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Reaction–diffusion front speed enhancement by flows
Andrej Zlatoš
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA Received 25 May 2009; received in revised form 27 May 2011; accepted 27 May 2011
Available online 20 July 2011
Abstract
We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction–diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from Audoly, Berestycki and Pomeau (2000) [1], Berestycki (2003) [3], Fannjiang, Kiselev and Ryzhik (2006) [11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size.
©2011 Elsevier Masson SAS. All rights reserved.
1. Introduction and the main results
It is well known that the presence of a fluid flow can significantly increase mixing properties of diffusion. The study of this phenomenon, sometimes callededdy diffusivity, has been the aim of a large body of mathematical and physical literature. Questions of long time–large scale behavior are usually addressed via techniques of homogenization theory (see, e.g., [14,21] and references therein). This approach is appropriate when one can wait a long time for mixing to take effect. The presence of other processes, however, may introduce additional time scales to the problem. One such process is reactive combustion, which happens on short time scales and therefore requires a different approach to the study of combustive mixing.
The effects of flows on combustion have recently been studied by various authors, both qualitatively and quantita- tively. The main effects are two-fold. A strong flow can speed up propagation of a reaction (such as a wind spreading a fire) [1,3,5,7,13,15–17,23,25,30,32] but also extinguish it (the “try to light a match in the wind” effect) [8,9,11,18, 27,29,31]. The models used are reaction–advection–diffusion equations, in which the first phenomenon is manifested by the enhancement of speed of their (pulsating) traveling front solutions, and the second by quenching of solutions with (large) compactly supported initial data.
In the present paper we study both these effects for stationary periodic flows. We consider the reaction–advection–
diffusion equation
Tt+u(x)· ∇T =T +f (T ) (1.1)
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0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2011.05.004
for the (normalized) temperature T (t, x)∈ [0,1]of a premixed combustible gas, with(t, x)∈R×Rd. The gas is advected by a periodic incompressible (i.e.,∇ ·u≡0) mean-zero vector fieldu∈C1,δ(Rd)(also calledflow). For the sake of simplicity, we will assume that all periods ofuare 1, that is, the cell of periodicity is thed-dimensional torus Td≡ [−12,12]d, with−12and 12identified. The general periodic case is identical. The non-negativereaction function f ∈C1,δ([0,1])accounts for the increase of temperature due to a chemical reaction such as burning, and is of the combustion type. That is, there isθ∈ [0,1)such thatf (s)=0 fors∈ [0, θ]∪{1},f (s) >0 fors∈(θ,1), andf is non- increasing on(1−δ,1)for someδ >0. Ifθ >0, thenf is anignition reaction(withignition temperatureθ), otherwise f is apositive reaction. A special case of the latter is theKolmogorov–Petrovskii–Piskunov(KPP)reaction[19] with 0< f (s)sf(0)for alls∈(0,1).
Apulsating traveling front in the direction of a unit vectore∈Rd is a solution of (1.1) of the formT (t, x)= U (x·e−ct, x), withc∈Rthefront speedandU:R×Rd→ [0,1]periodic in the second variable such that
s→−∞lim U (s, x)=1 and lim
s→+∞U (s, x)=0, (1.2)
uniformly in x∈Rd. It is well known that under our assumptions on u andf, for eache∈Rd there is a unique c∗e(u, f ) >0 such that a pulsating traveling front in directioneand with speedcexists if and only ifc=c∗e(u, f )for ignition reactions [28], resp.c∈ [ce∗(u, f ),∞)for positive reactions [2].
In both cases we will be interested in the fronts with the unique/minimal speeds c∗e(u, f ). These are the most physical ones because they determine the speed of spreading of solutions to the Cauchy problem for (1.1) with (large enough) compactly supported initial data. An exact formula for c∗e(u, f )has been obtained for general reactions in [10] (and for the special case of KPP reactions also earlier in [4]). Unfortunately, it is a complicated variational expression and it is not obvious how to use it to obtain simple general estimates once∗(u, f ).
Our goal is to derive simple estimates on the front speed c∗e(u, f ), and use them to answer open questions from [1,3,11] concerning speed-up of pulsating fronts and quenching of reaction by strong flows. We will provide such esti- mates in two dimensions, in terms of the size off and theeffective diffusivityDe(u)of the flowu(in the directione).
The latter quantity can be found in a much simpler way thanc∗e(u, f )using (1.5) and (1.6) below. It appears in the homogenization theory which, as mentioned above, is applicable to the study of long time behavior of the solutions of the related linear PDE
ψt+u(x)· ∇ψ=ψ. (1.3)
Despite the fact that the presence of reaction introduces a short time scale to the model, we will be able to show by other methods that in two dimensions (but not in three!), the effective diffusivity determines the front speed up to a bounded factor.
The long time behavior of the solutions of (1.3) is governed by theeffective diffusion equation ψ¯t= ∇ ·
D(u)∇ ¯ψ
, (1.4)
where the (x-independent positive symmetric)effective diffusivity matrixD(u)is obtained as follows. For anye∈Rd, letχe(x)be the periodic mean-zero solution of the cell problem
−χe+u· ∇χe=u·e (1.5)
onTd. ThenD(u)is given by e·D(u)e=
Td
(∇χe+e)·
∇χe+e
dx=e·e+
Td
∇χe· ∇χedx
for any e, e∈Rd. The effective spreading in the direction of a unit vectore∈Rd is now governed by theeffective diffusivity
De(u)≡e·D(u)e=1+ ∇χe2L2(Td). (1.6)
When the nonlinearity in (1.1) is weak (the reaction time scale is large) so that we have Tt+u(x)· ∇T =T +ε2f (T )
withε1, the long time–large space scalingt→t /ε2,x→x/εgives Tt+1
εu x
ε
· ∇T =T +f (T ).
The homogenized version of this equation is T¯t= ∇ ·
D(u)∇ ¯T
+f (T ),¯
with the unique/minimal front speed in directionedepending onDe(u)andf. Although the above approximation holds only on certain space–time scales, it does suggest a relation betweenc∗e(u, f )on one hand andDe(u)andf on the other. Our main result confirms this relation in two dimensions:
Theorem 1.1. Letf be any combustion-type reaction. There are C1(f ), C2(f ) >0 such that for any 1-periodic incompressible mean-zeroC1,δflowuonR2and any unit vectore∈R2,
C1(f )
De(u)c∗e(u, f )C2(f )
De(u). (1.7)
Remarks.1. We have the following estimates on the constants in (1.7). From the results of [25] and monotonicity of c∗e(u, f )inf it follows that
C2(f )C f (s)/s
∞
1+ f (s)/s
∞
(1.8) for someC >0. Ifmζ(f )=min{f (s)|s∈ [ζ,1−(1−8ζ )2]}>0 (for eachf as above there is suchζ ∈(0,1)), then from the proof of Theorem 1.1 it follows that
C1(f )Cζ
mζ(f ) 1+
mζ(f ) (1.9)
for someCζ >0. We note that this estimate is optimal up to a constant for smallmζ(f )(due to (1.8)), but also for largemζ(f ). Indeed,C1(f )is uniformly bounded above inf for KPP reactions [25] and thus for all reactions.
2. In particular,Cj(mf )∼√
m(j =1,2) asm→0 for any fixedf.
3. In the case of KPP reactions, this result has been proved in [23,25], which also implies the upper bound for general reactions (see Remark 1). This case is considerably simpler due to the fact thatc∗e(u, f )=ce∗(u,f ), where˜ f (s)˜ ≡f(0)sandc∗e(u,f )˜ is the minimal front speed for thelinear equation(1.1) withf˜in place off. (Fronts for f˜do not converge to 1 asx·e→ −∞but rather grow exponentially.)
The relation c∗e(u, f )∼C(f )√
De(u) is analogous to that in the case of u≡0 and constant diffusivity matrix D >0. Indeed, spatial scaling x→√
Dx shows that the unique/minimal front speed in the direction efor Tt =
∇ ·(D∇T )+f (T )isC(f )√
e·De(withC(f )≡c∗e(0, f )independent ofe). The problem withu≡0, however, is that the convergence of solutions of (1.3) to those of (1.4) occurs on large time scales, while solutions of (1.1) can be effectively estimated using (1.3) on short time scales only. We will therefore studyshort time diffusivityfor (1.3) in Section 2 and relate it toDe(u)in Theorem 2.1 below. The theorem, which applies in all dimensions, will be a key step towards overcoming this problem ford=2.
The restriction to two dimensions comes from Theorem 3.1 below, a relation between the speed and the width of a front. It turns out, in fact, that this result and Theorem 1.1 are false in dimensionsd3, and Theorem 5.1 shows that the bounds in (1.7) cannot hold with flow-independentC1(f )andC2(f )ford3!
This raises an interesting question about large deviations of the stochastic processXtxfrom (2.1), corresponding to (1.3). IfIe,u is the rate function forZtx,e≡(x−Xtx)·e(i.e., limt→∞t−1lnPΩ(Ztx,e> ct )= −Ie,u(c)forc >0 and anyx), thenIe,u(ce∗(u, f ))=f(0)holds for KPP reactions. From [25] we know that in two dimensions we have c∗e(u, f )/√
De(u)=2
f(0)+O(f(0)3/4)for smallf(0)and KPPf, with an(e, u)-independent error bound. This means thatIe,u(c)≈c2/4De(u)forc√
De(u), in the sense of 4De(u)Ie,u(c)/c2→1 asc/√
De(u)→0, uniformly ine, u. That is, effective diffusivityDe(u)yields a good approximation of the rate functionIe,u(c)forc√
De(u)in two dimensions. However, Theorem 5.1 shows that this is not true for general flows in more dimensions. At this point we do not know how to explain this difference.
Fig. 1. Streamlines of the cellular flowucell.
Motivated by a conjecture from [1,3] (see Corollary 1.3 below), we are particularly interested in the strong flow asymptotics of the unique/minimal speedc∗e(Au, f )for
Tt+Au(x)· ∇T =T +f (T ), (1.10)
with theflow profileuas above andflow amplitudeA∈Rlarge. A natural question is which flow profiles are able to arbitrarily speed up fronts provided their amplitude is large enough (see [1,3,13,17,23,25,27,30,32]). Having proved Theorem 1.1, this question in two dimensions becomes equivalent to the question of the asymptotics ofDe(Au). The latter is much simpler sinceDe(Au)can be computed via (1.5) and (1.6) withAuin place ofu. In particular, it has been proved in [6,12,25] (see, e.g., Proposition 1.2 in [25]) that in any dimension, lim supA→∞De(Au) <∞when the equation
u· ∇φe=u·e (1.11)
has a solutionφe∈H1(Td), and limA→∞De(Au)= ∞when (1.11) has no such solution. We therefore obtain the following characterization.
Corollary 1.2.Letu(x)be a1-periodic incompressible mean-zeroC1,δflow onR2, lete∈R2be a unit vector andf any combustion-type reaction.
(i) If (1.11)has a solutionφe∈H1(T2), then lim sup
A→∞ c∗e(Au, f ) <∞. (1.12)
(ii) If (1.11)has no solutions inH1(T2), then
Alim→∞c∗e(Au, f )= ∞. (1.13)
Remark.This result has been proved in [25] for KPP reactions but is new for generalf.
Of particular interest have recently been bothpercolatingandcellular flows. Percolating flows possess streamlines (solutions of the ODE X=u(X)) joining x·e= −∞ andx·e= ∞, a special case being shear flowsu(x)= v(x2, . . . , xd)e1. Existence of such streamlines has obviously a strong effect on speed-up of fronts. Cellular flows, on the other hand, possess only closed streamlines, a prototypical example beingucell(x)= ∇⊥(sin 2π x1sin 2π x2)whose streamlines are depicted in Fig. 1. Their effect on speed-up of fronts is therefore more subtle, with diffusion across a thin boundary layer near the flow separatrices playing an important role. The interest in these flows stems from them being ubiquitous in nature. They appear as a result of instabilities in fluids such as Rayleigh–Bénard instability in heat convection, Taylor vortices in a Couette flow between rotating cylinders, or heat expansion driven Landau–Darrieus instability.
Corollary 1.2 shows speed-up of fronts in the sense of (1.13) for both percolating and cellular flows but known estimates on the effective diffusivity and Theorem 1.1 yield more precise asymptotics. For percolating flows
0<lim inf
A→∞
ce∗(Au, f )
A lim sup
A→∞
c∗e(Au, f )
A <∞ (1.14)
has been conjectured in [1] and later proved for allf in [17], and can also be recovered from Theorem 1.2 in [32] and Theorem 1.1. The asymptoticc∗e(Aucell, f )∼A1/4for the above cellular flow has also been conjectured in [1,3] and obtained in [23] for KPP reactions, but the best result for general reactions has beenA1/5c∗e(Aucell, f )A1/4for e=e1[17].
Using Theorem 1.1 and the estimateDe(Au)∼A1/2from [20], we now obtain the conjectured asymptotic from [1,3] for general incompressible periodic cellular flows and allf. We also need to assume, as in [20], that the stream function of the flow has only non-degenerate critical points (otherwise the result is not true in general). That is, there is a periodicC2,δfunctionH:R2→Rwith only non-degenerate critical points such thatu= ∇⊥H≡(−Hx2, Hx1)and the complement of the level setH=0 has only bounded connected components (flow cells). Notice that this allows for almost arbitrary periodic geometry of the flow cells. We will call suchuperiodic non-degenerate cellular flows.
Corollary 1.3.Consider a1-periodic non-degenerate cellular flowuonR2, lete∈R2be a unit vector, andf any combustion-type reaction. Then
0<lim inf
A→∞
ce∗(Au, f )
A1/4 lim sup
A→∞
c∗e(Au, f )
A1/4 <∞. (1.15)
As mentioned above, we also address the closely related question of quenching of reaction by cellular flows.
Quenching occurs in the Cauchy problem for (1.1) with initial dataT (0, x)=T0(x)whenT (t,·)∞→0 ast→ ∞.
This happens for ignition reactions whenT0is small in some sense so thatT (τ,·)becomes uniformly smaller than the ignition temperature for someτ >0 (and thusT solves (1.3) fort > τby the maximum principle). But sometimes even large initial data can be quenched with the help of enhanced diffusion due to mixing by strong flows.
It has been proved in [11] that the flowAucell quenches initial data supported in a strip of widthLprovided the amplitudeAL4lnL. The authors also conjectured that the factor lnLcan be removed (heuristically, the minimal quenching amplitude should be such thatc∗e(Au, f )∼L, i.e., A∼L4). We prove this conjecture for all periodic non-degenerate cellular flows onR2, which are also symmetric across thex2axis (i.e., the stream functionH is odd inx1). We will call such periodic non-degenerate flows, which includeucell,symmetric.
Theorem 1.4.Consider a1-periodic symmetric non-degenerate cellular flowuonR2. There areγθ>0 (independent ofu)andCu,θ>0such that iff is an ignition reaction with ignition temperatureθ >0andf (s)/s∞γθ, then initial dataT0(x)∈ [0,1]supported in[−L, L] ×Rare quenched wheneverACu,θL4.
Remarks.1. We note that the boundγθ onf is in fact necessary. It is easy to show that iff is large enough, then even a single cell with Dirichlet boundary conditions can support the reaction for anyA[11].
2. As in [11], scaling shows that f (s)/s∞γθ can be replaced by the requirement that the period ofu be smaller thanf (s)/s1/2∞ γθ−1/2.
The rest of the paper is organized as follows. In Section 2 we introduce our main technical tool, Theorem 2.1, relat- ing short and long time diffusivity of (1.3) in any dimension. In Section 3 we prove Theorem 3.1, a relation between the speed and the width of a front in two dimensions, and then Theorem 1.1. In Section 4 we prove Theorem 1.4 and in Section 5 we provide a counterexample to Theorem 1.1 in three and more dimensions.
2. Short time diffusivity of periodic flows
In this section we show that there is a close relation between short- and long-time diffusivity of the parabolic operator in (1.3). The results contained here are valid in any dimension.
Consider the stochastic processXtxstarting atx∈Rdand satisfying the stochastic differential equation dXxt =√
2dBt−u Xtx
dt, X0x=x, (2.1)
whereBt is a normalized Brownian motion onRd withB0=0 (defined on a probability space (Ω,B∞,PΩ)). By Lemma 7.8 in [24], we have that ifψsolves (1.3) withψ (0, x)=ψ0(x), then
ψ (t, x)=
Rd
ψ0(y)kt(x, y) dy=EΩ
ψ0
Xxt
, (2.2)
withkt(x, y)the fundamental solution for (1.3) andEΩ expectation with respect toω∈Ω. That is,kt(x,·)is the density forXtx.
We also have that ifT solves (1.1) withT (0, x)=T0(x)andψ0(x)=T0(x)0, then by the comparison princi- ple [26], for allt, x,
0ψ (t, x)T (t, x)etf (s)/s∞ψ (t, x). (2.3)
We will use this relation to obtain short time upper estimates on the solution of (1.1).
We start the study of short time diffusivity for (1.3) with noting that uniformly inx∈Rd,
tlim→∞EΩ
|(Xxt −x)·e|2 2t
=De(u). (2.4)
This is based on the fact that EΩXxt −x
·e2
=
Rd
(y−x)·e2kt(x, y) dy=
Rd
(√
t y−x)·e2td/2kt(x,√ t y) dy
and the following estimates onkt(x, y):
td/2kt(x,√
t y)→k∗1(0, y) inL2 Rd
ast→ ∞, uniformly inx;
0kt(x, y)Ct−d/2e−|x−y|2/Ct for someu-dependentC >0.
The first is the standard homogenization limit (see, e.g., [14,21]) with kt∗(x, y)≡ 1
√det(D(u)) (4π t )d/2e−(y−x)·D(u)−1(y−x)/4t
the fundamental solution of (1.4), the second is the Nash–Aronson estimate for mean-zero flows (see, e.g., [22]). The limit (2.4) now follows from
Rd
|y·e|2k1∗(0, y) dy=
Rd
|√
D(u) z·e|2
(4π )d/2 e−z2/4dz=
Rd
|z·√
D(u) e|2
(4π )d/2 e−z2/4dz=2De(u).
The main result of this section is a lower bound on short time diffusivity for (1.3):
Theorem 2.1.There isC >0such that for anyτ1, any1-periodic incompressible mean-zero Lipschitz flowuand anyα >0there arex∈Rdandt∈ [0, τ]such that
PΩXxt −x
·eα
τ De(u)
1−Cα. (2.5)
Proof. We first note that ifΩ˜ ≡Td×Ω is equipped with the product probability measure (i.e., the first coordinate x∈Td is uniformly distributed), then (2.4) gives
tlim→∞EΩ˜
|(Xxt −x)·e|2 2t
=De(u), (2.6)
where the expectation is with respect to(x, ω)∈ ˜Ω. Then we have
Lemma 2.2.There isC >˜ 0such that for anyτ 1anduas in Theorem2.1, EΩ˜Xxτ−x
·e2
Cτ D˜ e(u). (2.7)
Proof. Let us first assumeτ=1. Forx∈Td, letX˜xt ≡Xxt mod 1∈Td be the process corresponding to (1.3) onTd. We note thatX˜xt is uniformly distributed overTdas a random variable onΩ˜. Indeed, ifB⊆Tdandψ0(x)=χB(x), then for eacht0,
PΩ˜X˜tx∈B
=
Td
ψ (t, x) dx=
Td
ψ (0, x) dx= |B|
because the evolution (1.3) preserves the total mass ofψ.
We next letYtx≡Xtx−Xtx−1andZxt ≡Ytx·e. Then periodicity ofuimplies Ytx=XX
x t−1
1 −Xxt−1=XX˜
x t−1
1 − ˜Xxt−1
in law. Since theX˜xt−1for allt 1 are identically distributed as random variables on Ω˜, the same is true for the increment displacementsYtxas well as theZxt. In particular, for eacht1,
EΩ˜Ztx2
=EΩ˜Z1x2
. (2.8)
We also have(XxN−x)·e=N
n=1Zxnfor anyN∈Nand so by (2.6), N
n,m=1
EΩ˜ ZxnZxm
=2N De(u)+o(N ). (2.9)
Since|EΩ˜(ZxnZxm)|EΩ˜(|Z1x|2)is obvious from (2.8) and the Schwarz inequality, we will obtain (2.7) forτ=1 if we can show the existence ofu-independentM∈Nandγ >0 such that
EΩ˜
ZnxZmx2−γ|m−n|EΩ˜Zx12
(2.10) whenever|m−n|M+1.
We denoteht(x, y)≡
j∈Zdkt(x, y+j )the fundamental solution for (1.3) onTd, withkt from (2.2). We then have
Td
ht(x, y) dy=
Td
ht(x, y) dx=1. (2.11)
By Lemma 5.6 in [9], there isM∈Nsuch that for alluas above,hM(·,·)−1∞12. The maximum principle gives ht(·,·)−1∞hs(·,·)−1∞fortsand this together with(ht−1)∗(hs−1)=ht+s−1 (from (2.11)) implies for allmMandγ≡(2M)−1,
hm(·,·)−1
∞2−γ (m+1). (2.12)
As a final prerequisite, we note that
(Td)2
j∈Zd
(y+j−x)kt(x, y+j ) dx dy=0. (2.13)
This can be obtained by taking the initial datumψ0(x)=χTd(x)in (1.3) onRdand evaluating d
dt
Rd
xψ dx=
Rd
xψ−x∇ ·(uψ ) dx=
Rd
uψ dx=
Td
u(x)
j∈Zd
ψ (t, x+j ) dx=0, where we used integration by parts (with the Nash–Aronson estimate forψ), the fact that
j∈Zdψ (t, x+j )≡1 and ubeing mean-zero. Thus for eacht0 (recall thatTd= [−12,12]d),
0=
Rd
xψ (t, x) dx=
Rd×Td
xkt(x, y) dx dy=
Rd×Td
(x−y)kt(x, y) dx dy
because
Rdykt(x, y) dx=y. Now (2.13) follows from periodicity:
0=
(Td)2
j∈Zd
(x+j−y)kt(x+j, y) dx dy= −
(Td)2
j∈Zd
(y−j−x)kt(x, y−j ) dx dy.
In what follows we denotexe≡x·eforx∈Rd. Form−nM+1 we have bykt∗ks=kt+s, EΩ˜
ZxnZxm=
Td×(Rd)4
(xn−xn−1)e(xm−xm−1)ekn−1(x, xn−1)k1(xn−1, xn)
×km−1−n(xn, xm−1)k1(xm−1, xm) dx dxn−1dxndxm−1dxm
=
(Td)5
j,l∈Zd
(xn+j−xn−1)e(xm+l−xm−1)ehn−1(x, xn−1)k1(xn−1, xn+j )
×hm−1−n(xn, xm−1)k1(xm−1, xm+l) dx dxn−1dxndxm−1dxm
. Here we have used the fact that
p,q∈Zd
(xm+p−xm−1−q)ekm−1−n(xn, xm−1+q)k1(xm−1+q, xm+p)
=
l∈Zd
(xm+l−xm−1)e
q∈Zd
km−1−n(xn, xm−1+q)k1(xm−1+q, xm+l+q)
=
l∈Zd
(xm+l−xm−1)ek1(xm−1, xm+l)
q∈Zd
km−1−n(xn, xm−1+q)
=
l∈Zd
(xm+l−xm−1)ehm−1−n(xn, xm−1)k1(xm−1, xm+l) (due to periodicity ofu) and similarly
p,q∈Zd
(xn+p−xn−1−q)ekn−1(x, xn−1+q)k1(xn−1+q, xn+p)
=
j∈Zd
(xn+j−xn−1)ehn−1(x, xn−1)k1(xn−1, xn+j ).
The integral with respect to x can now be eliminated along withhn−1(x, xn−1)because
Tdhn−1(x, xn−1) dx=1.
Notice that (2.13) gives
(Td)2
l∈Zd
(xm+l−xm−1)ek1(xm−1, xm+l) dxm−1dxm=0,
and so (2.12) and Schwarz inequality imply 2γ (m−n)EΩ˜
ZxnZxm
=2γ (m−n)
(Td)4
j,l∈Zd
(xn+j−xn−1)e(xm+l−xm−1)ek1(xn−1, xn+j )
×
hm−1−n(xn, xm−1)−1
k1(xm−1, xm+l) dxn−1dxndxm−1dxm
(Td)4
j,l∈Zd
(xn+j−xn−1)e2k1(xn−1, xn+j )k1(xm−1, xm+l) dxn−1dxndxm−1dxm 1/2
×
(Td)4
j,l∈Zd
(xm+l−xm−1)e2k1(xn−1, xn+j )k1(xm−1, xm+l) dxn−1dxndxm−1dxm 1/2
=
(Td)2
j∈Zd
(xn+j−xn−1)e2k1(xn−1, xn+j ) dxn−1dxn
1/2
×
(Td)2
l∈Zd
(xm+l−xm−1)e2k1(xm−1, xm+l) dxm−1dxm 1/2
=EΩ˜Zxn21/2
EΩ˜Zxm21/2
.
Now (2.8) yields (2.10), finishing the proof forτ =1. The general case is identical, this time withYnx beingXnτx − X(nx−1)τ, the sameγ,MandC, and 2N˜ replaced by 2N τin (2.9) (one actually getsC˜→2 asτ→ ∞). 2
We will now prove (2.5) withC≡10C˜−1/2by contradiction. Assume that for someuthere areτ1 andα >0 such that for anyx∈Rd and anyt∈ [0, τ],
PΩXtx−x
·e< α
τ De(u)
> Cα. (2.14)
We first claim that (2.14) implies for the aboveCand eachx∈Rd, PΩ
∀t∈ [0, τ]: Xxt −x
·e<4√ τ De(u)
C
1
2. (2.15)
Indeed, if this is not true, letx be such that the probability in (2.15) is less than 12. This means that there is a subset Ω⊆Ω of measure more than 12 such that if tj(ω)0 is the first time the (almost surely continuous in t) path Xtx=Xtx(ω)hits the set
Hj≡
y∈Rd (y−x)·e=2j α
τ De(u) ,
thentj(ω)τ for eachω∈Ω andj =0, . . . ,Cα2 . Now the strong Markov property of the processXtx, the fact thatτ−tj∈ [0, τ], and (2.14) imply that forj=0, . . . ,Cα2 the following conditional probability satisfies
PΩXτx−x
·e∈
(2j−1)α
τ De(u), (2j+1)α
τ De(u) Btj
> CαχΩ(ω),
withBtj theσ-algebra corresponding to the stopping timetj. Thus PΩXτx−x
·e∈
(2j−1)α
τ De(u), (2j+1)α
τ De(u)
>Cα 2
forj =0, . . . ,Cα2 . SinceCα2 (Cα2 +1) >1, this is a contradiction, thus proving (2.15).
Now (2.15) and the almost sure continuity ofXxt intshow for eachx∈Rdand eachj1, PΩ
Xτx−x
·e4j√ τ De(u)
C
1
2 j
.
That, however, means EΩ˜Xτx−x
·e2
16τ De(u) C2
j1
j2 1
2 j−1
− 1
2 j
=96τ De(u)
C2 <Cτ D˜ e(u), contradicting (2.7). This finishes the proof of Theorem 2.1. 2
3. Pulsating front speed in 2D
We will now prove Theorem 1.1. The upper bound in (1.7) is immediate from the results in [25]. Indeed, the same bound has been proved there for KPP reactions withC2(f )=C
f(0)(1+
f(0)). One thus only needs to apply this result to some KPPf˜such thatf f˜andf (s)/s∞= ˜f(0), and use the fact thatc∗e(u, f )c∗e(u,f )˜ due to f f˜.
We will now prove the lower bound in (1.7) using Theorem 2.1 coupled with the following bound on the width of the front in terms of its speed. We note that we can assumeTt>0. This has been proved in [2] for all ignition reactions (and also for positive reactions withf(0) >0), and the lower bound in (1.7) for any ignition reactionf˜f proves the lower bound forf because againc∗e(u,f )˜ c∗e(u, f ).
Theorem 3.1.There isC0>0such that for any1-periodic incompressible mean-zeroC1,δ flowu onR2, any unit vectore∈R2, and any combustion-type reactionf the following holds. Iff mχ[ζ,ξ]for somem >0and0< ζ <
ξ <1,ε∈(0, (ξ−ζ )/2), andT (t, x)is a pulsating front for(1.1)with speedc >0andTt >0, then there isz∈R such that for anyt∈Rany connected setBwith
B⊆
x∈R2x·ez+ct+cC0
m−1+ε−2
+2andT (t, x)ζ+ε
≡Bt+ or
B⊆
x∈R2x·ez+ct−cC0
m−1+ε−2
−2andT (t, x)ξ−ε
≡Bt− satisfiesdiam(B)101.
Remark.The above form of this result will be sufficient for our purposes. Its proof in fact shows that the set ofx such thatT (t, x)ζ +εresp.T (t, x)ξ −εcovers less than 1% of any unit square lying in the half-planex·e z+ct+cC0(m−1+ε−2)+2 resp.x·ez+ct−cC0(m−1+ε−2)−2 (and this bound decreases asC0increases).
That is, for any t, outside of a spatial strip of width 2cC0(m−1+ε−2)+4, values ofT (t,·)are mostly outside of [ζ +ε, ξ−ε]. This yields a bound on the width of the front in terms of its speed.
Proof. LetT (t, x)=U (x·e−ct, x)withU (s, x)satisfying (1.2), so that
−cUs+u· ∇xU+u·eUs=xU+Uss+2e· ∇xUs+f (U ). (3.1) Integrating this overΓ ≡R×Γ0≡R× [0,1]2and using 1-periodicity ofU inx, (1.2), andubeing incompressible and mean zero, we get
Γ
f
U (s, x)
ds dx=c. (3.2)
Similarly, multiplying (3.1) byUand integrating overΓ yields c
2+
Γ
|∇xU+eUs|2ds dx=
Γ
f
U (s, x)
U (s, x) ds dx.
The right hand side is bounded above bycthanks to (3.2) and so
Γ
f T (t, x)
dt dx=1, (3.3)
Γ
∇xT (t, x)2dt dx1
2. (3.4)
LetT (t )¯ ≡
Γ0T (t, x) dx∈ [0,1]so thatT¯t >0 becauseTt >0. Assume thatT (t )¯ ∈ [ζ +2ε, ξ−ε2]for somet. Then the Poincaré inequalityT (t,·)− ¯T (t )L2(Γ0)C1∇xT (t,·)L2(Γ0)for someC12 gives eitherT (t,·)−
