Pulsating Front Speed-up and Quenching of Reaction by Fast Advection
PULSATING FRONT SPEED-UP AND QUENCHING OF REACTION
BY FAST ADVECTION
ANDREJ ZLATOŠ
Abstract. We consider reaction-diffusion equations with combustion-type non-linearities
in two dimensions and study speed-up of their pulsating fronts by general periodic incom-
pressible flows with a cellular structure. We show that the occurence of front speed-up in
the sense limA→∞ c∗(A) = ∞, with A the amplitude of the flow and c∗(A) the (minimal)
front speed, only depends on the geometry of the flow and not on the reaction function. In
particular, front speed-up occurs for KPP reactions if and only if it does for ignition reac-
tions. We provide a sharp characterization of the periodic symmetric flows which achieve
this speed-up and also show that these are precisely those which, when scaled properly, are
able to quench any ignition reaction.
1. Introduction and Examples
In this paper we study the effects of strong incompressible advection on combustion. We
consider the reaction-advection-diffusion equation
Tt + Au(x) · ∇T = ∆T + f(T ), T (0, x) = T0(x) ∈ [0, 1] (1.1)
on D ≡ R× Td−1, with u a prescribed flow profile and A ≫ 1 its amplitude. Here T (t, x) ∈
[0, 1] is the normalized temperature of a premixed combustible gas and f is the burning rate.
We assume that u ∈ C1,ε(D) is a periodic incompressible (i.e., ∇·u ≡ 0) vector field which
is symmetric across the hyperplane x1 = 0. That is, u(Rx) = Ru(x) where R(x1, . . . , xd) =
(−x1, x2, x3, . . . , xd) is the reflection across x1 = 0. If the period of u in x1 is p, then this
implies that u is symmetric across each hyperplane x1 = kp, k ∈ Z. Hence u is a periodic
symmetric flow of cellular type (since u1(x) = 0 when x1 ∈ pZ) with [0, p] × Td−1 a cell of
periodicity.
The reaction function f ∈ C1,ε([0, 1]) is of combustion type. That is, there is θ0 ∈ [0, 1)
such that f(s) = 0 for s ∈ [0, θ0] ∪ {1} and f(s) > 0 for s ∈ (θ0, 1), and f is non-increasing
on (1−ε, 1) for some ε > 0. This includes the ignition reaction term with θ0 > 0 and positive
reaction term with θ0 = 0. In the latter case we single out the Kolmogorov-Petrovskii-
Piskunov (KPP) reaction [13] with 0 < f(s) ≤ sf ′(0) for all s ∈ (0, 1).
We will be interested in two effects of the strong flow Au on combustion: pulsating front
speed enhancement and quenching of reaction. This problem has recently seen a flurry of
activity — see [1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 22, 23, 24]. A pulsating front is a solution
of (1.1) of the form T (t, x) = U(x1 − ct, x), with c the front speed and U(s, x) periodic in x1
(with period p) such that
U(s, x) = 1 and lim
U(s, x) = 0,
http://arxiv.org/abs/0704.1164v2
2 ANDREJ ZLATOŠ
uniformly in x. It is well known [4] that in the case of positive reaction there is c∗(A),
called the minimal pulsating front speed, such that pulsating fronts exist precisely for speeds
c ≥ c∗(A). In the ignition reaction case the front speed is unique and we again denote it
c∗(A). In the present paper we will be interested in the enhancement of this (minimal) front
speed by strong flows.
We say that the flow Au quenches (extinguishes) the initial “flame” T0 if the solution of
(1.1) satisfies ‖T (t, ·)‖∞ → 0 as t → ∞. Here one usually considers compactly supported
initial data. The flow profile u is said to be quenching for the reaction f if for any compactly
supported initial datum T0 there is an amplitude A0 such that T0 is quenched by the flow
Au whenever A ≥ A0. We note that quenching never happens for KPP reactions — the
solutions of (1.1) for compactly supported non-zero T0 always propagate and the speed of
their spreading equals c∗(A) [4, 20].
In this paper we characterize those periodic symmetric incompressible flows in two di-
mensions which achieve speed-up of fronts and, if scaled properly, quenching of any ignition
reaction. For l > 0 we denote by lT the interval [0, l] with its ends identified, and we let
u(l)(x) ≡ u(x/l) be the scaled flow on R× lT (with cells of size lp× l).
Theorem 1.1. Let u be a C1,ε incompressible p-periodic flow on D = R × T which is sym-
metric across x1 = 0, and let f be any combustion-type reaction.
(i) If the equation
u · ∇ψ = u1 (1.2)
on pT× T has a solution ψ ∈ H1(pT× T), then
lim sup
c∗(A) <∞ (1.3)
and no u(l) is quenching for f .
(ii) If (1.2) has no H1(pT× T)-solutions, then
c∗(A) = ∞ (1.4)
and if f is of ignition type, then there is l0 ∈ (0,∞) such that the flow u(l) on R× lT is
quenching for f when l < l0 and not quenching when l > l0.
Remarks. 1. The proof shows that in (ii), l0 ≥ c‖f(s)/s‖−1/2∞ for some u-independent c > 0.
It can also be showed that the claim l0 > 0 in (ii) extends to some positive reactions that
are weak at low temperatures (more precisely, f(s) ≤ αsβ for some α > 0 and β > 3 — see
Corollary 4.4), in particular, the Arrhenius reaction f(s) = e−C/s(1 − s), C > 0. On the
other hand, if f(s) ≥ αsβ for some α > 0, β < 3, and all small s, then l0 = 0 for any u [22].
2. We note that l0 = ∞ is impossible for cellular flows in two dimensions — see [23] which
studies strongly quenching flows u, that is, quenching for any ignition reaction and any l.
3. Although we only consider periodic boundary conditions here, it is easy to see that
Theorem 1.1 remains valid for (1.1) on R×[0, 1] with Neumann boundary conditions, provided
u2(x) = 0 when x2 ∈ {0, 1}.
FRONT SPEED-UP AND QUENCHING 3
4. Although a part of our analysis — Sections 2 and 3 — is valid in any dimension, it
remains an open quenstion whether Theorem 1.1 also extends beyond two dimensions.
Theorem 1.1 has the following corollary:
Corollary 1.2. Let u be a C1,ε incompressible p-periodic flow on D = R × T which is
symmetric across x1 = 0. Then speed-up of pulsating fronts by u in the sense of (1.4) occurs
for ignition reactions if and only if it occurs for KPP reactions.
Remark. Although speed-up of KPP fronts has been studied extensively (see, e.g., [3, 5, 6,
10, 11, 15, 18, 24]), rigorous results on ignition front speed-up have so far been established
only in two dimensions for percolating flows and special cellular flows [11] (see below).
It is not surprising that the flows which achieve speed-up of fronts are precisely those which
quench large initial data. Fast fronts are long, the latter being due to short time–long distance
mixing by the underlying flow. Such mixing yields quenching, although possibly only away
from regions where the flow is relatively still (e.g., the centers of the cells in Figure 1 below).
If these regions are sufficiently small, for instance when the flow is scaled, then reaction
cannot survive inside them and global quenching follows. This relation of front speed to flow
mixing properties also illuminates Corollary 1.2.
Note that the above assumptions on u exclude the class of percolating flows (in particular,
shear flows u(x) = (α(x2, . . . , xd), 0, . . . , 0)) which possess streamlines connecting x1 = −∞
and x1 = +∞. In two dimensions, the conclusions of Theorem 1.1(ii) for these flows have
been established in [6, 7, 11, 12, 18]. Moreover, results from [5, 24] can be used to prove
linear pulsating front speed-up (namely, limA→∞ c∗(A)/A > 0) by percolating flows in the
presence of KPP reactions in any dimension.
As for cellular flows in two dimensions (the kind we consider here), the claims about
the front speed c∗(A) in Theorem 1.1 have been proved for KPP reactions in [18]. The
special case of the flow u(x) = ∇⊥H(x) ≡ (−Hx2 , Hx1) with the stream function H(x1, x2) =
sin 2πx1 sin 2πx2 has been addressed in [9, 11, 15], which proved (1.4) for any reaction and
quenching by u(l) for small enough l and ignition reactions. The streamlines of this flow are
depicted in Figure 1.
Figure 1. A cellular flow.
4 ANDREJ ZLATOŠ
We note that it is easy to show that (1.2) has no H1(T2)-solutions in this case [18], and so
one can recover these results from Theorem 1.1(ii). Our general method does not yield the
more precise asymptotics c∗(A) ∼ A1/4 in the KPP case [15] and A1/5 . c∗(A) . A1/4 in the
ignition case [11] for this particular flow.
We conclude this introduction with two more examples of types of flows to which Theo-
rem 1.1 applies.
Example 1.3. Checkerboard flows. Consider the cellular flow above vanishing in every
other cell as depicted in Figure 2, thus forming a checkerboard-like pattern. This flow is both
periodic (with period 2) and symmetric but it is not C1,ε. Let us remedy this problem by
letting the stream function be H(x1, x2) = (sin 2πx1 sin 2πx2)
α with α > 2 in the cells where
u does not vanish. Again, (1.2) has no H1(2T×T)-solutions [18], and so Theorem 1.1(ii) —
speed-up of fronts and quenching by u(l) — holds. Moreover, the same conclusion is valid for
other flows with this type of structure, even if the angle of contact of the “active” cells is π.
Figure 2. A checkerboard cellular flow.
Example 1.4. Flows with gaps. Consider again the cellular flow above but with a vertical
“gap” of width δ > 0, in which the flow vanishes, inserted in place of each vertical segment
{k}×T, k ∈ Z, such as shown in Figure 3. We again need to alter the stream function as we
did in the previous example in order to make the flow C1,ε. This time it is easy to see that
(1.2) has H1((1+ δ)T×T)-solutions [18], and so Theorem 1.1(i) — no speed-up of fronts and
no quenching by u(l) — holds in this case. The same conclusion is valid for other flows with
similar structures of streamlines, even when the gaps are replaced by channels in which the
flow moves “along” the channel only (see [18] for more details).
We also note that Sections 2 and 3 below yield the conclusions of Theorem 1.1(i) for cellular
flows with gaps in any dimension (using that gaps force Lemma 2.2(ii) to hold).
The rest of the paper consists of Section 2 where we prove a few preliminary lemmas, and
Sections 3 and 4 which contain the proof of Theorem 1.1.
The author would like to thank Sasha Kiselev, Tom Kurtz, and Greg Lawler for useful dis-
cussions. Partial support by the NSF through the grant DMS-0632442 is also acknowledged.
FRONT SPEED-UP AND QUENCHING 5
Figure 3. A cellular flow with gaps.
2. Some Preliminaries
In this and the next two sections we will assume the hypotheses of Theorem 1.1 with the
period p = 1 — the general case is handled identically. This implies that u is symmetric
across each hyperplane x1 = k, k ∈ Z. The analysis in this section and the next applies to
(1.1) on D = R× Td−1 for any d ∈ N.
Let us consider the stochastic process X
t starting at x ∈ D and satisfying the stochastic
differential equation
2 dBt −Au(XA,xt )dt, X
0 = x, (2.1)
where Bt is a normalized Brownian motion on D. We note that by Lemma 7.8 in [16], we
have that if
φt + Au(x) · ∇φ = ∆φ, φ(0, x) = φ0(x), (2.2)
φ(t, x) = E
. (2.3)
In particular, φ0(x) = χ[−L,L](x) gives
φ(t, x) = P
|XA,xt | ≤ L
, (2.4)
where we define |x| ≡ |x1| for x ∈ D. Also notice that if φ0 = T0 ∈ [0, 1], then by comparison
theorems [19] for any t, x,
0 ≤ T0(t, x) ≤ et‖f(s)/s‖∞φ(t, x) ≤ et‖f
′‖∞φ(t, x). (2.5)
Lemma 2.1. (i) If k ∈ Z and y1 = k then the distribution of XA,yt is symmetric across the
hyperplane x1 = k, that is,
t ∈ V ) = P
t ∈ R(V − (k, 0)) + (k, 0)
for each V ⊆ D.
(ii) If k ∈ Z and y1 ≥ k, then for any I ⊆ R+,
t )1 ∈ k + I
t )1 ∈ k − I
. (2.6)
When y1 ≤ k, the inequality in (2.6) is reversed.
6 ANDREJ ZLATOŠ
(iii) If L ∈ N, then
|XA,yt | ≤ L
. (2.7)
Proof. (i) and (ii) are obvious from the symmetry of u across x1 = k and from almost sure
continuity of X
t in t. To show (iii), it is sufficient to consider y1 > L. Applying (ii) with
k = jL for j = 1, . . . , ⌈y1/L⌉ − 1, we see that
t )1 ∈ [−L, L]
t )1 ∈ [(2j − 1)L, (2j + 1)L]
The claim follows. �
Next we prove the following key dichotomy.
Lemma 2.2. For any sequence {An}∞n=1 one of the following holds.
(i) For any t, ε > 0 and L <∞ there are x, n such that
|XAn,xt − x| ≤ L
< ε. (2.8)
(ii) For any t, ε > 0 there is L <∞ such that for any x, n,
|XAn,xt − x| ≤ L
> 1− ε. (2.9)
Proof. Let us first assume that there is t′ > 0 such that for any ε′ > 0 and L′ <∞ there are
x, n such that
|XAn,xt′ − x| ≤ L
< ε′. (2.10)
Given any ε > 0, L ∈ N, let m > 2/ε be an integer and let x, n be as in (2.10) with ε′ = 1/m,
L′ = (2m+ 1)L. Notice that by periodicity of u we can assume |x1| ≤ 1. For any t ≥ t′ we
|XAn,xt − x| ≤ L
|XAn,xt | ≤ 2L
|XAn,xt′ | ≤ 2mL
+ sup
|y|≥2mL
|XAn,yt−t′ | ≤ 2L
The first term is smaller than ε′ < ε/2 by (2.10) and the second is at most 1/m < ε/2 by
(2.7). This yields (i) for t ≥ t′. On the other hand, if (i) does not hold for some t ∈ (0, t′),
then there are ε, L such that for all x, n,
|XAn,xt − x| ≤ L
Choose m ∈ N so that mt ≥ t′. It follows that
|XAn,xmt − x| ≤ mL
for all x, n. But this contradicts (i) for mt, which has just been proven. Therefore (i) holds
for all t > 0 under the hypothesis above.
Now assume the opposite case to the one above. Namely, that for each t′ > 0 there are
ε′ > 0 and L′ <∞ such that for all x, n,
|XAn,xt′ − x| ≤ L
≥ ε′. (2.11)
We will show that then (ii) holds, thus finishing the proof.
FRONT SPEED-UP AND QUENCHING 7
For each t > 0 let
ε0(t) ≡ sup
|XAn,xt − x| ≤ L
Periodicity of u guaranties that
ε0(t) = sup
|x|≤1,n
|XAn,xt | ≤ L
≡ sup
ε1(t, L)
Notice that ε0(t) is non-increasing. Indeed, for L,m ∈ N and t ≥ t′,
ε1(t, L) ≤ ε1(t′, mL) +
(2.12)
by (2.7), and so ε0(t) ≤ ε0(t′) + 1/m for any m.
We will now show that ε0(t) = 1 for all t. To this end assume ε0(t) < 1 for some t. Let m
be large (to be chosen later), and let L be such that
ε1(t, L) > ε0(t)−
(2.13)
Consider any |x| ≤ 1, n such that
|XAn,xt | ≤ (2m+ 1)L
≤ ε0(t) +
. (2.14)
Such x, n do exists because of ε0(t) ≥ ε1(t, (2m+ 1)L). Then the set of Brownian paths for
which there is t′ ∈ [0, t] such that |XAn,xt−t′ | = (m+ 1)L has measure at least 1− ε0(t)− 1/m.
Since
|XAn,xt | ∈ [L, (2m+ 1)L]
∣ |XAn,xt−t′ | = (m+ 1)L for some t
′ ∈ [0, t]
≥ inf
t′∈[0,t]
′, mL) > ε0(t)−
by (2.12) and (2.13), this means
|XAn,xt | ≤ (2m+ 1)L
|XAn,xt | ≤ L
|XAn,xt | ∈ [L, (2m+ 1)L]
≥ ε1(t, L) +
1− ε0(t)−
ε0(t)−
2− ε0(t)−
ε0(t)−
Since ε0(t) < 1, this is larger than ε0(t) + 1/m when m is large enough. This, however,
contradicts (2.14). Therefore we must have ε0(t) = 1 for all t, which is (ii). �
We will also need the following result which is essentially from [8].
Lemma 2.3. For any d ∈ N, there is c > 0 such that for any Lipschitz incompressible flow
u, any A, and any t ≥ 0, the solution φ of (2.2) on Ω ≡ [0, 1]×Td−1 with Dirichlet boundary
conditions on ∂Ω satisfies
‖φ(t, ·)‖∞ ≤ 2e−ct‖φ0‖∞. (2.15)
8 ANDREJ ZLATOŠ
Proof. The maximum principle implies that it is sufficient to show that there is τ > 0 such
‖φ(τ, ·)‖∞ ≤
‖φ0‖∞.
uniformly in u and A. For incompressible flows on Td and mean-zero φ0 this follows from
Lemma 5.6 in [8]. The proof extends without change to our case, the Dirichlet boundary
condition replacing the mean-zero assumption when the Poincaré inequality is used. �
3. Proof of Theorem 1.1: Part I
Let us now assume that u and f are as in Theorem 1.1 and An → ∞ is such that Lemma
2.2(ii) holds. We will then show that the minimal front speeds c∗(An) are uniformly bounded
and the flows Anu do not quench large enough compactly supported initial data T0 for (1.1).
The analysis in this section applies to D = R× Td−1 for any d ∈ N.
Lemma 3.1. Consider the setting of Theorem 1.1 with D = R× Td−1, and let An → ∞ be
such that Lemma 2.2(ii) holds. Then c∗(An) are uniformly bounded above.
Proof. Choose L ∈ N that satisfies Lemma 2.2(ii) for t = 1 and ε = 1
. Let x be such that
x1 ∈ Z and consider XAn,xt from (2.1). Take τ0 = 0 and let τj be the first time such that
|XAn,xτj −X
| = 3L (recall that |x| = |x1|). We then have from (2.9) and (2.7),
P(τj − τj−1 ≤ 1) ≤
because 1
p+ (1− p) ≥ 3
implies p ≤ 1
. This means that for any large enough C, t ∈ N,
P(|XAn,xt − x| ≥ 3LCt) ≤ P(τCt ≤ t) ≤
)Ct−j
2(C−1)t
(5/4)C−1CC
2C−1(C − 1)C−1
≤ κ(C)t
with κ(C) ≡ 2Ce(2/3)C → 0 as C → ∞. We used here the fact that fewer than t of the
differences τj−τj−1 can exceed 1 in the second inequality, and Stirling’s formula in the fourth.
Let now T be the solution of (1.1) with A = An and T0 ≡ χR−×Td−1 . If φ solves (2.2) with
A = An and φ0 ≡ T0, then we have by (2.5) for x(s) ≡ (s, 0, . . . , 0),
T (t, x(3LCt)) ≤ et‖f ′‖∞φ(t, x(3LCt)) ≤ et‖f ′‖∞P(|XAn,x(3LCt)t − x(3LCt)| ≥ 3LCt) → 0
as t→ ∞, provided C is large enough. On the other hand, it is well known that T (t, x(ct)) →
1 as t→ ∞ when c < c∗(An) [4, 20, 21]. This means c∗(An) ≤ 3LC and we are done. �
Lemma 3.2. Consider the setting of Theorem 1.1 with D = R× Td−1, and let An → ∞ be
such that Lemma 2.2(ii) holds. Then there is compactly supported T0(x) ∈ [0, 1] such that the
solution T of (1.1) with A = An does not quench for any n.
FRONT SPEED-UP AND QUENCHING 9
Proof. By comparison theorems, we only need to consider f of ignition type — with θ0 > 0.
We again choose L ∈ N that satisfies Lemma 2.2(ii) for t = 1 and ε = 1
. We next note that
there is δ > 0 such that
P(|XAn,xt − x| ≥ t8/15) ≤ e−t
(3.1)
for all large enough t and all x ∈ D and n. Indeed, assume x1 ∈ Z and t ∈ Z (the general
case follows immediately from this), and let j(t) = inf{j | τj > t}, with τj from the proof of
Lemma 3.1. Then that proof shows that for C ∈ Z we have
P(j(t) > Ct) = P(τCt ≤ t) ≤ κ(C)t (3.2)
with κ(C) < 1 if C is large. On the other hand, symmetry of u across each hyperplane
x = k ∈ Z shows that Yj ≡ (XAn,xτj −X
)1 are iids with P
Yj = ±L
. This gives
P(|XAn,x
− x| ≥ L(Ct)9/17
∣ j(t) ≤ Ct) ≤ e−(Ct)δ
for some δ > 0 by
≈ (1 + jδ−
2 )(1− jδ−
(1− j2δ−1)−j1−2δ(1 + jδ−
2 )−j
(1− jδ−
]j2δ/2
≈ e−j2δ/2,
where we used Stirling’s formula again. This, the fact that |XAn,xτj(t) − X
t | ≤ L (by the
definition of τj and j(t)), and (3.2) yield (3.1) for large enough t (with a different δ > 0).
We will also need the conclusion of Lemma 3.1 in [9] which says that there is c̃ > 0 such
that for any x ∈ D, m ∈ Z, A ∈ R, incompressible u, and t ≥ 1 we have
∈ [m,m+ 1]
≤ c̃t−1/2. (3.3)
We note that [9] only considers d = 2, but the general case is identical.
Let us now take non-negative ψ0 ∈ C(R) ∩ C3([−2, 2]) such that
suppψ0 = [−2, 2],
ψ0(s) = ψ0(−s) and ψ0(0) = 2+θ03 ,
ψ0(s) =
(3− |s|)2 − 1
for |s| ∈ [1, 2],
ψ′0 is decreasing on [−1, 1].
Note that this means that ψ0 is non-negative, symmetric, non-increasing on R
+, and convex
where f(ψ0(s)) = 0. We then let
T0(x) ≡ ψ0
with a large M ∈ Z to be determined later. We will show using the properties of ψ0 that if
T solves (1.1) with A = An, then for τ ≡M3/2 we have
T (τ, x) ≥ T0(x) (3.4)
(which gives the desired result by comparison theorems).
10 ANDREJ ZLATOŠ
Let ε be such that ψ0(1 + ε) =
1+2θ0
and M such that εM +M4/5 ≤M − 2. Let φ be the
solution of (2.2) with φ0 ≡ T0 and assume first that x1 ∈ [(1 + ε)M, 2M −M4/5] ∩ Z. Let
x′ ≡ (x2, . . . , xd). Then by (2.3), monotonicity of ψ0 on R+, and symmetry of u,
φ(τ, x) ≥
m=−M4/5−1
P((XAn,xτ )1 ∈ [x1 +m, x1 +m+ 1])φ0(x1 +m+ 1, x′)
P((XAn,xτ )1 ∈ [x1 +m, x1 +m+ 1])(φ0(x1 +m+ 1, x′) + φ0(x1 −m, x′)). (3.5)
We have
x1 +m+ 1
x1 −m
= 2ψ0
+ψ′′0
and τ = M3/2 together with (3.1) implies that the sum of the P(·) terms in (3.5) is larger
than 1
(1− e−τδ) = 1
(1− e−M3δ/2). This and ψ′′0(s) = 1+θ03 for s ∈ (1, 2) yields
φ(τ, x) ≥ (1− e−M3δ/2)φ0
1 + θ0
(4c̃M1/4)−2 +O(M−3/5),
where we also used that (3.3) gives
|XAn,xτ − x| ≥
Since φ0(x)− φ0(x1 + 12 , x
′) = O(M−1), this means
φ(τ, x) ≥ φ0(x) + c′M−1/2 (3.6)
for some c′ > 0 and any large enough M .
The same argument applies for any τ ′ ∈ [τ/2, τ ] (with a uniform c′) in place of τ . This,
Lemma 2.3, and the fact that φ0 varies on a scale O(M
−1) on [⌊x⌋, ⌊x⌋+1]×Td−1 yield (3.6)
for any x1 ∈ [(1 + ε)M, 2M −M4/5], provided M is large enough. If x1 ∈ [2M −M4/5, 2M ],
then (3.6) follows in the same way because ψ0(s) >
[(3− |s|)2 − 1] for s ∈ (2, 3). And if
x1 > 2M , then (3.6) is immediate from φ(τ, x) ≥ 0.
Symmetry and T ≥ φ give (3.4) whenever |x| ≥ (1 + ε)M , so let us now consider |x| ≤
(1 + ε)M . As above we obtain for large M ,
φ(τ, x) ≥ φ0(x)− c′M−1/2, (3.7)
where c′ only depends on ‖ψ′′0‖∞. We now choose a convex g : R+ → R+ with g(s) ≤ f(s) for
s ≤ 3+θ0
and g(s) ≥ α for some α > 0 and all s ≥ 1+3θ0
. Define β > 0 so that if γ(0) = 2+θ0
and γ′(s) = g(γ(s)), then γ(β) = 3+θ0
. Next let f̃ ≡ β
g ≤ g when τ = M3/2 ≥ β and let
w : (R+)2 → R+ satisfy w(0, s) = s and
wt(t, s) = f̃(w(t, s)).
FRONT SPEED-UP AND QUENCHING 11
Notice that
w(τ, 2+θ0
) = 3+θ0
and w(τ, s) ≥ s+ αβ for s ≥ 1+3θ0
. (3.8)
It is easy to show using f̃ ′, f̃ ′′ ≥ 0 that ws, wss ≥ 0. It then follows that T̃ (t, x) ≡ w(t, φ(t, x))
is a sub-solution of (1.1) with A = An and T̃0 = T0 as long as ‖T̃ (t, ·)‖∞ ≤ 3+θ04 (so that
f̃(T̃ ) ≤ f(T̃ )). Since ‖φ‖∞ ≤ ψ0(0) = 2+θ03 , this is true for all t ≤ τ by (3.8) and wt, ws ≥ 0.
But then T (τ, x) ≥ T̃ (τ, x), while large enough M guarantees for |x| ≤ (1 + ε)M ,
φ(τ, x) ≥ φ0(x)− c′M−1/2 ≥ 1+2θ03 − c
′M−1/2 ≥ 1+3θ0
So for these x by (3.8),
T (τ, x) ≥ T̃ (τ, x) ≥ φ(τ, x) + αβ ≥ φ0(x)− c′M−1/2 + αβ ≥ φ0(x) = T0(x)
when M is large. This is (3.4) and thus concludes the proof. �
4. Proof of Theorem 1.1: Part II
We now assume that u and f are as in Theorem 1.1 and An → ∞ is such that Lemma 2.2(i)
holds. We will then show that lim supn→∞ c∗(An) = ∞, and that there is c > 0 such that if
f is of ignition type with ‖f(s)/s‖∞ ≤ c, then any compactly supported initial datum T0 for
(1.1) is quenched by some flow Anu. The analysis in this section applies in two dimensions
only, so we will consider d = 2 and D = R× T.
Lemma 4.1. Consider the setting of Theorem 1.1 with D = R×T and let An → ∞ be such
that Lemma 2.2(i) holds. Then lim supn→∞ c∗(An) = ∞.
Proof. Assume that c∗(An) ≤ c0 <∞ for all n and let T be a pulsating front solution of (1.1)
with A = An and speed c∗(An), that is,
T (t+ c∗(An)
−1, x1 + 1, x2) = T (t, x1, x2),
T (t,±∞, x2) = 12 ∓
uniformly in x2
(4.1)
(recall that u has period 1 in x1). We note that [2] shows
Tt(t, x) ≥ 0. (4.2)
Integrating (1.1) over [0, c∗(An)
−1]×D and using (4.1) and incompressibility of u, we obtain
∫ c∗(An)
f(T (t, x)) dxdt.
Next we multiply (1.1) by T and again integrate as above to get
∫ c∗(An)
T (t, x)f(T (t, x))− |∇T (t, x)|2 dxdt ≤ 1−
∫ c∗(An)
|∇T (t, x)|2 dxdt.
This means that for some t ∈ [0, c∗(An)−1] (which we take to be 0 by translating T in time),
f(T (0, x)) dx ≤ 2c0, (4.3)
12 ANDREJ ZLATOŠ
|∇T (0, x)|2 dx ≤ c0. (4.4)
We will now show that (4.1)–(4.4) force the reaction zone (front width) to be bounded in the
following sense. LetD−ε be the rightmost cell [m
ε , m
ε +1]×T such that infx∈D−ε T (0, x) ≥ 1−ε
(i.e., m−ε is the largest integer for which this condition holds). We also let D
ε be the leftmost
cell [m+ε , m
ε + 1]× T such that supx∈D+ε T (0, x) ≤ 1− ε. Obviously m
ε < m
ε . We will now
show that for each small ε > 0 there is Lε <∞ such that for each n we have
m+10ε −m−ε ≤ Lε. (4.5)
Assume for a moment that (4.5) holds. Periodicity and (2.8) tell us that there are n and
x ∈ D−ε such that
|XAn,xτ − x| ≥ Lε
for τ ≡ ε‖f ′‖−1∞ > 0. Since x1 ≥ m−ε ≥ m+10ε − Lε, symmetry of u implies
(XAn,xτ )1 ≥ m+10ε
Using (2.5) and (2.3) we have
T (τ, x) ≤ eτ‖f ′‖∞
1− 10ε
< 1− ε ≤ T (0, x)
if ε > 0 is small. This contradicts (4.2), so our assumption c∗(An) ≤ c0 <∞ must be invalid.
Thus the proof will be finished if we establish (4.5) for all small ε > 0.
Let us consider an arbitrary small ε > 0 such that f is bounded away from zero on
[1− 13ε, 1− ε
] and assume, towards contradiction, that for each L ∈ N there is n such that
m+10ε −m−ε ≥ 10L. (4.6)
Let T0(x) ≡ T (0, x),
T̄0(x) ≡
[⌊x1⌋,⌊x1⌋+1]×T
T0(x) dx,
and denote Dj ≡ [m−ε + j,m−ε + j + 1] × T. Then (4.4) and Poincaré inequality (with
constant C) imply that for each small δ > 0 and L ≡ ⌈Cc0/δ⌉, at least 7L of the cells Dj ,
j = L, . . . , 9L, satisfy
‖T0 − T̄0‖2L2(Dj) ≤ C‖∇T0‖
L2(Dj)
≤ δ. (4.7)
Hence there are at least ⌊3L
⌋ disjoint 5-tuples of consecutive cells satisfying (4.7). Then (4.3),
f bounded away from zero on [1−13ε, 1− ε
], and T̄0(Dj) decreasing in j (by (4.2)) imply that
for some j0 ∈ [L, 9L] we must have either (4.7) and T̄0(Dj) ≤ 1−12ε for j = j0−2, . . . , j0+2,
or (4.7) and T̄0(Dj) ≥ 1 − ε2 for j = j0 − 2, . . . , j0 + 2 (provided δ is small enough and L
large).
Let us assume the case T̄0(Dj) ≤ 1− 12ε for j = j0− 2, . . . , j0+2, j0 ∈ [L, 9L]. Then (4.2)
and (4.6) say that there must be y ∈ Dj0 such that for t ≥ 0,
T (t, y) ≥ T0(y) ≥ 1− 10ε. (4.8)
FRONT SPEED-UP AND QUENCHING 13
Let S−2γ ⊂ Dj0−2∪Dj0−1∪Dj0 be the square of a small side 2γ > 0 (to be chosen later) centered
at y− ≡ y − (1, 0) and denote by Γ− the intersection of S−2γ with the connected component
Ω− of the set {x | T0(x) ≥ 1−11ε} containing y− (recall that that T0(y−) ≥ T0(y) ≥ 1−10ε).
If Γ− has diameter less than γ (in particular, Γ− = Ω− ⊆ S−2γ), then for Γ ≡ Γ− + (1, 0),
all x ∈ ∂Γ, and all t ≤ c∗(An)−1,
T (t, x) ≤ T (0, x− (1, 0)) ≤ 1− 11ε
by (4.1) and (4.2). It follows by comparison that T (t, x) ≤ et‖f ′‖∞(R(t, x) + 1 − 11ε) where
R(t, x) solves (2.2) on S2γ ≡ S−2γ + (1, 0) with Dirichlet boundary conditions and R(0, x) =
11εχΓ(x). But then the uniform bound in Lemma 2.3 and parabolic scaling in (t, x) gives
that for any t > 0 there is small enough γ > 0 such that ‖R(t, x)‖∞ ≤ ε2 , and if t is chosen
small enough (and γ accordingly), then T (t, y) < 1 − 10ε follows. This clearly contradicts
(4.8).
If instead (for the chosen γ) the set Γ− ⊂ Dj0−2 ∪ Dj0−1 ∪ Dj0 has diameter at least γ,
then T̄0(Dj) ≤ 1 − 12ε and inf T0(Γ−) ≥ 1 − 11ε imply that the second inequality in (4.7)
must be violated for at least one of j = j0 − 2, j0 − 1, j0, provided δ > 0 is chosen small
enough (depending on γ, ε). Indeed — if ‖∇T0‖2L2(Dj) is small enough, then T must be close
to 1− 11ε on some vertical line passing through Γ−, and then T must be close to 1− 11ε on
most horizontal lines inside Dj by the same argument. This contradicts T̄0(Dj) ≤ 1− 12ε.
Finally, if we instead assume T̄0(Dj) ≥ 1 − ε2 for j = j0 − 2, . . . , j0 + 2 and T (t, y) ≤
T0(y − (1, 0)) ≤ 1− ε for small t ≥ 0, a similar argument again leads to contradiction. This
means that (4.6) cannot hold for small ε > 0 and (4.5) follows. The proof is finished. �
Lemma 4.2. Consider the setting of Theorem 1.1 with D = R×T. There is c > 0 such that
if f is of ignition type with ‖f(s)/s‖∞ ≤ c and An → ∞ is such that Lemma 2.2(i) holds,
then for any compactly supported T0(x) ∈ [0, 1] there is n such that the solution T of (1.1)
with A = An quenches.
Remark. We note that c is from Lemma 2.3 and can be easily evaluated from its proof.
Proof. By comparison theorems, it is sufficient to consider initial data T0(x) ≡ χ[−L,L](x1)
for all L ∈ N. Let φ be the solution of (2.2) with A = An and initial datum φ0 ≡ T0. We
first claim that for each τ, δ > 0 there is n and a continuous curve h : [0, 1] → [0, 1]× T such
that (h(0))1 = 0 and (h(1))1 = 1 , and for all s ∈ [0, 1] and t ≥ τ ,
φ(t, h(s)) ≤ δ. (4.9)
To this end we let ψ be the solution of (2.2) with initial condition ψ0 ≡ χ[−K−2,K](x1) where
K ≥ 3Lδ−1. By periodicity of u and (2.8), there must be n (which will be kept constant from
now on) and y ∈ [−1, 0]× T such that
ψ(τ, y) = P
(XAn,yτ )1 ∈ [−K − 2, K]
The maximum principle for (2.2) implies that the connected component of the set
{(t, x) ∈ [0, τ ]×D |ψ(t, x) ≤ δ
14 ANDREJ ZLATOŠ
containing (τ, y) must intersect
{x ∈ D |ψ(0, x) ≤ δ
} = (R \ [−K − 2, K])× T.
Since by symmetry ψ(t, x1, x2) = ψ(t,−2−x1, x2) for x1 ≥ 0, this means that there is a curve
h(s) joining {0} × T and {K} × T such that for each s there is τs ≤ τ with
ψ(τs, h(s)) = P
(XAn,h(s)τs )1 ∈ [−K − 2, K]
Lemma 2.1(iii) and the definition of K then mean that for all t ≥ τ ,
φ(t, h(s)) = P
|XAn,h(s)t | ≤ L]
which is (4.9) (after reparametrization of h and restriction to s ∈ [0, 1]).
Symmetry of u and φ0 implies that (4.9) holds for h(s) extended to s ∈ [−1, 1] by h(−s) =
(−(h(s))1, (h(s))2). Finally, (4.9) applies to h(s) extended periodically (with period 2) onto
R. This last claim holds because φ(t, x) ≥ φ(t, x + (2, 0)) when x1 ≥ −1 (and φ(t, x) ≥
φ(t, x − (2, 0)) when x1 ≤ 1), which in turn follows because φ(t, x) − φ(t, x + (2, 0)) solves
(2.2) with initial datum that is symmetric across x1 = −1 and non-negative on [−1,∞)× T
(and hence stays such by the symmetry of u).
This means that ‖φ(t+ τ, ·)‖∞ ≤ ‖ψ(t, ·)‖∞+ δ where ψ is the solution of (2.2) on 2T×T
with ψ0 ≡ 1 and ψ(t, h(s)) = 0 for all t > 0 and s ∈ [0, 2]. Since the Poincaré inequality
and the proof of Lemma 2.3 extend to this setting with the same universal constant c > 0,
we obtain that ‖φ(t, ·)‖∞ ≤ δ + 2e−c(t−τ). If now ‖f(s)/s‖∞ = c′ < c and τ, δ > 0 are
chosen small enough depending on c − c′ (and n accordingly), we obtain ‖T (t0, ·)‖∞ ≤
′t0(δ+2ecτe−ct0) ≤ θ0 for some t0. The maximum principle then implies ‖T (t, ·)‖∞ ≤ θ0 for
any t ≥ t0 and quenching follows. �
The proof of Theorem 1.1 is now based on the last four lemmas and this result from [18]:
Lemma 4.3. Assume the setting of Theorem 1.1 with f a KPP nonlinearity and D = R×T.
(i) If (1.2) on 2T× T has a solution ψ ∈ H1(2T× T), then (1.3) holds.
(ii) If (1.2) has no H1(2T× T)-solutions, then (1.4) holds.
Proof of Theorem 1.1. If (1.2) has a solution ψ ∈ H1(2T × T), then c∗(An) is bounded for
any KPP f and any An → ∞, and so Lemma 4.1 gives Lemma 2.2(ii). Lemmas 3.1 and
3.2 now give (i) for any f . Note that if each sequence An does not quench some compactly
supported initial datum T0 for (1.1) with A = An, then there is T0 that is not quenched by
any A. This holds because if each T0(x) ≡ χ[−n,n](x1) is quenched by some An, then this
sequence would yield a contradiction.
If, on the other hand, (1.2) has no H1(2T×T)-solutions, then c∗(An) → ∞ for any KPP f
and any An → ∞, and so Lemma 3.1 gives Lemma 2.2(i). Lemma 4.1 now gives (1.4) for any
f . The claim about the existence of l0 follows from the fact that T solves Tt − Au(l) · ∇T =
∆T + f(T ) on R× lT if and only if S(t, x) ≡ T (l2t, lx) solves St − Alu · ∇S = ∆S + l2f(S)
on R × T. Comparison theorems and f ≥ 0 then show that if u(l) is quenching for f , then
so is u(l̃) for any l̃ < l. This only guarantees l0 ∈ [0,∞], but l0 < ∞ follows from Theorem
FRONT SPEED-UP AND QUENCHING 15
8.2 in [23] and the fact that the flow u leaves the bounded domain [0, p] × T invariant. For
ignition reactions Lemma 4.2 shows l0 > 0 — if each T0 is quenched by at least one Anu for
any sequence An → ∞, then each T0 is quenched by Au for all large A. �
Finally, we provide the following extension of Theorem 1.1(ii) to some positive reactions.
Corollary 4.4. The claim l0 > 0 in Theorem 1.1(ii) holds for any combustion-type reaction
satisfying f(s) ≤ αsβ for some α > 0, β > 3, and all s ∈ [0, 1].
Proof. By the proof of Theorem 1.1, it is sufficient to show that there is l > 0 such that u
is quenching for l2f(s). The proof is essentially identical to that of Theorem 8.3 in [23]. We
let IA ≡
‖φ(t, ·)‖β−1∞ dt where φ is the solution of (2.2) and φ0(x) ≡ T0(x). It follows
from [14] (see also [22, Lemma 2.1]) that u is quenching for l2f(s) when for each compactly
supported T0 there is A0 such that l
2α(β − 1)IA < 1 whenever A ≥ A0. So fix T0 and
notice that the bound ‖φ(t, ·)‖∞ ≤ c̃|suppT0|t−1/2 for t ≥ 1, which follows from (3.3), gives
‖φ(t, ·)‖β−1∞ dt ≤ 1 if t0 is chosen appropriately (depending on c̃|suppT0|). For t ≤ t0 we
use the bound ‖φ(t, ·)‖∞ ≤ 5e−ct, which follows from the proof of Lemma 4.2 (with the same
c) provided A0 is chosen large enough so that δ in that proof is smaller than e
−ct0 for each
A ≥ A0 (and τ is such that ecτ ≤ 2). This choice is possible because each sequence An → ∞
has a term An guaranteeing δ < e
−ct0 . Hence for A ≥ A0 we have
‖φ(t, ·)‖β−1∞ dt ≤
(5e−ct)β−1 dt ≡ C <∞.
Now let l > 0 be such that l2α(β − 1)(1 + C) < 1, and we are done. �
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Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Email: zlatos@math.uchicago.edu
	1. Introduction and Examples
	2. Some Preliminaries
	3. Proof of Theorem ??: Part I
	4. Proof of Theorem ??: Part II
	References
