equation K–S corresponds to the lowest order of approximation near
the stability threshold. The method in Section 6 delves into the core of the model.
The paper is organized as follows: In Section 2, we consider one-dimensional models,
the κ − θ system equivalent to a ultra-parabolic equation, and its quasi-steady version,
the Q–S equation. Next, we focus on two-dimensional models in a strip of R2 . In Section
3, we discuss the Near-Equidiffusional Flames (NEF) theory that has played an important role in the study of instabilities responsible for the formation of non-steady cellular
structure described by free-interface problems. Turing (see [35]) already observed that
spatially inhomogeneous patterns evolve by diffusion driven instability when equidiffusion does not hold. Next, in Section 4, we consider a model in gas-solid combustion
which also can be interpreted in terms of diffusive instability. We observe in Section
5 that away from the stability threshold, the structure of the front equation may be far
more involved. In this respect we have introduced a generalization of the K–S equation
as a damped wave equation ([11]):
1
4( ϕtt + ϕyyyy )+( I − 8Dyy ) ϕt +(α − 1) ϕyy + ( ϕy )2 = 0.
2
(1.5)
Finally, we devote Section 6 to a model aside from combustion theory: we consider a
solid-liquid free-interface model, with a solidification front. The main issue is that the
front equation is retrieved from the two-dimensional model as a natural solvability condition. Eventually, the equation K–S, with ν = 3, is recovered asymptotically.
2 The κ-θ model
Equation (1.1) is not a unique low-dimensional model generating cellular instability. The
geometrically-invariant κ-θ model enjoys similar properties (see Frankel et al. [20]):
Vn = 1 + κ + θ,
Dt θ = θss − ακ − θ,
(2.1)
where Dt θ is the Lagrangian time derivative of the (reduced) interface temperature θ
along the “flow” generated by the normal velocity field.
96
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
2.1 Weakly nonlinear κ-θ model [3]
For a mildly distorted planar flame, propagating along the x-axis, the flame interface
may be described by an explicit function of the transversal coordinate y, namely x =
−t + ϕ(y,t). As a result one obtains a weakly nonlinear version of the κ-θ model:
1
ϕt + ϕ2y = ϕyy − θ,
2
θt + ϕy θy = θyy + αϕyy − θ,
(2.2)
with periodic boundary conditions at y = ±ℓ/2. Problem (2.2) is a parabolic system with
smooth nonlinearities. Existence in the large can be proved thanks to the particular structure of the system: differentiating the ϕ-equation with respect to y, the coefficients of ϕyy
and θy are the same (equal to ϕy ) in both equations, see [3, Section 2].
It turns out that the κ-θ system is uniformly closed to the K–S equation when α = 1 + ε,
where ε is a small positive parameter. The trick in [3] is to reduce the κ − θ system to
a scalar equation by expressing θ = ϕyy − ϕt − 12 ϕ2y from the first equation in (2.2), and
substituting it into the second equation, to obtain an equivalent ultra-parabolic equation:
h
1
1 i
(2.3)
ϕt + ϕtt + ϕyyyy +(α − 1) ϕyy − 2ϕyyt + [ I − ∂2y + ∂t ] ϕ2y = ϕy ϕyy − ϕt − ϕ2y .
2
2
y
√
The rescaling ϕ = εψ, t = τ/ε2 , y = η/ ε yields after division by ε3 a perturbed equation:
h
i
1
1
ψτ + ε2 ψττ + ψηηηη + ψηη − 2εψηητ + [1 − ε∂2η + ε2 ∂τ ]ψη2 = εψη ψηη − εψτ − εψη2 ,
2
2
η
(2.4)
whose formal limit is clearly:
1
Φτ + Φηηηη + Φηη + (Φη )2 = 0,
2
namely the K–S equation in the rescaled coordinates with ν = 1. The main result (see [3,
Theorem 3.2]) is the existence of a constant C independent of 0 < ε ≤ ε 0 , such that:
√
max | ϕ(t,y)− εΦ(y ε,tε2 )| ≤ Cε2 ,
√
for |y| ≤ ℓ0 /2 ε, 0 ≤ t ≤ T/ε2 . Here ℓ0 and T are fixed; moreover, the initial data must be
chosen accordingly ([3, (23)]).
2.2 The Q–S equation [5]
It has been observed that, not far from the instability threshold, the time derivative in the
second equation of the κ-θ model has a relatively small effect on the solution. Based on
this observation, we consider a quasi-steady κ-θ model as follows (see [5]):
Vn = 1 + κ + θ,
θss − ακ − θ = 0,
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
97
and its weakly nonlinear version:
1
ϕt + ϕ2y = ϕyy − θ,
2
θyy + αϕyy − θ = 0.
(2.5)
The inversion θ = α( I − ∂2y )−1 ϕyy is well defined on any reasonable function space to be
considered for this problem. Therefore, we may re-write the system (2.5) as a single
nonlocal equation:
1
ϕt − ϕyy + α( I − ∂2y )−1 ϕyy + ϕ2y = 0,
(2.6)
2
called the Q–S equation (for quasi-steady), together with natural periodic boundary conditions at y = ±ℓ/2.
A basic feature of (2.6) is that there is a competition between the operators − Dyy
(stabilizing) and α( I − Dyy )−1 Dyy (destabilizing). This situation is reminiscent of the K–S
equation; however, the main benefit of Q–S over K–S is that the destabilizing operator is
bounded, albeit nonlocal.
Eq. (2.6) can be re-written as a fourth-order equation (see [2, Theorem 2.1]):
( I − ∂2y )
1 2
ϕt + ϕy + ϕyyyy +(α − 1) ϕyy = 0,
2
(2.7)
which reads after the same rescaling as in Section 2.1:
( I − ε∂2η )
1 2
ψτ + ψη + ψηηηη + ψηη = 0,
2
(2.8)
hence, the convergence to K–S, see [2, Section 3].
From a more physical viewpoint, all three of these models, K–S, κ − θ and Q–S share
the same basic quality revealed by linear stability analysis, namely long-wave destabilization, which is suppressed by the dominant dissipative principal term for small wave
lengths (see the discussion in [3] and Vukadinovic [36]).
Although the Q–S equation was introduced as an ad hoc truncation of the full κ − θ
model (2.2), it represents an interesting dynamical system in its own right. Its dynamics
is essentially finite-dimensional: Q–S possesses a universal absorbing set and a compact attractor; furthermore, the attractor is of a finite Hausdorff dimension (see [4] and
Vukadinovic [36]).
3 Near-Equidiffusional Flames (NEF)
3.1 The thermal-diffusional model
A paradigm in premixed flame combustion is the two-dimensional thermal-diffusional
model, a simplified model that involves only two equations: the heat equation for the
98
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
system’s temperature and the diffusion equation for the deficient reactant’s concentration
(see, e.g., Buckmaster and Ludford [13]):
∂Y
= Le−1 ∆Y − ω (Y,T ).
∂t
∂T
= ∆T + ω (Y,T ),
∂t
(3.1)
The parameter Le is the Lewis number, corresponding to the ratio of thermal and molecular diffusivities. The scaled reaction rate ω (Y,T ) is as usual given by the Arrhenius
law
ω = BYexp(− E/RT ),
(3.2)
where E is the activation energy and R is the gas constant. A measure of the activation
energy is the Zel’dovich number, β.
Due to the distributed nature of the reaction rate, ω, it is still difficult to theoretically
explore the system (3.1), (3.2). 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 free-interface, x = ξ (t,y), the flame front (see [13, p. 218]).
3.2 Derivation of the NEF model [32]
The laminar flames of low-Lewis-number premixtures are known to display thermaldiffusive instability responsible for the formation of non-steady cellular structure. The
NEF model (see Matkowsky and Sivashinsky [32]) combines the limit of large activation
energy, β ≫ 1, with the limit of Lewis number near unity. The NEF theory is characterized
(see Buckmaster and Ludford [14]) by the requirements: (i) Le−1 = 1− β−1 l, where l = O(1)
is the reduced Lewis number; and (ii) H = H f + O( β−1 ), where H is the enthalpy, Y + T,
and H f is its value at x =−∞. The second requirement corresponds to using the following
expansions for T and Y:
T =T0 + β−1 T1 + β−2 T2 + ...,
Y =( H f − T0 )+ β−1 ( H1 − T1 )+ β−2 ( H2 − T2 )+ ... .
Within this framework, the thermal-diffusional model (3.1), (3.2) gives raise to a freeinterface problem for T0 and H1 . According to the notation of [33], T0 is replaced by
θ, H1 by 2S, and −l/2 by α which reflects the physico-chemical characteristics of the
combustible as in (1.1). Here we consider only the case where α is positive, i.e., the case
of high mobility of the deficient reactant.
The NEF system for θ, S and the flame front x = ξ (t,y) reads:
∂θ
=∆θ, x < ξ (t,y),
∂t
∂S
=∆S − α∆θ,
∂t
θ = 1, x ≥ ξ (t,y),
x 6= ξ (t,y).
(3.3)
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
99
At the front, θ and S are continuous, while the following jump conditions occur for the
normal derivatives:
∂S
∂θ
∂θ
= − exp(S),
=α
.
(3.4)
∂n
∂n
∂n
When x → ±∞, θ (t, −∞,y) = S(t, −∞,y) = S(t, +∞,y) = 0; however, polynomial growth
as x → +∞ is allowed in the literature. If the domain is a strip, R ×(−ℓ/2, ℓ/2), then
periodic boundary conditions are usually assumed at y =±ℓ/2. This system has a planar
travelling wave (TW) solution, with velocity −1, which reads in the coordinate z = x + t:
θ = ez , S = αzez , z ≤ 0,
θ = 1, S = 0, z ≥ 0.
(3.5)
(See Marion and Ducrot [16] for two-dimensional TWs.) It is standard to fix the interface at the origin by setting ξ (t,y) = −t + ϕ(t,y), x′ = x − ξ (t,y) = z − ϕ(t,y). In this new
framework:
θt +(1 − ϕt )θx ′ = ∆ ϕ θ, x′ < 0,
θ ( x′ ) = 1, x′ > 0,
St +(1 − ϕt )Sx ′ = ∆ ϕ S − α∆ ϕ θ, x′ 6= 0,
(3.6)
where ∆ ϕ = (1 +( ϕy )2 ) Dx ′ x ′ + Dyy − ϕyy Dx ′ − 2ϕy Dx ′ y . The first condition in (3.4) reads:
q
∂θ
2
1 +( ϕy )
= − exp(S).
∂x′
In view of the stability analysis of the TW, it is usual to introduce the perturbations
θ = θ + u, S = S + v of the planar TW, and consider the system for (u,v, ϕ). By stability we
mean orbital stability with asymptotic phase. Instability in the space R2 has been studied
in [12], i.e. when the parameter α exceeds the critical value, αc = 1, at which the threshold
of stability occurs. Lorenzi [29, 30] studied the stability in the strip R ×(−ℓ/2, ℓ/2) with
different boundary conditions at y = ±ℓ/2; in any case, the critical value, αc , tends to 1
as ℓ → +∞. The method used is quite general (see [10] for an abstract presentation) and
allows the elimination of the front ϕ. It leads to a class of fully nonlinear problems, which
can be studied by the techniques of Lunardi [31].
3.3 Formal derivation of the K–S equation [33]
In the pioneering paper [33], Sivashinsky performed a formal asymptotic analysis near
the threshold of stability, αc = 1. Setting α = 1 + ε, he defined a convenient set of rescaled
dependent and independent variables based on physical considerations, which provides
a distinguished limit as ε → 0:
√
t = τ/ε2 , y = η/ ε, ϕ = εψ, u = ε2 ũ, v = ε2 ṽ.
(3.7)
(For a discussion about relevant powers of ε, see van den Berg et al. [2] in the radiative
case.) Then, Sivashinsky sought a formal ansatz ( [33, p. 75]):
ũ = ũ0 + εũ1 + ...,
ṽ = ṽ0 + εṽ1 + ...,
ψ = ψ0 + εψ1 + ....
100
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
However, at the zeroth order, it is not possible to “close” the nonlinear system for the
triplet (ũ0 , ṽ0 ,ψ0 ). This situation is quite common in singular perturbation theory when
the zeroth order can not be fully determined (see, e.g., Eckhaus [17]). In such a case,
one must go to the first order, which is indeed linear. Most often, the latter demands a
solvability condition, for example based on the Fredholm alternative, which provides the
missing relation for the zeroth order. Here, the solvability condition for the linear system
for (ũ1 , ṽ1 ,ψ1 ) reads:
1
0
0
ψτ0 + 4ψηηηη
+ ψηη
+ (ψη0 )2 = 0,
2
i.e., ψ0 verifies the K–S equation in the rescaled coordinates with ν = 4.
3.4 A quasi-steady version of the NEF model [9]
Near 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 moving front. As in Section 2.2, one defines a quasi-steady NEF
model [9], writing (for simplicity) x instead of x′ = x − ξ (t,y):
(1 − ϕt )θx = ∆ ϕ θ, x < 0, θ ( x) = 1, x > 0,
(1 − ϕt )Sx = ∆ ϕ S − α∆ ϕ θ, x 6= 0.
(3.8)
Then, the problem for the perturbation (u,v, ϕ) of the planar TW reads:
(1 − ϕt )u x − ∆ ϕ u − ϕt θ x = (∆ ϕ − ∆)θ,
u = 0, x > 0,
x < 0,
(1 − ϕt )vx − ∆ ϕ (v − αu)− ϕt S x = (∆ ϕ − ∆)(S − αθ ),
x 6= 0,
where (∆ ϕ − ∆)(θ ) = (( ϕy )2 − ϕyy )θ x and (∆ ϕ − ∆)(S − αθ ) = α(( ϕy )2 S x − ϕyy S). We retain
only linear and second-order terms for the perturbation of the front, ϕ, and first-order
terms for the perturbations of temperature, u, and enthalpy, v. The skipped terms contribute to higher order perturbations only. After some computations, this leads to the
equations:
u x − ∆u − ϕt θ x = ( ϕy )2 − ϕyy θ x ,
2
x < 0,
vx − ∆(v − αu)− ϕt S x = ( ϕy ) S x − ϕyy S,
x 6= 0,
(3.9)
with the interface conditions at x = 0:
u(0) = [v] = 0,
1
v(0)− u x (0) = ( ϕy )2 ,
2
[vx ] = −αu x (0).
(3.10)
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
101
System (3.9)-(3.10) is simple enough to be integrated explicitly via a discrete Fourier
transform in the variable y; therefore, it allows a separation of the dependent variables.
One gets to a self-consistent pseudo-differential equation for the front, ϕ, which reads:
( Xk2 + αXk − α) ϕbt (t,k) = (−4λ2k +(α − 1)λk ) ϕb(t,k)
1
+ ( Xk3 − 3Xk2 − 4αXk + 4α)([
ϕy )2 (t,k), k = 0,1,... ,
4
(3.11)
eigenvalues of the operator Dyy with
where the −λk ’s are the non-positive √
pperiodic boundary conditions at y = ±ℓ/2 and Xk = 1 + 4λk is the symbol of operator 1 − 4Dyy .
One can rewrite (3.11) as a fourth-order equation on the interval (−ℓ/2, ℓ/2):
∂
B ϕ = S ( ϕ)+ F ( ϕy )2 ,
∂t
(3.12)
where S is a fourth-order differential operator with H♯4 (the usual Sobolev space consisting of ℓ-periodic functions) as a domain:
S ( ϕ) = − ϕyyyy −(α − 1) ϕyy .
Operators B and F are pseudo-differential (or Fourier multipliers) with symbols, respectively,
bk = Xk2 + αXk − α,
1
f k = ( Xk3 − 3Xk2 − 4αXk + 4α).
4
Therefore,
B = I − 4Dyy + α
q
I − 4Dyy − I ,
q
3
1
3
1 − 4Dyy − I .
F = ( I − 4Dyy ) 2 − ( I − 4Dyy )− α
4
4
We point out that the nonlinear term in (3.12) is also of the fourth-order, therefore it is
a fully nonlinear parabolic equation, in contrast to K–S which is a semilinear parabolic
equation. If we invert operator B, then (3.12) is written as a second-order fully nonlinear
parabolic equation.
It is easy to compute the critical value αc = 1 + 16π 2 /ℓ2 at which the threshold of
stability holds, αc → 1 as ℓ → ∞ (see [9, Theorem 2.1]). The main issue is the link between
(3.12) and K–S. Setting α =1+ ε and performing the change of dependent and independent
variables in (3.7), we see that ψ solves the perturbed equation:
q
o
∂ n
I − 4εDηη +(1 + ε)
I − 4εDηη − 1 ψ + 4Dηηηη ψ + Dηη ψ
∂τ
q
o
3
1n
− ( I − 4εDηη ) 2 − 3( I − 4εDηη )− 4(1 + ε)
1 − 4εDηη − I ( Dη ψ)2 = 0.
4
(3.13)
102
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
Clearly, at the limit ε → 0, one retrieves the equation K–S. However, this must be demonstrated in a rigorous way. The idea is to link the small parameter, ε, and the width
√ of the
strip, which will blow up as ε → 0; for ℓ0 > 0, we take ℓ of the form, ℓε = ℓ0 / ε; hence,
αc = 1 + ε16π 2 /ℓ20 . Thus, ℓ0 becomes the new bifurcation parameter. Obviously, ℓ0 > 4π in
order to have αc ∈ (1,1 + ε), i.e., α > αc ; otherwise, the origin is stable and the dynamics is
trivial. The main result is the following (see [9, Theorem 2.2]):
Theorem 3.1. Let Φ0 ∈ H♯m be a periodic function of period ℓ0 . Further, let Φ be the periodic
solution of (1.2) (with period ℓ0 ) on a fixed time interval [0,T ], satisfying the initial condition
Φ(0, ·) = Φ0 . Then, if m is large enough, there exists ε 0 = ε 0 ( T ) ∈ (0,1) such that, for 0 < ε ≤ ε 0 ,
2
Problem
√ (3.12) admits a unique classical solution ϕ on
√ [0,T/ε ], which
√ is periodic with period
ℓ0 / ε with respect to y, and satisfies ϕ(0,y) = εΦ0 (y ε), |y| ≤ ℓ0 /2 ε.
Moreover, there exists a positive constant C such that, for any ε ∈ (0,ε 0 ],
√
| ϕ(t,y)− εΦ(tε2 ,y ε)| ≤ Cε2 ,
0≤t≤
ℓ0
T
, |y| ≤ √ .
2
ε
2 ε
In other words, starting from the same configuration, the solution of (3.12) 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 a special type, compatible
with Φ0 and (1.2) at τ = 0.
4 A gas-solid model with pattern formation [6, 24, 26]
Next, we consider a model in gas-solid combustion, proposed by Kagan and Sivashinsky [26]. This model was motivated by the experimental studies of Zik and Moses [39],
who observed a striking fingering pattern in flame spread over thin solid fuels. The phenomenon was interpreted in terms of the diffusive instability similar to that occurring in
laminar flames of low-Lewis-number premixtures (see Section 3.2). Hereafter, we use the
same notation.
The free-interface system for the scaled temperature θ, the excess enthalpy S, the prescribed flow intensity U (0 < U < 1), and the moving front x = ξ (t,y), reads:
Uθx = ∆θ, x < ξ (t,y), θ = 1, x ≥ ξ (t,y),
∂θ
+ USx = ∆S − α∆θ, x 6= ξ (t,y),
∂t
(4.1)
together with the jump conditions (3.4) at the free interface. Here α is a real number
(see [26, p. 274]). System (4.1) is rather unusual since, in contrast to the NEF system (3.2),
it is the temperature’s time derivative which appears in the enthalpy equation. There is
an additional transport equation for the solid product (char) Σ:
Σ = 0, x ≤ ξ (t,y),
Σt = 0, x > ξ (t,y).
(4.2)
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
103
System (4.1)(4.2) admits a planar solution traveling at constant velocity −V, where V =
−U lnU, which reads in the coordinate z = x + Vt:
θ = exp(Uz), z ≤ 0,
θ = 1, z ≥ 0,
S = (α − lnU )Uzexp(Uz)+(lnU ) exp(Uz), z ≤ 0,
Σ = −(lnU )−1 , z > 0.
Σ = 0, z < 0,
S = lnU, z ≥ 0,
(4.3)
We follow the lines of Section 3.4, considering a quasi-steady version of (4.1) in the coordinate x′ = z − ϕ(t,y) where ϕ(t,y) = ξ (t,y)+ Vt:
Uθx ′ = ∆ ϕ θ, x′ < 0,
θ = 1, x′ ≥ 0,
(V − ϕt )θx′ + USx′ = ∆ ϕ S − α∆ ϕ θ,
x′ 6= 0.
Omitting the prime, the perturbations θ = θ + u, S = S + v verify, retaining only linear and
second-order terms for ϕ, first-order terms for u and v:
Uu x − ∆u = (∆ ϕ − ∆)θ, x < 0,
u = 0, x > 0,
Vu x − ∆(v − αu)+ Uvx − ϕt θ x = (∆ ϕ − ∆)(S − αθ ),
x 6= 0,
with (∆ ϕ − ∆)θ = (U ( ϕy )2 − ϕyy )U exp(Ux) and (∆ ϕ − ∆)(S − αθ ) = ( ϕy )2 (α − lnU )U 2 (1 +
Ux) exp(Ux)− ϕyy (α − lnU )U 2 xexp(Ux) if x < 0, 0 otherwise. The interface conditions at
x = 0 reads:
u(0) = [v] = 0,
1
Uv(0)− u x (0− ) = ( ϕy )2 U,
2
[vx ] = −αu x (0− ).
(4.4)
It turns out that the equation for the front (in Fourier coordinates) reads:
1
1
( Xk U ) ϕbt (t,k) = (U 2 − Xk2 )( Xk2 − γU 2 ) ϕb(t,k)+ ( Xk3 − 3UXk2 − 4γU 2 Xk + 4γU 3 )([
ϕy )2 (t,k)
4
4
1
=(−4λ2k +(γ − 1)U 2 λk ) ϕb(t,k)+ ( Xk3 − 3UXk2 − 4γU 2 Xk + 4γU 3 )([
ϕy )2 (t,k),
4
p
where Xk = U 2 + 4λk , k = 0,1,.... Defining the pseudo-differential operators B,S and
F through their symbols, respectively,
bk = Xk U,
sk = −4λ2k +(γ − 1)λk U 2 ,
1
f k = ( Xk3 − 3UXk2 − 4γU 2 Xk + 4γU 3 ),
4
it comes
1
B = U (U 2 I − 4Dyy ) 2 ,
q
3
1 2
3
2
2
2
2
F = (U I − 4Dyy ) − U (U I − 4Dyy )− γU
U I − 4Dyy − U ,
4
4
104
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
while the realization of S in L2 is the operator −4Dyyyy −(γ − 1)U 2 Dyy .
Then, the front ϕ solves the fourth-order, fully nonlinear equation, on the interval
(−ℓ/2, ℓ/2):
d
B ( ϕ) = S ( ϕ)+ F (( ϕy )2 ).
(4.5)
dt
If we invert operator B, one now obtains a third-order fully nonlinear equation (see [6,
Section 3]).
We define a small perturbation parameter ε > 0 by α = 1 + lnU + ε, and rescale the
variables in the spirit of (3.7):
√
t = τ/ε2 U 2 , y = η/ εU, ϕ = (ε/U )ψ.
√
Taking ℓ = ℓ0 / εU, ℓ0 > 4π, one links ε and the period ℓ, which blows up as ε → 0. After
division by ε3 and U 3 it comes:
∂ q
I − 4εDηη ψ + 4Dηηηη ψ + Dηη ψ
∂τ
q
o
3
1n
− ( I − 4εDηη ) 2 − 3( I − 4εDηη )− 4(1 + ε)
I − 4εDηη − I ( Dη ψ)2 = 0,
4
(4.6)
which is clearly a perturbation of the K–S equation. The analogue of Theorem 3.1 can be
proved (see [6] for details).
Regarding the char, we refer to [24, Part II]. Letting Σ = Σ + σ be the perturbation of
the solid product, the value at the fixed interface is rescaled as σ(t,0,y) = ε2 U σ̃(τ,0,η ). It
turns out that
ψτ
1
2
+ (ψη ) + o(ε),
(4.7)
V σ̃ (τ,0,η ) = ṽ(τ,0,η )+ ε −
lnU 2
where v = ε2 ṽ is the perturbation of the enthalpy, which verifies:
o
q
1n
( I − 4εDηη )
I − 4εDηη + I ṽ(0)
2
q
o
∂ q
1n
=ε
I − 4εDηη ψ + γDηη ψ + ε 4(1 + ε)
I − 4εDηη − I ( Dη ψ)2 .
∂τ
4
(4.8)
Formulae (4.7) and (4.8) are convenient for numerical computations of the char, see [24,
Section 8]. A rich dynamics appears for large time t and spatial coordinate y, where a
chaotic fingering evolution is observed.
5 A strongly damped wave equation [11]
When α − 1 is positive but not necessarily small; namely, away from the stability threshold, the structure of the front equation is far more involved, as already reflected by the
ultra-parabolic equation (2.3) that is equivalent to the weakly nonlinear κ − θ model in
Section 2.1.
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
105
Moreover, a full derivation in Fourier-Laplace variables of the front equation in the
NEF model shows that (3.12) becomes a wave equation, with a linear part whose leading
term reads (see [11, Appendix]):
4( ϕtt + ϕyyyy )+( I − 8Dyy ) ϕt +(α − 1) ϕyy .
A paradigm model is a nonlinear wave equation with a strongly damping operator
I − 8Dyy acting on ϕt :
1
4( ϕtt + ϕyyyy )+( I − 8Dyy ) ϕt +(α − 1) ϕyy + ( ϕy )2 = 0,
2
(5.1)
on a strip R ×[−ℓ/2, ℓ/2] with periodic boundary conditions and initial conditions ( ϕ0 , ϕ1 )
for ϕ and ϕt respectively.
Whether or not the null solution is stable is the first question regarding (5.1); we expect a threshold of stability at the same critical value αc = 1+ 16π 2 /ℓ2 . The trick is to write
(5.1) as a first-order system and study the semigroup associated with the linear operator.
In wave problems, the semigroup is a priori only strongly continuous. However, in the
case of a damped wave equation, and if the damping is strong enough, the semigroup
may be analytic (see Carvalho and Cholewa [15]). At this point, we give a quick snapshot
(see [11, Section 3] for details).
Eq. (5.1) may be rewritten in the abstract form:
1
ϕtt = Bϕt + Aϕ − ( ϕy )2 ,
8
(5.2)
where A = − Dyyyy −(α − 1) /4Dyy , and B = 2Dyy −(1/4) I is the damping operator. We
split ϕ = Πϕ +( I − Π) ϕ = r + u, where Πϕ stands for the mean value of ϕ over (−l/2, ℓ/2),
with initial data split accordingly. The system for (u,r) reads:
1
utt = But + Au − ( I − Π)((uy )2 ),
8
1
1
rtt =− rt (t)− Π((uy )2 ).
4
8
(5.3)
Clearly the equation for r is immediately solved once u is known. Hence, the core of the
analysis is the first equation in (5.3), which is written as a first order system for U:=(u,ut )
U t = A U + F ( U ),
U(0) = ( I − Π)( ϕ0 , ϕ1 ).
(5.4)
where, for u = (u,v),
0 I
A u :=
u,
A B
0
F (u) :=
.
− 18 ( I − Π)((uy )2 )
The main property of operator A with domain D (A ) = ( I − Π) H♯4 ×( I − Π) H♯2 is that it
generates an analytic semigroup. Its spectrum consists of real eigenvalues only and it
contains positive eigenvalues if and only if α > αc (see [11, Proposition 3.1]).
106
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
Although the wave model was intended in the case where α − 1 is not necessarily
small, the second question about (5.1) addresses the consistency of the model when α =
1 + ε. Clearly, ψ = ϕ/ε verifies:
1
2
(5.5)
4ε2 ψττ +( I − 8εDηη )ψτ + 4ψηηηη + ψηη + (ψη ) = 0,
2
√
in the rescaled variables τ = tε2 , η = y ε, and the latter equation approaches K–S as ε → 0
(see [11, Section 4] for a proof). Finally, we note that the perturbed equation (5.5) has
received some attention from a numerical viewpoint (see, e.g., [11, Section 4], Zhang et
al. [37, 38]).
6 A solid-liquid interface model [7, 8]
Here we consider a two-dimensional, Stefan-like, free-interface problem, which is a simplified rescaled version of a solid-liquid interface model introduced by Frankel [19, 21].
The solidification front is represented by x = ξ (t,y). The (supercooled) liquid phase is for
x < ξ (t,y); the solid phase is for x > ξ (t,y). The dynamics of heat is described by the heat
conduction equation in the strip R ×(−ℓ/2, ℓ/2),
θt = ∆θ,
x 6= ξ (t,y),
(6.1)
with periodic boundary conditions. At x = −∞, the temperature of the liquid is normalized to 0. At the front, x = ξ (t,y), there are two interface conditions: first, the balance of
energy at the interface is given by the jump:
ξ
∂θ
= Vn = q t ,
(6.2)
∂n
1 + ξ 2y
where Vn is the normal velocity; second, according to the Gibbs-Thompson law, the nonequilibrium interface temperature is defined by:
θ = 1 − γκ + r(Vn ),
(6.3)
where the melting temperature has been normalized to 1, κ is the interface curvature,
and the positive constant γ represents the solid-liquid surface tension. For simplicity it
is assumed that r − 1 is linear; hence, (6.3) becomes:
θ = 1 − γκ + 1 + Vn .
(6.4)
The system admits a one-dimensional TW solution θ (z) = ez for z ≤ 0, and θ (z) = 1 for
z > 0.
A weakly nonlinear, quasi-steady version of (6.1), (6.2), (6.4) is obtained by assuming
a slightly distorted planar front propagating along the y-axis, ξ (t,y) = −t + ϕ(t,y). Therefore, the problem is reformulated for ϕ, and the perturbation of the temperature u = θ − θ;
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
107
the curvature κ is replaced by the second-order derivative and Vn by −1 + ϕt + 12 ( ϕy )2 .
Besides, at least near the instability threshold, the time derivative in the temperature
equation is neglected, which leads to the system (see Section 3.4 and [7]):
u x − ∆u − ϕt θ x = ( ϕy )2 − ϕyy ex χ(−∞,0) = ( ϕy )2 − ϕyy θ x ,
(6.5)
with the interface conditions at x = 0:
∂u
( a) ϕ t =
−( ϕy )2 ,
∂x
1
(b) ϕt = u(0)+ γϕyy − ( ϕy )2 .
2
(6.6)
The results are as follows (see [7]):
First, there exists a γc < 1, γc → 1 as ℓ → +∞,
2ℓ2
γc = √
,
ℓ ℓ2 + 16π 2 +ℓ2 + 16π 2
such that, for γ > γc the TW is orbitally stable, and unstable for 0 < γ < γc .
Second, there is a second-order parabolic equation for the front ϕ in the interval (−ℓ/2, ℓ/2)
ϕt = L ϕ + G (( ϕy )2 ),
(6.7)
where both L√and G are pseudo-differential operators, whose symbols read (with the
notation Xk = 1 + 4λk ):
lk = − λk
γXk2 + γXk − 2
,
Xk2 + 2Xk − 1
gk = −
1 Xk2 + 3Xk − 2
,
2 Xk2 + 2Xk − 1
k = 0,1,... .
The method is based on three main steps: (i) definition of a suitable linear onedimensional operator L1 in the x variable only, with kernel; (ii) projection with respect
to the x coordinate only, the front equation appearing as a natural solvability condition; (iii)
lifting of the condition (6.6)a and a Lyapunov-Schmidt method via (6.6)b.
Let us enlighten (i). It is possible to eliminate ϕt in (6.5) thanks to (6.6)b,
1
2
u x − ∆u − u(·,0, ·)+ γϕyy − ( ϕy ) θ x = (∆ ϕ − ∆)θ,
2
or, equivalently:
1
2
u x − ∆u − u(·,0, ·)θ x =
( ϕy ) −(1 − γ) ϕyy θ x ,
2
therefore the linear operator at stake is
L1 : u 7→ Dxx u − Dx u + u(0)θ x ,
with two jump conditions, namely continuity of u and jump of u x across x = 0 (see [7,
Theorem 2.1] for the properties of the realization L1 of L1 ). The fact that operator L1 has
108
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
a kernel is the key point of the method; it induces a solvability condition which is the
mathematical counterpart of the formal condition in Section
√3.3.
Next, we introduce the usual rescaling t = τ/ε2 , y = η/ ε, u = ε2 v, ϕ = εψ. Following
the three main steps as above [8], it turns out that ψ eventually solves the fourth-order
equation:
∂
Bε ψ = Sε ψ + Fε ((ψη )2 ),
∂τ
(6.8)
13ε
ψηη + O(ε2 ),
3
Sε ψ = −3ψηηηη − ψηη + O(ε),
1
Fε ((ψη )2 ) = − ((ψη )2 )+ O(ε).
2
(6.9)
where:
Bε ψ = ψ −
The non-explicit terms in (6.9) are quite involved (see [8, 4.1.5]). Clearly, (6.8) reduces to
the equation K–S (with ν = 3) when we set ε = 0 (see [8] for a detailed proof).
Acknowledgments
The goal of this survey paper has been to present a concise overview of a long-term research project devoted to the mathematical bases of the Kuramoto-Sivashinsky equation
in combustion theory. I want to express my appreciation to my main coauthors Joost Hulshof, Luca Lorenzi, Alessandra Lunardi and Grisha Sivashinsky, whose contributions to
this project have been essential. I also wish to thank Michael Frankel (in memoriam),
Lina Hu, Victor Roytburd, Jie Shen and Chuanju Xu for their involvement.
References
[1] H. Berestycki, S. Kamin, G. I. Sivashinsky, Metastability in a flame front evolution equation,
Interf. Free Bound. 3 (2001), 361-392.
[2] J. B. van den Berg, C.-M. Brauner, J. Hulshof, A. Lunardi, The speed law for highly radiative
flames in a gaseous mixture with large activation energy, SIAM J. Appl. Math. 66 (2005),
408-432.
[3] C.-M. Brauner, M. L. Frankel, J. Hulshof, A. Lunardi, G. I. Sivashinsky, On the κ − θ model of
cellular flames: existence in the large and asymptotics, Discr. Cont. Dyn. Syst. Ser. S 1 (2007),
27-39.
[4] C.-M. Brauner, M. L. Frankel, J. Hulshof, V. Roytburd, Stability and attractors for QuasiSteady model of cellular flames, Interf. Free Bound. 8 (2006), 301-316.
[5] C.-M. Brauner, M. L. Frankel, J. Hulshof, G. I. Sivashinsky, Weakly nonlinear asymptotics of
the κ − θ model of cellular flames: the Q-S equation, Interf. Free Bound. 7 (2005), 131-146.
[6] C.-M. Brauner, L. Hu, L. Lorenzi, Asymptotic analysis in a gas-solid combustion model with
pattern formation, Chin. Annal. Math. Ser. B 34 (2013), 65-88.
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
109
[7] C.-M. Brauner, J. Hulshof, L. Lorenzi, Stability of the Travelling Wave in a 2D weakly nonlinear Stefan problem, Kin. Relat. Models 2 (2009), 109-134.
[8] C.-M. Brauner, J. Hulshof, L. Lorenzi, Rigorous derivation of the Kuramoto-Sivashinsky
equation in a 2D weakly nonlinear Stefan problem, Interfaces Free Bound. 13 (2011), 73-103.
[9] C.-M. Brauner, J. Hulshof, L. Lorenzi, G. I. Sivashinsky, A fully nonlinear equation for the
flame front in a quasi-steady combustion model, Discrete Contin. Dyn. Syst. Ser. A 27 (2010),
1415-1446.
[10] C.-M. Brauner, J. Hulshof, A. Lunardi, A general approach to stability in free boundary
problems, J. Differential Equations 164 (2000), 16-48.
[11] C.-M. Brauner, L. Lorenzi, G. I. Sivashinsky, C. Xu, On a strongly damped wave equation
for the flame front, Chin. Ann. Math. Ser. B 31 (2010), 819-840.
[12] C.-M. Brauner, A. Lunardi, Instabilities in a two-dimensional combustion model with free
boundary, Arch. Ration. Mech. Anal. 154 (2000), 157-182.
[13] J. D. Buckmaster, G. S. S. Ludford, Theory of laminar flames, Cambridge University Press,
Cambridge, 1982.
[14] J. D. Buckmaster, G. S. S. Ludford, Lectures on mathematical combustion, CBMS-NSF Conference Series Applied Mathematics, SIAM, 1983.
[15] A. N. Carvalho, J. W. Cholewa, Strongly damped wave equations in W 1,p (Ω)× L p (Ω), Discrete Contin. Dyn. Syst. Supp. 2007, 230-239.
[16] A. Ducrot, M. Marion, Two-dimensional travelling wave solutions of a system modelling
near equidiffusional flames, Nonlinear Anal. 61 (2005), 1105-1134
[17] W. Eckhaus, Asymptotic analysis of singular perturbations, North-Holland, 1979.
[18] P. C. Fife, Dynamics of internal layers and diffusive interfaces, 53, SIAM, Philadelphia, 1988.
[19] M. L. Frankel, On the weakly nonlinear evolution of a perturbed solid-liquid interface, Phys.
D 27 (1987), 260-266.
[20] M. L. Frankel, P. V. Gordon, G. I. Sivashinsky, On desintegration of near-limit cellular flames,
Phys. Lett. A 310 (2003), 389-392.
[21] M. L. Frankel, V. Roytburd, G. I. Sivashinsky, Complex dynamics generated by a sharp interface model of self-propagating high temperature synthesis, Combust. Theory Model. 2
(1998), 1-18.
[22] M. L. Frankel, G. I. Sivashinsky, On the nonlinear thermal-diffusive theory of curved flames,
J. Physique 48 (1987), 25-28.
[23] D. A. Frank-Kamenetskii, On the diffusion theory of heterogeneous reactions, Zh. Fiz. Khim
13 (1939), 756.
[24] L. Hu, C.-M. Brauner, J. Shen, G. I. Sivashinsky, Modeling and simulation of fingering pattern formation in a combustion model, Math. Models Methods Appl. Sci. 25 (2014), 685-720.
[25] J. M. Hyman, B. Nicolaenko, The Kuramoto-Sivashinsky equation: a bridge between PDEs
and dynamical systems, Phys. D 18 (1986), 113-126.
[26] L. Kagan, G. I. Sivashinsky, Pattern formation in flame spread over thin solid fuels, Combust.
Theor. Model. 12, 2008, 269-281.
[27] Y. Kuramoto, Diffusion-Induced Chemical Turbulence, Springer Series in Synergetics 6,
Berlin (1980), 134-146.
[28] Y. Kuramoto, T. Tsusuki, Progr. Theoret. Phys. 54 (1975), 687-699; ibid. 55 (1976), 356-369;
Kuramoto, ibid. Supp. 64 (1978), 346-367.
[29] L. Lorenzi, A free boundary problem stemmed from combustion theory. I. Existence, uniqueness and regularity results, J. Math. Anal. Appl. 274 (2002), 505-535.
[30] L. Lorenzi, A free boundary problem stemmed from combustion theory. II. Stability, insta-
110
C.-M. Brauner / J. Math. Study, 49 (2016), pp. 93-110
bility and bifurcation results, J. Math. Anal. Appl. 275 (2002), 131-160.
[31] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser,
Basel, 1995.
[32] B. J. Matkowsky, G. I. Sivashinsky, An asymptotic derivation of two models in flame theory
associated with the constant density approximation, SIAM J. Appl. Math. 37 (1979), 686-699.
[33] G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl.
Math. 39 (1980), 67-82.
[34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied
Mathematical Sciences 68, 2nd ed., Springer, 1997.
[35] A. M. Turing, The chemical basis of morphogenesis, Phil. Transl. Roy. Soc. Lond. B237 (1952),
37-72.
[36] J. Vukadinovic, Global dissipativity and inertial manifolds for diffusive Burgers equations
with low-wavenumber instability, Discrete Contin. Dyn. Syst. 29 (2011), 327-341.
[37] J. Zhang, X. Fan, Numerical methods for the Kuramoto-Svashinsky equation, J. Northeast
Normal U. 47 (2015), 45-52.
[38] J. Zhang, W. Li, X. Fan, X. Yu, Numerical approximations of the spectral discretization of
flame front model, J. Math. Study 48 (2015), 346-361.
[39] O. Zik, E. Moses, Fingering instability in combustion: an extended view, Phys. Rev. E 60
(1999), 518-531.