Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton–Jacobi models

Ann. I. H. Poincaré – AN 30 (2013) 1049–1068

www.elsevier.com/locate/anihpc

Sharp asymptotic growth laws of turbulent flame speeds in cellular flows by inviscid Hamilton–Jacobi models

Jack Xin, Yifeng Yu

Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA Received 10 July 2012; accepted 24 November 2012

Available online 23 January 2013

Abstract

We study the large time asymptotic speeds (turbulent flame speedss_T) of the simplified Hamilton–Jacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L¹type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L²type) inviscid HJ model of Majda–Souganidis derived from the Kolmogorov–Petrovsky–Piskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995)[10], we compare the turbulent flame speedss_T’s from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization.

Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:70H20; 76M50; 76M45; 76N20

Keywords:Inviscid Hamilton–Jacobi equations; Cellular flows; Sharp front speed asymptotics

1. Introduction

Turbulent combustion is a complex nonlinear and multiscale dynamic phenomenon[25,29,24,23,13,14,21,26]. The first principle approach requires a system of reaction–diffusion–advection equations coupled with the Navier–Stokes equations. Progress in theoretical understanding and efficient modeling of the turbulent flame propagation however often relies on simplified models such as the advective Hamilton–Jacobi equations (HJ) and passive scalar reaction–

diffusion–advection equations (RDA), as documented in books[25,21,27]and research papers[1,2,7,8,10,14,16,20, 22–24,26,29].

✩ The first author was partially supported by NSF grants DMS-0911277, DMS-1211179. The second author was partially supported by DMS- 0901460 and NSF CAREER award DMS-1151919.

* Corresponding author.

E-mail addresses:jxin@math.uci.edu(J. Xin),yyu1@math.uci.edu(Y. Yu).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.11.004

Fig. 1. Left: G-equation model. Right: Majda–Souganidis model.

•G-equation model:A popular phenomenological approach in turbulent combustion is the level set formulation[20]

of flame front motion laws with the front width ignored[21]. The motion law is in the hands of a modeler based on theory and experiments. The simplest motion law is that the normal velocity of the front (Vn) is equal to a constantsl

(the laminar speed) plus the projection of fluid velocity V (x, t )along the normal^−→n. See Fig.1 (left picture). The laminar speed is the flame speed due to chemistry (reaction–diffusion) when fluid is at rest. As the fluid becomes active, the flame front will be wrinkled by the fluid velocity. However it is observed under suitable conditions that the front location eventually moves to leading order at a well-defined steady speeds_T in each specified direction, which is the so-called “turbulent burning velocity”. The study of the existence and properties of the turbulent flame speeds_T is a fundamental problem in turbulent combustion theory and experiments[25,22,21]. Let the flame front be the zero level set of a functionG(x, t ), then the normal direction isDG/|DG|, the normal velocity is−G_t/|DG|. The motion law becomes the so-called G-equation in turbulent combustion[25,21]:

G_t+V (x, t )·DG+s_l|DG| =0. (1.1)

Chemical kinetics and diffusion rates are all included in the laminar speedslwhich is provided by a modeler. Formally under the G-equation model, for a specified unit directionp,

s_T(p)= − lim

t→+∞

G(x, t ) t .

HereG(x, t )is the solution of Eq. (1.1) with initial dataG(x,0)=p·x. The existence of sT has been rigorously established in[28]and[4]independently for incompressible periodic flows (the assumptions in[4]are more general), and[15]for two-dimensional incompressible random flows. Very recently, Cardaliaguet and Souganidis[5]proved the homogenization of the G-equation for stationary ergodic flows in any dimension. For simplicity, we assume that V =V (x)is spatially periodic and has no time dependence. Thens_T =s_T(p)is equal to the effective Hamiltonian of the following cell problem

sl|p+DG| +V (x)·(p+DG)=sT(p). (1.2)

HeresT(p)is the unique number such that the above equation admits periodic approximate solutions. ChangeVtoAV for some positive constantA(turbulence intensity). A very important problem in turbulent combustion is to study the dependence of the turbulent flame speed on A(i.e.,sT =sT(A)). Interestingly, for the cellular flow (a prototypical flow in dynamo and convection-enhanced diffusion problems[6,9]):

V (x)=V (x1, x2)=(−H_x2, H_x1), H=sinx1sinx2, (1.3) it is known[1,17,19]thats_T =O(A/logA)in the limit of largeA, which also shows the weak bending phenomenon.

•RDA model:The passive scalar reaction–diffusion–advection (RDA) model is:

u_t+V (x)·Du=κu+ 1

τ_rf (u), x∈Rⁿ, (1.4)

whereurepresents the reactant temperature or concentration,Dis the spatial gradient operator,V (x)is a prescribed fluid velocity,f is a nonlinear reaction function;κ is the molecular diffusion constant,τ_r>0 is reaction time scale.

In this paper, we will choose the Kolmogorov–Petrovsky–Piskunov–Fisher (KPP–Fisher) type reaction functionf.

A typical example isf (u)=Cu(1−u)for someC >0. Under this model, the turbulent flame speedc^∗_T(p)along a given directionpis defined to be the large time spreading rate of solution from compactly supported non-negative initial data[16]. In case of periodic flow, it also agrees with the minimal traveling wave speed[3,27]. It is known even for more general time dependent (stationary and ergodic) flows[26,3,16,27]thatc_T^∗(p)obeys a variational principle in terms of the large time growth rate of a viscous quadratically nonlinear Hamilton–Jacobi equation (QHJ). In spatially periodic flows,

c^∗_T(p)=inf

λ>0

τ_r⁻¹f(0)+H^∗(pλ)

λ , (1.5)

whereH^∗is the effective Hamiltonian of the following cell problem

−κW+V (x)·(p+DW )+ |p+DW|²=H^∗(p).

• Majda–Souganidis model: Turbulent combustion always involves small scales. Following the model proposed by Majda and Souganidis[13], we write V =V (^xα)for some α∈(0,1]. In[13], the authors actually considered more general case, i.e.,V is in a scale-separation formV =V (x, t,^xα,^tα). The flame thickness is in general much smaller than the turbulence scale. As in[13], we setκ=dandτ_r=. See Fig.1(right picture). Suppose thatT = T (x, t )is the solution of (1.4) with compactly supported initial dataT (x,0). The limiting behavior ofT =Tis[13]:

lim→0T=0 locally uniformly in{(x, t): Z <0}andT→1 locally uniformly in the interior of{(x, t): Z=0}, whereZ∈C(Rⁿ× [0,+∞))is the unique viscosity solution of the variational inequality

max

Zt− ˆH (DxZ)−f(0), Z

=0, (x, t )×Rⁿ×(0,+∞),

with initial dataZ(x,0)=0 in the support of T (x,0), and Z(x,0)= −∞otherwise. The effective Hamiltonian Hˆ = ˆH (p)is defined as a solution of the following cell problem: for eachp∈Rⁿ, there are a unique numberH (p)ˆ and a functionF (x)ˆ ∈C^0,1(Tⁿ)such that

a(α)dFˆ+d|p+DFˆ|²−V (x)·(p+DF )ˆ = ˆH (p), (1.6) a(α)=1 if α=1 anda(α)=0 ifα∈(0,1). The setΓt =∂{x∈Rⁿ: Z(x, t ) <0}can be viewed as a front which moves with normal velocity which is the turbulent flame speed predicted by Majda–Souganidis model:

v_n=c_T(n).

In order to be consistent with the G-equation, we choosento be the unit normal vector pointing to the propagation direction, i.e.,n= −_|^DZ_DZ|. Then

c_T(p)=inf

λ>0

f(0)+H (pλ)

λ , (1.7)

with

H (p)= ˆH (−p).

Note whenα=1 (viscous case),H is the same asH^∗in (1.5) andc_T(p)=c^∗_T(p). Ifα∈(0,1),H is the effective Hamiltonian of the following inviscid QHJ (F-equation):

d|p+DF|²+V (x)·(p+DF )=H (p). (1.8)

In this paper, we show the difference between the inviscids_T andc_T in cellular flows (1.3) through their sharp asymptotics at largeA. Therefore we sets_l=2

df(0). In the paper[10]by Embid, Majda and Souganidis, the com- parison ofs_T andc_T for periodic shear flows on non-zero mean showedc_T > s_T under certain conditions of the mean flow. At the end of[10], the authors raised the following question: “It is very interesting to develop further compar- isons of enhanced flame speeds between the complete nonlinear averaging theory summarized in Section2and the averaged G-equation from Section3for more realistic periodic flow fields such as arrays of vortices”. The following is our main result which provides an answer to this question at least for two-dimensional cellular flow. For clarity of presentation, we assumes_l=d=1 andf(0)=¹₄as in[10].

Theorem 1.1.For theV given by(1.3), scaleV (x)toAV (x). Consider the inviscid G-equation front speeds_T (1.2) and the inviscid KPP front speedc_T (1.7)–(1.8)in cellular flows(1.3). Letp=(p₁, p₂)be a unit vector. There exist positive constants0< C₁C₂independent ofAandpsuch that forA4,

Aπ(|p₁| + |p₂|)

2 logA+C₂ s_T(p, A)Aπ(|p₁| + |p₂|)

2 logA+C₁ . (1.9)

Also, there exist positive constantsC₃,C₄andA₀4independent ofAandpsuch that whenAA₀ Aπ(|p₁| + |p₂|)

2 logA−log logA+C₃ c_T(p, A) Aπ(|p₁| + |p₂|)

2 log(A)−2 log logA−C₄. (1.10)

Note that (1.9) and (1.10) imply that

A→+∞lim

log(A)sT(p, A)

A = lim

A→+∞

log(A)cT(p, A)

A =π

2

|p1| + |p2| ,

and

cT(p, A)−sT(p, A)=O

Alog logA log²A

, asA→ +∞.

For generaln-dimensional incompressible flow, we have the following relatively rough comparison.

Theorem 1.2.Assume thatV (x):Tⁿ→Rⁿis periodic and incompressible. Suppose thats_T(p, A)=O(_logA^A ). Then

c_T(p, A)=O A

logA

.

The proof of Theorem 1.1 is much more delicate and the symmetric structure of the stream function H = sinx₁sinx₂around hyperbolic critical points will play essential role. We would like to mention that for the 2d cellular flow, the turbulent flame speed predicted by the RDA model (1.5) obeys the growth law ofO(A¹⁴)[18].

Remark 1.1(Difference between cellular flow and shear flow).According to inf-max formulas, s_T(p, A)= inf

φ∈C¹(Tⁿ)max

Tⁿ

|p+Dφ| +AV (x)·(p+Dφ)

(1.11) and

H (p, A)= inf

φ∈C¹(Tⁿ)

maxTⁿ

|p+Dφ|²+AV (x)·(p+Dφ)

. (1.12)

HereTⁿis then-dimensional flat Torus. For the specific shear flowV (x₁, x₂)=(v(x₂),0)andp=(1,0). It is easy to see thats_T(p, A)=1+Amax_T1vandH (λp, A)=λ²+Aλmax_T1v. Then

cT(p, A)=1+Amax

T¹ v=sT(p, A).

NotecT will never exceedsT no matter how largeAis. This is different from the cellular flow case.

The paper is organized as follows. In Section2, we provide some straightforward comparison results based on inf- max formulas. In particular, we show thats_T c_T and limA→+∞sT

A =limA→+∞cT

A. In Section3, we give the proof of Theorem1.1by estimating the travel time of the controlled characteristics in the Lagrangian representation of the G-equation, and the Lax formula[12]of the quadratically nonlinear F-equation (1.8). In order to establish the upper bound ofc_T(p, A), we prove an almost sharp estimate of sup_R2|DF|based on the (Eulerian) corrector equation (1.8) withV replaced byAV through very delicate analysis. In Section4, we prove Theorem1.2. Concluding remarks are in Section5.

Assumption and Notations.

(1) Throughout this section,A4 and|p| =1. Also,C,Cˆ,C,˜ C₁andC₂represent positive constants independent ofAandp. Moreover, we setd=s_l=1 andf(0)=¹₄. For convenience, we also use notations likeO(Φ(A)) which meansCΦ(A). HereΦ(A)is a function ofA.

(2) To simplify notations, we omit the dependence onpand write

⎧⎨

⎩

s_T(p, A)=α_A, H (p, A)=β_A, cT(p, A)=γA.

(3) Tⁿisn-dimensional flat Torus.f ∈C^k(Tⁿ)means that f ∈C^k(Rⁿ)and is periodic. Sometimes we also iden- tifyTⁿwith the cube[−¹₂,¹₂]ⁿ.

2. Some simple comparisons based on inf-max formulas

The following says that the G-equation model always predicts slower turbulent flame speeds than the Majda–

Souganidis model.

Lemma 2.1.

α_A(p)γ_A(p)β_A(p)+1 4.

Proof. The right inequality is obvious by choosingλ=1 in (1.7). Let us prove the left inequality. Sincet²t−¹₄, according to the inf-max formulas (1.11)–(1.12),

β_A(p)α_A(p)−1 4.

Combining with the degree 1 homogeneity ofα_A(p)with respect to thepvariable, the above lemma holds. 2 The following theorem says thatα_A/A,β_A/Aandγ_A/Ahave the same asymptotic limit.

Theorem 2.1.Givenp∈Rⁿ. Denote c_p= inf

φ∈C¹(Tⁿ)

maxTⁿ

V (x)·(p+Dφ) .

Then

A→+∞lim α_A

A = lim

A→+∞

β_A

A = lim

A→+∞

γ_A A =c_p.

In particular, G-equation and Majda–Souganidis models predict the bending effect simultaneously.

Proof. The proof is simple. Owing to the inf-max formula (1.11), αA

A c_p.

Now fix >0 and chooseφ∈C¹(Tⁿ)such that maxTⁿ

V (x)·(p+Dφ)

+c_p. Then

α_A A 1

Amax

Tⁿ |Dφ| +c_p+.

Fig. 2. An illustration of a controlled trajectoryξtraveling between two vortices of the cellular flow in the proof of Lemma3.1.

Hence

A→+∞lim α_A

A =c_p.

The proof forβ_Ais similar. The proof forγ_Afollows immediately from Lemma2.1. 2 3. Proof of Theorem 1.1

Throughout this section, we assume thatV (x)=V (x1, x2)=(−Hx₂, Hx₁)forH=sinx1sinx2. Moreover, in the section, a function f is periodic iff (x+2πk) =f (x)for any k∈Z². The cellular flow is written in the scaled formA(x). The solution of Eq. (1.1) with initial datau₀(x)is given by a control representation formula:

G(x, t )= inf

ξ∈Wx

u₀ ξ(t )

,

whereWxis the collection of allξ∈W^1,∞([0, t];Rⁿ)such thatξ(0)=xand:

ξ(τ )+A V ξ(τ )

=y(τ ),∀y(τ )∈L^∞ [0, t]

, |y|1,

where the function y is the dynamic control. The formula (3.13) is an extension of the well-known Lax formula [12,11,27]for strictly convex Lagrangian (L^q,q >1) to theL^∞type Lagrangian due toL¹type Hamiltonian in the G-equation[16,28,4,15]. We are going to use this formula to boundα_A. Due to the symmetry of the stream function and the inf-max formula (1.11),

α_A(p)=α_A(−p)= lim

t→∞

1 t sup

ξ∈Wx

p·ξ(t )

. (3.13)

Note that the limit is independent of the choice ofx (i.e., the initial positionξ(0)). Moreover, the symmetry is just used for convenience and is not essentially necessary. We shall first estimate the travel time of a controlled trajectoryξ passing through the two vortices in cellular flow as shown in Fig.2. The controly(τ )makes possible the passage of theξ-trajectory around the saddle points(π,0)and(π, π ).

Lemma 3.1.GivenA4, there exist a positive constantTand a Lipschitz continuous curveξ: [0,+∞)→R²such that

(i)

˙ξ (t )+AVξ(t )1 a.e.; (ii) T^logA_A^+C and fork∈N∪ {0}

ξ kT

= π

2,0

+k π

2,π 2

.

HereC >0is a constant which is independent ofA.

Proof. Throughout the proof, C represents a positive constant independent ofA. Due to symmetry, it suffices to construct a pathξ: [0, T] →R²such thatξ(0)=(^π₂,0),ξ(T)=(π,^π₂)(see Fig.2), and

TlogA+C

A .

In order to save notations,x1(t)andx2(t)in the following different steps represent different curves.

Step 1.Letη1: [0,∞)→Rsatisfyη(0)=^π₂ andη(t)˙ =Asinη+1. Suppose thatη1(t1)=^2π₃ . Clearly

t₁=

2π

3

π 2

dx

Asinx+1 C A.

Step 2.Fortt₁, letη₂: [t₁,∞)→R²satisfyingη₂(t₁)=(^2π₃,0)and

˙

η₂(t)= −AV η₂(t)

+ DH

|DH|. DenoteL1=min_|x−(^2π

3 ,0)|¹₂|DH (x)|>0,L2=min_|x

2−^π₃|¹₂||DH (x)|>0 andL=min{L1, L2,1}. Lett2be the moment whenH (η₂(t₂))=_4A^L. Our claim is thatt₂−t_14A¹ . In fact, fort∈ [t₁, t₁+_4A¹ ],

η₂(t)− 2π

3 ,0 1

2

andH (η₂(_4A¹ ))_4A^L. Hence our claim holds. Denoteη₂(t)=(x₁(t), x₂(t)). Note that fort∈ [t₁, t₂]

˙

x₁(t)Acosx₂(t)sinx₁(t)−14×cosπ 6 sinπ

6 −1>0.

Hence x₁(t₂)∈

2π 3 , π

. (3.14)

Also it is clear thatx₂(t₂)∈(0,¹₂).

Step 3.Fortt₂, letη₃: [t₂,+∞)→R²satisfyη₃(t₂)=η₂(t₂)and

˙

η3(t)= −AV η3(t)

.

Assume thatη₃(t)=(x₁(t), x₂(t)). Chooset₃to be the first time such thatη₃(t₃)·e₂=^π₃. Our claim is t₃logA+C

A .

In fact fort∈ [t2, t3], sinx₁(t)sinx₂(t)=λ

forλ=_4A^L, whereLis the same as in Step 2. Sincex2changes from ^C_A to^π₃ and|cosx1(t)|¹₂fort∈ [t2, t3],

t3−t2=

π

3

Cλ

1 Asinu

1−_sin^λ2²u

du

π Cλ

31Asinu du+ C A³

π

3

Cλ

1 u³du

C+logA

A .

Hence our claim holds.

Step 4.Fortt₃, defineη₄: [t₃,+∞)→R²such thatη₄(t₃)=η₃(t₃)and

˙

η4(t)= −AV η4(t)

− DH

|DH|.

Chooset₄ such thatH (η₄(t₄))=0. We claim that t₄−t_34A¹. Again we denoteη₄(t)=(x₁(t), x₂(t)). Note that fort∈ [t₃, t₃+_4A¹ ],|x₂(t)−^π₃|¹₂ andH (η₄(t₃+_4A¹))H (η₄(t₃))−_4A^L =0.Lis the same as in Step 2. Hence our claim holds. Also, it is clear thatx₁(t₄)=π. Moreover, for t∈ [t₃, t₄],x˙₁(t)0. Owing to (3.14) and Step 3, πx₁(t) >^2π₃ fort∈ [t₃, t₄]. Therefore

˙

x₂(t) > A1 2 ×1

2−1>0.

Hencex₂(t₄)∈(^π₃,^π₃+¹₂)⊂(^π₃,^π₂).

Step 5.Fortt₄, letη₅: [t₄,+∞)→R²beη₅=(0, x2(t)),η₅(t₄)=η₄(t₄)and

˙

x₂(t)=Asinx₂(t)+1.

Chooset₅be the first moment thatx₂(t₅)=^π₂. Clearly,t₅−t₄^C_A. Finally, we define that fort∈ [0, t5]

ξ=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

η1(t) for 0tt1, η2(t) fort1tt2, η3(t) fort2tt3, η4(t) fort₃tt₄, η₅(t) fort₄tt₅. Thent₅^logA_A^+C. 2

Estimate ofβ_A. Next we will start to estimateβ_A. Applying similar estimates in the proof of Lemma3.1to a closed streamline (i.e., closed level curves ofH), we have:

Lemma 3.2.Suppose thatLis a closed streamline. Letξ be the controlled trajectoryξ˙=√

AV (ξ )+_A^{1V (ξ )}_|V| andT the time thatξ travels through the wholeL. Then

T ClogA

√A , T 0

˙ξ−√

AV (ξ )²(t) dt T A². HereCis a positive constant which is independent ofA.

Proof. We want to emphasize again that throughout the proofC represents a positive constant which is independent ofA. We only need to prove that

T ClogA

√A . (3.15)

Without loss of generality, let us assume thatLlies within the cell[0, π] × [0, π]and is the level curve{H=a}.

Case 1. a∈ [0,¹₂]. It suffices to establish (3.15) forL1=L∩ [0,^π₂] × [0,^π₂] ∩ {x1x2}. The other portions are similar. For the controlled trajectoryξ=(x₁(t), x₂(t))inL1, we have that

˙

x₁(t)−√

Asinx₁(t)cosx₂(t)− 1 A√

2−

√2A

2 sinx₁(t)− 1 A√

2. Hence the traveling time withinL1is no more than

π

4

0

√ 1

2A

2 sinx₁+ ¹

A√ 2

dx1=ClogA

√A .

Case 2.a∈ [¹₂,1]. Then

˙ξ (t )C√ A√

1−a.

Note that total length ofLisC√

1−a. Therefore, the traveling time is at most√^C A. 2

Lemma 3.3.Leta be a positive constant. Suppose that|x−y|a. Then there existT >0and η∈W^1,∞([0, T]) satisfying| ˙η−√

AV (η(t ))|1a.e.,η(0)=x,η(T )=yandT C(^logA√

A +alogA). Also T

0

η(t)˙ −√

AVη(t )²dtCa+ T

A². (3.16)

HereCis a positive constant depending only onV (i.e., independent ofA,a,xandy).

Proof. Throughout the proof, positive constantsC,C₁andC₂depend only onV (i.e, independent ofA,a,x andy).

SupposeH (x)=s₁ andH (y)=s₂. Without loss of generality, we may assume thatx andy are both in the cell [0, π] × [0, π], 0s₁s₂1. Thens₂−s₁a.

Case 1.Assume thats₂> s₁sin^π₄sin^π₄ =¹₂. Defineη₁(t): [0,∞)→R²asη₁(0)=yand

˙

η1(t)=AV η1(t)

−DH (η₁)

|DH| .

Denotet1as the first moment whenH (η1(t1))=s1. I claim that t₁C|y−x|.

In fact whenH¹₂, there exists 0< C₁< C₂such that C₁√|DH|

1−H C₂. Hence

d√

1−H (η₁(t))

dt −C₁<0 and

|x−y|C₂

1−H (x)−

1−H (y)

=C2

1−s1− 1−s2

C1C2t1.

Then we defineη₂(t): [t₁,+∞)→ +R²asη₂(t₁)=η₁(t₁)and

˙

η₂(t)=√ AV

η₂(t) .

Lett₂be the first moment such thatη₂(t₂)=x. By Case 2 in the proof of Lemma3.2,t₂^√C_A.

Case 2.s₂ ¹₂. According to Lemma 3.2, it suffices to show that two streamlines {H =s₁}and{H=s₂}can be connected by a controlled curve within timeC(^logA√

A +alogA). Let us define a controlled trajectory asη(0)=y and

˙ η(t)=√

AV η(t )

+α(η).

Herea(η)satisfies:

a(η)=

−^{DH (η)}_|DH| ifη /∈W,

1 A

V (η)

|V| ifη∈W

Fig. 3. An illustration of a controlled trajectoryηconnecting two points in the quarter cell (enclosing a single vortex) of the cellular flow in the proof of Lemma3.3. The quarter cell (large square) containing the closed streamlines is[0, π]², the smaller squares in the four corners are of size π/4 byπ/4.

whereWis the union of four corners (see Fig.3), i.e., W=

4 i=1

W_i.

HereW1= [0,^π₄] × [0,^π₄],W2= [0,^π₄] × [^3π₄, π],W3= [^3π₄ , π] × [^3π₄ , π]andW4= [^3π₄, π] × [0,^π₄].A4 will guarantee that eitherx˙1(t)=0 orx˙2(t)=0 depending regions between differentWi. Suppose that there areNtimes that the controlled trajectoryηtravels between corners before it reaches the streamline{H =s1}. Denoteti as the traveling time for 1iN. Then

C N i=1

t_is₂−s₁.

For 2iN−1, by considering thex₁orx₂components, we have that t_i=O

1

√A

.

Hence

N−2C√

A(s₂−s₁).

The traveling time of the control within each corner is at mostO(^logA√

A). Therefore the total travel time is at most C(s₂−s₁)+CNlogA

√A C

alogA+logA

√A

.

(3.16) follows easily from the construction ofη. Combining Case 1 and 2, the lemma holds. 2

Lemma 3.4.Suppose thatF_A∈W^1,∞(R²)is a periodic viscosity solution of (1.8)withV (x)replaced byAV (x), i.e.,

|p+DF_A|²+AV (x)·(p+DF_A)=β_A. (3.17)

Denoteω_A=esssup_Tn|p+DF_A|andα˜_Aas the effective Hamiltonian of the following modified G-equation ω_A|p+DG˜| +AV (x)·(p+DG)˜ = ˜α_A.

Then

β_Aα˜_A.

Proof. Suppose thatφ∈C¹(Tⁿ)and φ(x₀)−F_A(x₀)=max

Rⁿ (φ−F_A).

Thenp+Dφ(x0)²+AV (x0)·

p+Dφ(x0) βA. Also, it is easy to see that

p+Dφ(x₀)ω_A. Hence

ω_Ap+Dφ(x₀)+AV (x₀)·

p+Dφ(x₀) β_A. So

maxRⁿ

ωA|p+Dφ| +AV·(p+Dφ) βA. By the inf-max formula (1.11),

˜

α_A= inf

φ∈C¹(Tⁿ)max

Rⁿ

ω_A|p+Dφ| +AV·(p+Dφ)

β_A. 2

Lemma 3.5.Suppose thatF_A∈W^1,∞(R²)is a periodic viscosity solution of (3.17)andMis the set whereF_Ais differentiable. Then

sup

x∈M|DF_A|O(√ A ).

Proof. To simplify notations, we drop theAdependence and writeF=F_A. Throughout this proof,Cdenotes a con- stant depending only onV. ChooseF such that

(−π,π )×(−π,π )F dx=0. Denotew(x)=^p·x_A^+F. Then

|Dw|²+V (x)·Dw=β_A A².

Choosex₀∈ [−π, π] × [−π, π]such thatwis differentiable atx0. Our goal is to show that DF (x0)C√

A.

Letξ(t ),t∈(−∞,0), be the backward characteristics withξ(0)=x0. Then w(x0)−w

ξ(t )

=tβ_A A²+1

4 0 t

ξ˙−V (ξ )²ds. (3.18)

SinceDw(x₀)exists, suchξ is unique. Also,ξ satisfies the Euler–Lagrange equation

ξ¨=DV (ξ )· ˙ξ−(˙ξ−V )·DV (ξ ) (3.19)

and the equality

Dw(ξ )=ξ˙−V (ξ )

2 . (3.20)

Note that the initial velocityξ(0)is determined. Accordingly,|˙ξ|,|¨ξ|C. Let us assume that M= max

t∈[−1,0]ξ˙−V (ξ ). Then owing to (3.19)

t∈[−min1,0]˙ξ (t )−Vξ(t )CM.

Due to the inf-max formula (1.12), it is clear thatβ_ACA. Choosingt= −1 in (3.18), we deduce that 1

4 0

−1

ξ˙−V (ξ )²dsC

A+w(x₀)−w ξ(−1)

.

Hence M²C

A+C

w(x₀)−w ξ(−1)

. (3.21)

Now we need to estimatew(x₀)−w(ξ(−1)). Letγ (t):R→R²satisfy γ (t )˙ =V

γ (t) , γ (−1)=ξ(−1).

Then

d|ξ(t )−γ (t)|

dt Cξ(t )−γ (t)+M.

So γ (0)−ξ(0)CM.

DenoteU=^p·√x+F

A =√

Aw. Then

|DU|²+√

AV (x)·DU=β_A A. Then

U ξ(0)

−U ξ(−1)

=U ξ(0)

−U γ (0)

+U γ (0)

−U γ (−1)

. Note for any Lipschitz continuous curves(t)andt1t2

U s(t₂)

−U s(t₁)

(t₂−t₁)β_A

A +1

4

t₂

t₁

s(t)˙ −√

AVs(t)²dt.

Then owing to Lemmas3.2and3.3, U

ξ(0)

−U γ (0)

C

logA

√A +MlogA β_A

A + 1 A²

+CM.

Also by choosings(t)=γ (√ At ), U

γ (0)

−U γ (−1)

β_A

A√ A. Therefore

U (x0)−U ξ(−1)

C

logA

√A +MlogA β_A

A + 1 A²

+CM+ β_A A√

A. (3.22)

Now we claim β_AO

A logA

.

In fact, sinceβACA, by (3.21), (3.22) andw=^√U_A, M²C

logA

A +MlogA

√A

.

ThenM^logA√

A and sup

M|DF|O(√

AlogA).

Therefore our claim follows from Lemma3.4and the known fact thatαA=O(_logA^A ). Using (3.21), (3.22) andw=^√U_A again,

M²C 1

A+ M

√A

.

HenceM^√C_A and the lemma holds. 2

By looking at points whereV (x)vanishes and the inf-max formula, it is easy to see that sup

x∈M|DFA| βA=O

A logA

.

It remains an open problem whether sup_x∈M|DFA| =O(

A logA).

Lemma 3.6.There exist positive constants0< C₁C₂independent ofAandpsuch that forA4, Aπ(|p1| + |p2|)

2 logA+C2 α_AAπ(|p1| + |p2|) 2 logA+C1

. (3.23)

There also exist positive constantsK₀4,K₁andK₂independent ofAandpsuch that whenAK₀ A(|p₁| + |p₂|)π

logA+log logA+K2β_AA(|p₁| + |p₂|)π logA−K1

. (3.24)

Proof. By the symmetry of cellular flow and the inf-max formulas (1.11)–(1.12), α_A(p1, p2)=αA(±p1,±p2) and βA(p1, p2)=βA(±p1,±p2).

Hence we may assume p1, p20. The symmetry is just used for convenience and is not essentially necessary.

We first prove (3.23). The left inequality in (3.23) follows easily from Lemma3.1and the control formula (3.13) by choosing ξ(0)=(^π₂,0). Now let us prove the right inequality by choosing ξ(0)=(0,0) in (3.13). Suppose ξ(t )=(x₁(t), x₂(t))∈W. Then fori=1,2

˙

xi(t)A|sinxi| +1.

Without loss of generality, we assume thatx₁(T )x₂(T )andx₁(T )∈(^kπ₂,^(k+₂^1)π]. Then we have that

T k

π

2

0

1

Asinx+1dxklogA+log^π₂

A .

This immediately leads to the right inequality.

Next we verify (3.24). The right inequality of (3.24) follows easily from Lemmas3.4and3.5. In fact, using same notations as in the statement of Lemma3.4, we write

ω_A=esssup_T2|p+DF_A|. Owing to Lemma3.5,ω_AC√

A. ChooseAlarge enough such that_ω^A

A4. Then due to (3.23) and Lemma3.4, β_Aα˜_A=ω_AαA

ωA A(|p₁| + |p₂|)π 2 log_ω^A

A+C1

A(|p₁| + |p₂|)π logA−K₁ for some positive constantK₁.

Now we prove the left inequality. It is well known that β_A(p)= − lim

t→+∞

F (x, t ) t , whereF (x, t )is the solution of equation

F_t+ |DF|²+AV (x)·DF=0, F (x,0)=p·x.

Sinceβ_A(p)=β_A(−p), according to the Lax formula[12,11,27]:

βA(p)=βA(−p)= lim

t→+∞

1 t sup

ξ

p·ξ(t )−1 4 t 0

˙ξ (s)+AV (ξ )²ds

, (3.25)

whereξ runs over all Lipschitz continuous curveξ: [0, t] →R²satisfyingξ(0)=x. The limit does not depend on the choice ofx. Again, the symmetry is used here just for convenience and is not essentially necessary. Hence to prove the left part, it suffices to construct a Lipschitz continuous curveξ: [0,∞)→R²such thatξ(0)=(^π₂,0)and there exists a positive constantT satisfying

(i) for allk∈N ξ(kT )=

π 2,0

+k

π 2,π

2

; (ii)

T =logA+log logA+C

2A ;

(iii) Moreover, the traveling cost kT

0

˙ξ+AVξ(t )²dt kC logA. In fact, by (3.25) and (i)–(iii), we have that

β_A A(p₁+p₂)π

logA+log logA+C − CA˜ log²A.

Here bothCandC˜ are positive constants independent ofAandp. Then whenAis large enough, β_A A(p1+p2)π

logA+log logA+C+ ˜C.

Now let us start to construct such aξ. Owing to the symmetry, it suffices to construct a Lipschitz continuous curve ξ : [0,∞)→R²such thatξ(0)=(^π₂,0),ξ(T )=(^π₂,^π₂),

T =logA+log logA+C 2A

with the traveling cost of:

T 0

˙ξ+A V (ξ )²dt C logA.

The shape ofξis similar to that in the proof of Lemma3.1, shown in Fig.2. We defineξ as follows:

ξ=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

˜

η₁(t) for 0ts₁,

˜

η2(t) fors1ts2,

˜

η₃(t) fors₂ts₃,

˜

η₄(t) fors₃ts₄,

˜

η₅(t) fors₄ts₅.

To save notations,x1(t)andx2(t)in the following different steps represent different curves.

Step 6.Definition ofη˜₁=(x₁(t),0). Letx₁(t): [0,+∞)→Rsatisfyx₁(0)=^π₂ and

˙

x₁(t)=Asinx₁(t).

Lets1be the moment whenx1(s1)=^2π₃. Clearly,s1^C_A and the cost is 0.

Step 7.Definition ofη˜₂: [s₁,+∞)→R². Letη˜₂(s₁)=(^2π₃,0)and

˙˜

η₂(t)= −AV ξ(t )

+

√A

√logA DH

|DH|.

Now lets₂=s₁+_8A¹ . Then it is clear that fors∈ [s₁, s₂] η₂(s)−

2π 3 ,0

<1

2 (3.26)

and for someCˆ∈ [min_{|x−(^2π

3,0)|¹2}|DH|,max_{|x−(2π

3,0)|¹2}|DH|], H

η2(s2)

= Cˆ 8√

A√

logA. (3.27)

Denoteη˜₂=(x₁(t), x₂(t)). Owing to (3.26), fort∈ [s₁, s₂]

˙

x1(t)Acosπ 6sinπ

6 −√ A >0.

Hence x₁(s₂)∈

2π 3 ,5π

6

. (3.28)

The cost is

s2

s1

η˙˜₂+AV (η˜₂)²(t) dt C logA.

Step 8.Definition ofη˜3=(x1(t), x2(t)): [s2,+∞)→R². Letη˜3(s2)= ˜η2(s2)and

˙˜

η₃(t)= −AV (η˜₃).

Lets₃be the moment whenx₂(s₃)=^π₃. Then fort∈(s₂, s₃), we have:

sinx₁(t)sinx₂(t)=λ forλ=₈^√_A√^Cˆ

logA, whereCˆ is the same as in (3.27). Therefore

˙

x₂(t)= −Asinx₂(t)cosx₁(t)=Asinx₂(t)

1− λ² sin²x₂.