A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians

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ScienceDirect

Ann. I. H. Poincaré – AN 32 (2015) 183–200

www.elsevier.com/locate/anihpc

A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians

Filippo Cagnetti

^a

, Diogo Gomes

^a,b

, Hiroyoshi Mitake

^c

, Hung V. Tran

^d,^∗

aCenter for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

bKing Abdullah University of Science and Technology (KAUST), CSMSE Division, Thuwal 23955-6900, Saudi Arabia cDepartment of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan dDepartment of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA

Received 22 October 2013; accepted 29 October 2013 Available online 16 November 2013

Abstract

We investigate large-time asymptotics for viscous Hamilton–Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.

MSC:35B40; 35F55; 49L25

Keywords:Large-time behavior; Hamilton–Jacobi equations; Degenerate parabolic equations; Nonlinear adjoint methods; Viscosity solutions

1. Introduction

In this paper we obtain new results on the study of the large time behavior of Hamilton–Jacobi equations with possibly degenerate diffusion terms

u_t+H (x, Du)=tr

A(x)D²u

inTⁿ×(0,∞), (1.1)

whereTⁿis then-dimensional torusRⁿ/Zⁿ. HereDu, D²uare the (spatial) gradient and Hessian of the real-valued unknown functionudefined onTⁿ× [0,∞). The functionsH:Tⁿ×Rⁿ→RandA:Tⁿ→Mⁿ_sym^×ⁿ are the Hamil- tonian and the diffusion matrix, respectively, whereMⁿ_sym^×ⁿ is the set of n×n real symmetric matrices. The basic hypotheses that we require are thatHis uniformly convex in the second variable, andAis nonnegative definite.

* Corresponding author.

E-mail addresses:cagnetti@math.ist.utl.pt(F. Cagnetti),dgomes@math.ist.utl.pt(D. Gomes),mitake@math.sci.fukuoka-u.ac.jp(H. Mitake), hung@math.uchicago.edu(H.V. Tran).

http://dx.doi.org/10.1016/j.anihpc.2013.10.005

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Our goal in this paper is to study the large time behavior of viscosity solutions of(1.1). Namely, we prove that u(·, t )−(v−ct )

L^∞(Tⁿ)→0 ast→ ∞, (1.2)

where(v, c)is a solution of theergodic problem H (x, Dv)=tr

A(x)D²v

+c inTⁿ. (1.3)

In view of the quadratic or superquadratic growth of the Hamiltonian, there exists a unique constantc∈Rsuch that (1.3)holds true for somev∈C(Tⁿ)in the viscosity sense. We notice that in the uniformly parabolic case (Ais positive definite),vis unique up to additive constants. It is however typically the case thatvis not unique even up to additive constants whenAis degenerate, which makes the convergence(1.2)delicate and hard to be achieved. We will state clearly the existence result of(1.3), which itself is important, in Section2.

It is worth emphasizing here that the study of the large-time asymptotics for this type of equations was only available in the literature for the uniformly parabolic case and for the first order case. There was no results on the large-time asymptotics for(1.1)with possibly degenerate diffusion terms up to now as far as the authors know.

In the last decade, a number of authors have studied extensively the large time behavior of solutions of (first order) Hamilton–Jacobi equations (i.e.,(1.1)withA≡0), whereHis coercive. Several convergence results have been established. The first general theorem in this direction was proven by Namah and Roquejoffre in[18], under the assumptions:p→H (x, p)is convex,H (x, p)H (x,0)for all(x, p)∈Tⁿ×Rⁿ, and maxx∈TⁿH (x,0)=0. Fathi then gave a breakthrough in this area in[10]by using a dynamical systems approach from the weak KAM theory.

Contrary to[18], the results of[10]use uniform convexity and smoothness assumptions on the Hamiltonian but do not require any condition on the structure above. These rely on a deep understanding of the dynamical structure of the solutions and of the corresponding ergodic problem. See also the paper of Fathi and Siconolfi[11]for a beautiful characterization of the Aubry set. Afterwards, Davini and Siconolfi in[7]and Ishii in[12]refined and generalized the approach of Fathi, and studied the asymptotic problem for Hamilton–Jacobi equations onTⁿ and on the whole n-dimensional Euclidean space, respectively. Besides, Barles and Souganidis[2]obtained additional results, for possibly non-convex Hamiltonians, by using a PDE method in the context of viscosity solutions. Barles, Ishii and Mitake [1]simplified the ideas in[2]and presented the most general assumptions (up to now). In general, these methods are based crucially on delicate stability results of extremal curves in the context of the dynamical approach in light of the finite speed of propagation, and of solutions for time large in the context of the PDE approach. It is also important to point out that the PDE approach in[2,1]does not work with the presence of any second order terms.

In the uniformly parabolic setting (i.e.,Auniformly positive definite), Barles and Souganidis[3]proved the long- time convergence of solutions. Their proof relies on a completely distinct set of ideas from the ones used in the first order case as the associated ergodic problem has a simpler structure. Indeed, the strong maximum principle holds, the ergodic problem has a unique solution up to constants. The proof for the large-time convergence in[3]strongly depends on this fact.

It is clear that all the methods aforementioned (for both the casesA≡0 andAuniformly positive definite) are not applicable for the general degenerate viscous cases because of the presence of the second order terms and the lack of both the finite speed of propagation as well as the strong comparison principle. We briefly describe the key ideas on establishing(1.2)in Subsection1.1. Here the nonlinear adjoint method, which was introduced by Evans in[8], plays the essential role in our analysis. Our main results are stated in Subsection1.2.

1.1. Key ideas

Let us now briefly describe the key ideas on establishing(1.2). Without loss of generality, we may assume the ergodic constant is 0 henceforth. In order to understand the limit ast→ ∞, we introduce a rescaled problem. For ε >0, setu^ε(x, t)=u(x, t /ε). Then(u^ε)_t(x, t)=ε⁻¹u_t(x, t /ε),Du^ε(x, t)=Du(x, t/ε), andu^εsolves

εu^ε_t +H x, Du^ε

=tr

A(x)D²u^ε

inTⁿ×(0,∞), u^ε(x,0)=u₀(x), onTⁿ.

By this rescaling,u^ε(x,1)=u(x,1/ε)and we can easily see that to prove(1.2)is equivalent to prove that u^ε(·,1)−v

L^∞(Tⁿ)→0 asε→0.

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To show the above, we first introduce the following approximation:

εw_t^ε+H

x, Dw^ε

=tr

A(x)D²w^ε

+ε⁴w^ε inTⁿ×(0,∞), u^ε(x,0)=u₀(x), onTⁿ.

Then, we observe thatw^εis smooth, and w^ε(·,1)−u^ε(·,1)

L^∞(Tⁿ)→0 asε→0.

It is thus enough to derive the convergence ofw^ε(·,1)asε→0. To prove this, we show that εw_t^ε(·,1)

L^∞(Tⁿ)→0 asε→0, (1.4)

which is a way to prove the convergence(1.2). Indeed,(1.2)is a straightforward consequence of(1.4)by using the stability of viscosity solutions. We notice that this principle appears in the papers of Fathi[10], Barles and Souganidis [2] in the case A≡0 in a completely different way. More precisely, Barles and Souganidis [2] first realized the importance of(1.4), and they gave a beautiful proof of the fact that max{u_t,0},min{u_t,0} →0 as t → ∞in the viscosity sense. In view of this fact, they succeeded to deal with some cases of non-convex Hamilton–Jacobi equations.

On the other hand, we emphasize that the proofs in[10,2]do not work at all for the second order cases, and therefore, one cannot apply it to(1.1). One of our key contributions in this paper is the establishment of(1.4)in the general setting.

In order to prove(1.4), we use the nonlinear adjoint method introduced by Evans[8]and give new ingredients on the averaging action ast→ ∞(or equivalently as ε→0 by rescaling), as clarified below. LetLw^ε be the formal linearized operatorof the regularized equation aroundw^ε, i.e.,

Lw^εf:= ∂

∂η ε

w^ε+ηf

t+H

x, Dw^ε+ηDf

−tr A(x)

D²w^ε+ηD²f

−ε⁴

w^ε+ηf

η=0, for anyf ∈C²(Tⁿ×(0,∞)). Then we consider the following adjoint equation:

L^∗_wεσ^ε=0 inTⁿ×(0,1), σ^ε(x,1)=δx₀ onTⁿ,

whereL^∗_wε is the formal adjoint operator ofLw^ε, andδx₀ is the Dirac delta measure at some pointx0∈Tⁿ. We then see thatσ^ε(·, t )is a probability measure for allt∈(0,1)and conservation of energy holds, namely,

d dt

Tⁿ

H

x, Dw^ε

−tr

A(x)D²w^ε

−ε⁴w^ε

σ^ε(x, t) dx=0.

The conservation of energy in particular gives us a different and completely new way to interpretεw^ε_t(·,1)as

εw_t^ε(x0,1)=

1

0 Tⁿ

H

x, Dw^ε

−tr

A(x)D²w^ε

−ε⁴w^ε

σ^ε(x, t) dx dt. (1.5)

The most important part of the paper is then about showing that the right hand side of(1.5)vanishes asε→0, which requires new ideas and estimates (seeLemmas 2.8 and 3.7). We also notice that the averaging action appears implicitly in(1.5)and plays the key role here. More precisely, if we rescale the above integral back to its actual scale, it turns out to be

1 T

T

0 Tⁿ

H

x, Dw^ε

−tr

A(x)D²w^ε

−ε⁴w^ε

σ^ε(x, t) dx dt, (1.6)

whereT =1/ε→ ∞.

The nonlinear adjoint method for Hamilton–Jacobi equations was introduced by Evans[8]to study the vanishing viscosity process, and gradient shock structures of viscosity solutions of non-convex Hamilton–Jacobi equations.

Afterwards, Tran[21]used it to establish a rate of convergence for static Hamilton–Jacobi equations and was able to relax the convexity assumption of the Hamiltonians in some cases. Cagnetti, Gomes and Tran[4]then used it to study

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the Aubry–Mather theory in the non-convex settings and established the existence of Mather measures. See also[5,9]

for further developments of this new method in the context of Hamilton–Jacobi equations. We notice further thatσ^ε here is strongly related to the Mather measures in the context of the weak KAM theory. See[4]for more details.

1.2. Main results

We state the assumptions we usethroughoutthe paper as well as our main theorems.

For some givenθ, C >0, we denote byC(θ, C)the class of all pairs of(H, A)satisfying (H1) H∈C²(Tⁿ×Rⁿ), andD_pp² H2θ In, whereInis the identity matrix of sizen, (H2) |D_xH (x, p)|C(1+ |p|²),

(H3) A(x)=(a^ij(x))∈Mⁿsym^×ⁿwithA(x)0, andA∈C²(Tⁿ).

Theorem 1.1 (Main Theorem 1). Assume that (H, A)∈C(θ, C). Letu be the solution of (1.1)with initial data u(·,0)=u₀∈C(Tⁿ). Then there exists(v, c)∈C(Tⁿ)×Rsuch that(1.2)holds, where the pair(v, c)is a solution of the ergodic problem(1.3).

We also consider the weakly coupled system (u_i)_t+H_i(x, Du_i)+

m j=1

c_iju_j=tr

A_i(x)D²u_i

inTⁿ×(0,∞),fori=1, . . . , m, (1.7) where(H_i, A_i)∈C(θ, C)is, for eachi, the Hamiltonian and diffusion matrixH_i:Tⁿ×Rⁿ→RandA_i:Tⁿ→Mⁿ_sym^×ⁿ andu_iare the real-valued unknown functions onTⁿ×[0,∞)fori=1, . . . , m. The coefficientsc_ijare given constants for 1i, jmwhich are assumed to satisfy

(H4) c_ii>0,c_ij0 fori=j,_m

j=1c_ij=0 for anyi=1, . . . , m.

We remark that (H4) ensures that(1.7)is a monotone system.

Under these conditions, we prove

Theorem 1.2 (Main Theorem 2). Assume that (Hi, Ai)∈C(θ, C), and cij satisfies (H4) for 1i, j m. Let (u1, . . . , um) be the solution of (1.7) with initial data (u01, . . . , u0m)∈ C(Tⁿ)^m. There exists (v1, . . . , vm, c)∈ C(Tⁿ)^m×Rsuch that

u_i(·, t )−(v_i−ct )

L^∞(Tⁿ)→0 ast→ +∞, fori=1, . . . m, (1.8)

where(v₁, . . . , v_m, c)is a solution of the ergodic problem for systems:

H_i(x, Dv_i)+ m j=1

c_ijv_j=tr

A_i(x)D²v_i

+c inTⁿ, fori=1, . . . , m.

The study of the large-time behavior of solutions to the weakly coupled system (1.7)of first order cases (i.e.

A_i ≡0 for 1im) was started by[14]and[6]independently under rather restrictive assumptions for Hamilto- nians. Recently, Mitake and Tran[16]were able to establish convergent results under rather general assumptions on Hamiltonians. Their proof is based on the dynamical approach, inspired by the papers[7,12], together with a new representation formula for the solutions. See [19]for a PDE approach inspired by[2]. We also refer to [15]for a related work on homogenization of weakly coupled systems of Hamilton–Jacobi equations with fast switching rates.

We prove Theorem 1.2 by following the ideas described above. We notice that the coupling terms cause some additional difficulties and are needed to be handled carefully. We in fact establish a new estimate (seeLemma 3.7(ii) and Subsection3.4) to control these coupling terms in order to achieve(1.4). This is completely different from the single case.

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After this paper was completed, we learnt that Ley and Nguyen[13]obtained recently related convergence results for some specific degenerate parabolic equations. They however assume rather restrictive and technical conditions on the degenerate diffusions so that they could combine the PDE approaches in[2]and[3]to achieve the results. On the other hand, they can deal with a type of fully nonlinear case, which is not included in ours.

This paper is organized as follows: Sections2and3are devoted to proveTheorems 1.1 and 1.2respectively. We provide details and explanations to the method described above in Subsections2.1,2.3,3.1, and3.3and give key estimates in Subsections2.2and3.2.

2. Degenerate viscous Hamilton–Jacobi equations

In this section we study degenerate viscous Hamilton–Jacobi equations. To keep the formulation as simple as possible, we first consider the equation

(C)

u_t+H (x, Du)=a(x)u inTⁿ×(0,∞), u(x,0)=u0(x) onTⁿ,

whereu0∈C(Tⁿ). Throughout this section wealwaysassume that(H, aIn)∈C(θ, C), i.e.,a is supposed to be in C²(Tⁿ)witha0. We remark that all the results proved for this particular case hold with trivial modifications for the general elliptic operator tr(A(x)D²u), except estimate(2.5)andLemma 2.8(ii). The corresponding results will be considered in the end of this section.

The next three propositions concern basic existence results, both for (C) and for the associated stationary problem.

The proofs are standard, hence omitted. We refer the readers to the companion paper[17]by Mitake and Tran for the detailed proofs ofPropositions 2.2 and 2.3.

Proposition 2.1.Let u0∈C(Tⁿ). There exists a unique solutionuof (C)which is uniformly continuous onTⁿ× [0,∞). Furthermore, ifu0∈Lip(Tⁿ), thenu∈Lip(Tⁿ× [0,∞)).

Proposition 2.2.There exists a unique constantc∈Rsuch that the ergodic problem (E) H

x, Dv(x)

=a(x)v+c inTⁿ,

admits a solutionv∈Lip(Tⁿ). We callcthe ergodic constant of (E).

In order to get the existence of the solutions of the ergodic problem (E), we use the vanishing viscosity method as described by the following proposition.

Proposition 2.3.For everyε∈(0,1)there exists a unique constantH_εsuch that the following ergodic problem:

(E)ε H x, Dv^ε

=

ε⁴+a(x)

v^ε+H_ε inTⁿ

has a unique solutionv^ε∈Lip(Tⁿ)up to some additive constants. In addition,

|H_ε−c|Cε², Dv^ε

L^∞(Tⁿ)C, (2.1)

for some positive constantC independent ofε. Herecis the ergodic constant of(E).

In view of the quadratic or superquadratic growth of the HamiltonianH, we can get(2.1)by the Bernstein method.

See the proof of[17, Proposition 1.1]for details. By passing to some subsequences if necessary,v^ε−v^ε(x₀)for a fixedx₀∈Tⁿconverges uniformly to a Lipschitz functionv:Tⁿ→Rwhich is a solution of (E) asε→0.

We proveTheorem 1.1in a sequence of subsections by using the method described in Introduction.

2.1. Regularizing process and proof ofTheorem 1.1

We only need to study the case wherec=0, wherecis the ergodic constant, by replacing, if necessary,H and u(x, t )byH−candu(x, t )+ct, respectively. Therefore, from now on, we always assume thatc=0 in this section.

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Also, without loss of generality we can assume thatu₀∈Lip(Tⁿ), since the general caseu₀∈C(Tⁿ)can be obtained by a standard approximation argument. In particular, thanks toProposition 2.1, we see thatuis Lipschitz continuous onTⁿ× [0,∞).

As stated in Introduction, we consider a rescaled problem. Settingu^ε(x, t)=u(x, t /ε)forε >0, whereuis the solution of (C), one can easily check thatu^εsatisfies

(C)ε

εu^ε_t +H x, Du^ε

=a(x)u^ε inTⁿ×(0,∞), u^ε(x,0)=u₀(x) onTⁿ.

Notice however that in this way we do not have a priori uniform Lipschitz estimates onε, since the Lipschitz bounds onugive us that

u^ε_t

L^∞(Tⁿ×[0,1])C/ε, Du^ε

L^∞(Tⁿ×[0,1])C. (2.2)

In general, the functionu^ε is only Lipschitz continuous. For this reason, we add a viscosity term to(C)ε, and we consider the regularized equation

(A)^η_ε

εw^ε,η_t +H

x, Dw^ε,η

=

a(x)+η

w^ε,η inTⁿ×(0,∞), w^ε,η(x,0)=u0(x) onTⁿ,

forη >0. The advantage of considering(A)^ηε lies in the fact that the solution is smooth, and this will allow us to use the nonlinear adjoint method. The adjoint equation corresponding to(A)^ηε is

(AJ)^η_ε

−εσ_t^ε,η−div D_pH

x, Dw^ε,η σ^ε,η

=

a(x)+η σ^ε,η

inTⁿ×(0,1),

σ^ε,η(x,1)=δx₀ onTⁿ,

whereδ_x₀ is the Dirac delta measure at some pointx₀∈Tⁿ. Lemma 2.4(Elementary properties ofσ^ε,η). We haveσ^ε,η0and

Tⁿ

σ^ε,η(x, t) dx=1 for allt∈ [0,1].

This is a straightforward result of adjoint operator and easy to check. Heuristically, the vanishing viscosity method gives that the rate of convergence ofw^ε,ηtow^εasη→0 is

viscosity coefficient/

the coefficient ofw^ε,η_t

and therefore, we naturally expect that we need to chooseη=ε^α withα >2. We mostly chooseη=ε⁴hereinafter, and therefore we specially denote(A)^ε_ε⁴,(AJ)^ε_ε⁴by(A)ε,(AJ)εandw^ε,ε⁴,σ^ε,ε⁴ byw^ε,σ^ε.

The proof ofTheorem 1.1is based on the two following results,Proposition 2.5andTheorem 2.7.

Proposition 2.5.Letw^ε be the solution of(A)ε. There existsC >0independent ofεsuch that w^ε(·,1)

C¹(Tⁿ)C, u^ε(·,1)−w^ε(·,1)

L^∞(Tⁿ)Cε.

The proof ofProposition 2.5can be derived using standard arguments. Nevertheless, we give the proof below, since some of the estimates involved will be used later.

Before starting the proof, we state a basic property of the functiona∈C²(Tⁿ).

Lemma 2.6.There exists a constantC >0such that

Da(x)²Ca(x) for allx∈Tⁿ. (2.3)

Proof. In view of[20, Theorem 5.2.3],a^1/2∈Lip(Tⁿ). It is then immediate to get(2.3)by noticing that, fora(x) >0, D

a^1/2

(x)= Da(x)

2a^1/2(x). 2

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Proof of Proposition 2.5. Since we are assuming that the ergodic constant for (E) is 0 now andH is quadratic or superquadratic onp, we can easily get the first estimate. We only prove the second one.

Letw^ε,ηbe the solution of(A)^ηε and setϕ(x, t )= |Dw^ε,η|²/2. Then,ϕsatisfies εϕ_t+D_pH·Dϕ+D_xH·Dw^ε,η=(η+a)

ϕ−D²w^ε,η² +

Da·Dw^ε,η w^ε,η. We next notice that forδ >0 small enough, in light ofLemma 2.6,

ax_kw_x^ε,η

k w^ε,ηC|Da| ·w^ε,ηC

δ +δ|Da|²D²w^ε,η²C+1

2aD²w^ε,η². (2.4)

Hence

εϕ_t+D_pH·Dϕ+1

2(η+a)D²w^ε,η²(η+a)ϕ+C.

Multiply the above byσ^εand integrate over[0,1] ×Tⁿto yield

1

0 Tⁿ

a(x)+ηD²w^ε,η²σ^ε,ηdx dtC (2.5)

for someC >0.

Note thatw^ε,ηis differentiable with respect toηby standard regularity results for elliptic equations. Differentiating the equation in(A)^ηεwith respect toη, we get

ε w_η^ε,η

t+D_pH

x, Dw^ε,η

·Dw_η^ε,η=w^ε,η+

a(x)+η

w_η^ε,η, inTⁿ,

wherefηdenotes the derivative of a functionf with respect to the parameterη. Multiplying the above byσ^ε,ηand integrating by parts on[0,1] ×Tⁿyield

εw_η^ε,η(x0,1)=

1

0 Tⁿ

w^ε,ησ^ε,ηdx dt,

where we used the fact thatw^ε,ηη (x,0)≡0. Thanks to(2.5), by the Hölder inequality

εw^ε,η_η (x₀,1)C 1

0 Tⁿ

D²w^ε,η²σ^ε,ηdx dt 1/2

C

√η. By choosing properly the pointx0we have thus

w^ε,η_η (·,1)

L^∞(Tⁿ) C ε√η, which gives

w^ε,η(·,1)−u^ε(·,1)

L∞(Tⁿ)=w^ε,η(·,1)−w^ε,0(·,1)

L∞(Tⁿ)C√η ε . Finally, observing thatw^ε=w^ε,ε⁴, choosingη=ε⁴we get the result. 2

Next theorem gives(1.4)in the special case of problem(A)ε. Theorem 2.7.We have

εlim→0

εw^ε_t(·,1)

L^∞(Tⁿ)=0.

More precisely, there exists a positive constantC, independent ofε, such that εw_t^ε(·,1)

L^∞(Tⁿ)=H

·, Dw^ε(·,1)

−

ε⁴+a(·)

w^ε(·,1)

L^∞(Tⁿ)Cε^1/4.

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The proof ofTheorem 2.7is postponed to the end of this section. We can now give the proof ofTheorem 1.1.

Proof of Theorem 1.1. By Proposition 2.5, we can choose a sequence {εm} →0 such that{w^ε^m(·,1)}converges uniformly to a continuous functionv. In view ofTheorem 2.7,vis a solution of (E), and thus a (time independent) solution of the equation in(C)ε. We lettm=1/εmand useProposition 2.5to deduce that

u(·, tm)−v

L^∞(Tⁿ)→0 asm→ ∞.

Let us show that the limit does not depend on the sequence{tm}m∈N. Now, for anyx∈Tⁿ,t >0 such thattmt <

tm+1, we use the comparison principle to yield that u(x, t )−v(x)u

·, tm+(t−tm)

−v(·)

L^∞(Tⁿ)u(·, tm)−v(·)

L^∞(Tⁿ). Thus,

tlim→∞u(x, t )−v(x) lim

m→∞u(·, t_m)−v(·)

L^∞(Tⁿ)=0, which gives the conclusion. 2

2.2. Convergence mechanisms: Degenerate equations

We show now the following key lemma, which provides integral bounds on first and second order derivatives of the differencew^ε−v^εon the support ofσ^εanda, wherev^εis a solution of(E)ε, andw^εandσ^εare solutions of

(A)ε

εw^ε_t +H

x, Dw^ε

=

a(x)+ε⁴

w^ε inTⁿ×(0,∞), w^ε(x,0)=u₀(x) onTⁿ,

(AJ)ε

−εσ_t^ε−div DpH

x, Dw^ε σ^ε

=

a(x)+ε⁴ σ^ε

inTⁿ×(0,1),

σ^ε(x,1)=δx₀ onTⁿ,

respectively.

Lemma 2.8(Key estimates). There exists a positive constantC, independent ofε, such that the following hold:

(i) ₁

0

Tⁿ(¹_ε|D(w^ε−v^ε)|²+ε⁷|D²(w^ε−v^ε)|²)σ^εdx dtC, (ii) ₁

0

Tⁿa²(x)|D²(w^ε−v^ε)|²σ^εdx dtC√ ε.

Proof. Subtracting Eq.(A)εfrom(E)ε, thanks to the uniform convexity ofH, we get 0=ε

v^ε−w^ε

t+H x, Dv^ε

−H

x, Dw^ε

−

ε⁴+a(x)

v^ε−w^ε

−H_ε

ε

v^ε−w^ε

t+D_pH

x, Dw^ε

·D

v^ε−w^ε +θD

v^ε−w^ε²−

ε⁴+a(x)

v^ε−w^ε

−H_ε. Multiply the above inequality byσ^εand integrate by parts on[0,1] ×Tⁿto deduce that

θ

1

0 Tⁿ

D

w^ε−v^ε²σ^εdx dtHε−

1

0 Tⁿ

ε

v^ε−w^ε σ^ε

tdx dt

+

1

0 Tⁿ

εσ_t^ε+div D_pH

x, Dw^ε σ^ε

+ ε⁴+a

σ^ε

(v−w) dx dt

=H_ε+ε

Tⁿ

w^ε−v^ε σ^εdx

t=1

t=0

=H_ε+ε

w^ε(x0,1)−v^ε(x0)

−ε

Tⁿ

u0(x)−v^ε(x)

σ^ε(x,0) dx

(9)

=Hε+εw^ε(x0,1)−ε

Tⁿ

v^ε(x0)−v^ε(x)

σ^ε(x,0) dx−ε

Tⁿ

u0(x)σ^ε(x,0) dx

H_ε+Cε+CεDv^ε

L^∞(Tⁿ)−ε

Tⁿ

u₀(x)σ^ε(x,0) dxCε,

where we usedPropositions 2.5, 2.3(recall that we setc=0) in the last two inequalities. We hence get

1

0 Tⁿ

D

w^ε−v^ε²σ^εdx dtCε, (2.6)

which is the first part of (i).

Next, subtract(A)εfrom(E)εand differentiate with respect tox_i to get ε

v^ε−w^ε

xit+DpH x, Dv^ε

·Dv_x^ε

i−DpH

x, Dw^ε

·Dw^ε_x

i+Hx_i

x, Dv^ε

−Hx_i

x, Dw^ε

− ε⁴+a

v^ε−w^ε

x_i−ax_i

v^ε−w^ε

=0.

Letϕ(x, t )= |D(v^ε−w^ε)|²/2. Multiplying the last identity by(v^ε−w^ε)x_i and summing up with respect toi, we achieve that

εϕ_t+D_pH

x, Dw^ε

·Dϕ+ D_pH

x, Dv^ε

−D_pH

x, Dw^ε

·Dv_x^ε

i

v_x^ε

i−w^ε_x

i

+

D_xH x, Dv^ε

−D_xH

x, Dw^ε

·D

v^ε−w^ε

−

ε⁴+a(x)

ϕ−D²

v^ε−w^ε²

− Da·D

v^ε−w^ε

=0.

By using various bounds on the above as in the proof ofProposition 2.5, we derive that εϕ_t+D_pH

x, Dw^ε

·Dϕ−

ε⁴+a(x) ϕ+

ε⁴+a(x)/2D²

v^ε−w^ε² C+CD²v^ε+1D

v^ε−w^ε². (2.7)

The last term in the right hand side of(2.7)is a dangerous term. We now take advantage of(2.5)and(2.6)to handle it. Using the fact thatDv^εL^∞andDw^εL^∞are bounded, we have

CD²v^εD

v^ε−w^ε²CD²

v^ε−w^εD

v^ε−w^ε²+CD²w^εD

v^ε−w^ε² ε⁴

2D²

v^ε−w^ε²+ C ε⁴D

v^ε−w^ε²+CD²w^ε. (2.8)

Combine(2.7)and(2.8)to deduce εϕt+DpH

x, Dw^ε

·Dϕ−

ε⁴+a(x)

ϕ+ε⁴ 2D²

v^ε−w^ε²

CD

v^ε−w^ε²+C ε⁴D

v^ε−w^ε²+CD²w^ε. (2.9)

We multiply(2.9)byσ^ε, integrate over[0,1] ×Tⁿ, to yield that, in light of(2.5)and(2.6),

ε⁴

1

0 Tⁿ

D²

w^ε−v^ε²σ^εdx dtCε+ C ε⁴ε+C

1

0 Tⁿ

D²w^εσ^εdx dt

C

ε³+C 1

0 Tⁿ

D²w^ε²σ^εdx dt

1/2 1 0 Tⁿ

σ^εdx dt 1/2

C

ε³+C ε² C

ε³.

(10)

Finally, we prove (ii). Settingψ (x, t )=a(x)|D(v^ε−w^ε)(x, t)|²/2=a(x)ϕ(x, t)and multiplying(2.7)bya(x) we get

εψ_t+D_pH

x, Dw^ε

·Dψ− D_pH

x, Dw^ε

·Da ϕ−

ε⁴+a(x) ψ +

ε⁴+a(x)

(aϕ+2Da·Dϕ)+a(x)

ε⁴+a(x)/2D²

v^ε−w^ε² Ca(x)D²v^ε+1D

v^ε−w^ε².

We use the facts thatDa,aare bounded onTⁿto simplify the above as follows εψt+DpH

x, Dw^ε

·Dψ−

ε⁴+a(x)

ψ+a(x)

ε⁴+a(x)/2D²

v^ε−w^ε²

CD

v^ε−w^ε²−2

ε⁴+a(x)

Da·Dϕ+Ca(x)D²v^εD

v^ε−w^ε². (2.10)

Next, we have to control the last two terms on the right hand side of(2.10). Observe first that forδ >0 small enough 2ε⁴+a(x)

Da·DϕC

ε⁴+a(x)

|Da|D²

v^ε−w^εD

v^ε−w^ε

δ

ε⁴+a(x)

|Da|²D²

v^ε−w^ε²+C δD

v^ε−w^ε²

1

8

ε⁴+a(x)a(x)D²

v^ε−w^ε²+CD

v^ε−w^ε², (2.11)

where we usedLemma 2.6in the last inequality. On the other hand, a(x)D²v^εD

v^ε−w^ε² a(x)D²w^εD

v^ε−w^ε²+a(x)D²

v^ε−w^εD

v^ε−w^ε

√

εa(x)D²w^ε²+ C

√εD

v^ε−w^ε²+a(x)² 8 D²

v^ε−w^ε²+CD

v^ε−w^ε². (2.12) We combine(2.10),(2.11), and(2.12)to deduce that

εψt+DpH

x, Dw^ε

·Dψ−

ε⁴+a(x)

ψ+a(x)² 4 D²

v^ε−w^ε²

C+Cε⁻^1/2D

v^ε−w^ε²+ε^1/2a(x)D²w^ε².

We multiply the above inequality byσ^ε, integrate overTⁿ× [0,1]and use(2.5)to yield the result. 2 Remark 1.Let us give some comments on the estimates inLemma 2.8.

1. In casea≡0, estimate (i) gives us much better control ofD(w^ε−v^ε)andD²(w^ε−v^ε)on the support ofσ^ε. More precisely, a priori estimates only imply thatD(w^ε−v^ε)andε⁴(w^ε−v^ε)are bounded. By using the adjoint equation, we can get further formally thatε⁻^1/2D(w^ε−v^ε)andε^7/2D²(w^ε−v^ε)are bounded on the support ofσ^ε. We notice that while we need to require the uniform convexity ofH to obtain the first term in (i), the second term is achieved without any convexity assumption onH. A version of the second term in (i) was first derived by Evans[8].

2. If the equation in (C) is uniformly parabolic, i.e.,a(x) >0 for allx∈Tⁿ, then the second term of (i) is not needed anymore as estimate (ii) is much stronger. On the other hand, ifais degenerate, then (ii) only provides estimation of|D²(w^ε−v^ε)|²σ^εon the support ofaand it is hence essential to use the second term in (i) to control the part wherea=0.

2.3. Averaging action and proof ofTheorem 2.7

Lemma 2.9(Conservation of energy). The following hold:

(i) _dt^d

Tⁿ(H (x, Dw^ε)−(ε⁴+a(x))w^ε)σ^εdx=0, (ii) εw^ε_t(x₀,1)=₁

0

Tⁿ(H (x, Dw^ε)−(ε⁴+a(x))w^ε)σ^εdx dt.

(11)

We stress the fact that identityLemma 2.9(ii) is extremely important. As stated in Introduction, if we scale back the time, the integral in the right hand side becomes(1.6), that is the averaging action ast→ ∞. Relation (ii) together withLemma 2.8allow us to conclude the proof ofTheorem 2.7.

Proof of Lemma 2.9. We only need to prove (i) as (ii) follows directly from (i). This is a straightforward result of adjoint operators and comes from a direct calculation:

d dtTⁿ

H

x, Dw^ε

−

ε⁴+a(x) w^ε

σ^εdx

=

Tⁿ

D_pH

x, Dw^ε

·Dw_t^ε−

ε⁴+a(x) w_t^ε

σ^εdx+

Tⁿ

H

x, Dw^ε

−

ε⁴+a(x) w^ε

σ_t^εdx

= −

Tⁿ

div D_pH

x, Dw^ε σ^ε

+

ε⁴+a(x) σ^ε

w^ε_tdx−

Tⁿ

εw^ε_tσ_t^εdx=0. 2

We now can give the proof ofTheorem 2.7, which is the main principle to achieve large time asymptotics, by using the averaging action above and the key estimates inLemma 2.8.

Proof of Theorem 2.7. Let us first choosex₀such that εw^ε_t(x0,1)=H

x0, Dw^ε(x0,1)

−

ε⁴+a(x0)

w^ε(x0,1)

=H

·, Dw^ε(·,1)

−

ε⁴+a(·)

w^ε(·,1)

L^∞(Tⁿ). Thanks toLemma 2.9andProposition 2.3,

εw_t^ε(·,1)

L^∞(Tⁿ)=H

·, Dw^ε(·,1)

−

ε⁴+a(·)

w^ε(·,1)

L^∞(Tⁿ)

=

1

0 Tⁿ

H

x, Dw^ε

− ε⁴+a

w^ε σ^εdx dt

1

0 Tⁿ

H

x, Dw^ε

− ε⁴+a

w^ε

− H

x, Dv^ε

− ε⁴+a

v^εσ^εdx dt+ |Hε|

1

0 Tⁿ

CD

w^ε−v^ε+

ε⁴+a

w^ε−v^εσ^εdx dt+Cε².

We finally use the Hölder inequality andLemma 2.8to get

εw_t^ε(·,1)

L^∞(Tⁿ)C 1

0 Tⁿ

D

w^ε−v^ε²σ^εdx dt 1/2

+Cε⁴ 1

0 Tⁿ

D²

+C 1

0 Tⁿ

a²(x)D²

+Cε²Cε^1/4. 2

2.4. General case

In this subsection we consider the general case(1.1). As pointed out before we only need to address the analogs to estimate(2.5)andLemma 2.8(ii). These are