Stochastic homogenization of interfaces moving by ... · PDF fileCoercive Hamiltonians:Lions-Papanicolau-Varadhan ’88, Evans ’89, Ishii ’99 Stochastic media:Souganidis, Rezakhanlou-Tarver

Stochastic homogenization of interfaces moving byoscillatory normal velocity

Adina CIOMAGA

joint work with P.E. Souganidis and H.V. Tran

Universite Paris DiderotLaboratoire Jacques Louis-Lions

LJLL SeminarOctober 17, 2014

Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 1 / 24

Outline

1 Average behavior of moving interfacesMoving interfaces and Hamilton Jacobi equationsKnown results and main contribution

2 Homogenization of oscillating frontsThe macroscopic metric problemHomogenization of the metric problemEffective Hamiltonian

3 Open questions and future work


Average behavior of moving interfaces Moving interfaces and Hamilton Jacobi equations

Average behavior of moving interfaces

Figure 1: Oscillating interface (red) and its average behavior (black).

Γεt : V (x/ε, t) = a(x/ε)nε(x/ε, t) + κ(x/ε, t) (1)

Question

Understand the average behavior, as ε→ 0, of Γεt ,

Γεt := x ∈ Rn : uε(x , t) = 0 .



Homogenization of oscillating fronts

Problem

Average behavior of solutions of Hamilton-Jacobi equations of the formuεt + a

(xε , ω

)|Duε| = 0 in Rn × (0,∞)

uε(x , ω, 0) = u0(x) in Rn.(HJε)

The moving interface is Γε(ω, t) = x ∈ Rn; uε(x , ω, t) = 0 .

Assumptions

1 ω is an element of a probability space (Ω,A,P) and describes astationary-ergodic environment.

2 a(·, ω) changes sign, hence the corresponding Hamiltonian

H(x , p, ω) = a(x , ω)|p|

is non-coercive, and non-convex.



Assumptions on the velocity

Probability space (Ω,F ,P), endowed with an ergodic group of measure preservingtransformations (τz)z∈Rn . Assume a(·, ·) satisfies

(A1) is stationary with respect to the group (τz)z∈Rn , that is, for every y , z ∈ Rn

and ω ∈ Ω,a(y , τzω) = a(y + z , ω).

(A2) is bounded and equi-Lipschitz continuous, with Lipschitz constant L > 0,that is, for every y , z ∈ Rn and ω ∈ Ω,

|a(y , ω)− a(z , ω)| ≤ L|y − z |.

(A3) for any ω ∈ Ω the function a(·, ω) : Rn → R changes signs.



Periodic environments

Figure 2: Periodic configurations of environments: the velocity a(·) is positiveinside white regions and negative on black regions.



Random environments

Figure 3: Random configurations of environments: the velocity a(·) is positiveinside white regions and negative on black regions.


Average behavior of moving interfaces Known results and main contribution

Known results I

The above Hamiltonian is coercive and convex if

inf a(·, ω) = a0 > 0.

Theorem (Convex and coercive Hamiltonians)

There exists a continuous H and an event Ω ⊂ Ω of full probability such that foreach ω ∈ Ω the unique solution uε = uε(·, ω) ∈ C (Rn) of (HJε) converges locallyuniformly in Rn as ε→ 0 to the unique solution u of

ut + H(Du) = 0 in Rn × (0,∞)

u(·, 0) = u0 in Rn.(HJ)



Known results II

Coercive Hamiltonians: Lions-Papanicolau-Varadhan ’88, Evans ’89, Ishii ’99

Stochastic media: Souganidis, Rezakhanlou-Tarver ’00, Lions-Souganidis’05, Kosygina-Rezakhanlou-Varadhan ’06, Armstrong-Souganidis ’12,Armstrong-Tran, ’13.

Partial coercivity / reduction property: Alvarez - Bardi ’07, Alvarez - Ishii’01, Barles ’07, Imbert - Monneau ’08, Cardaliaguet ’09.

Non-convex Hamiltonians: (G-equation) Cardaliaguet-Nolen-Souganidis ’11Cardaliaguet and Souganidis ’12, (level-set convex) Armstrong-Souganidis’13, (double-well potential) Amstrong-Yu-Tran ’13.

Non-coercive, non-convex Hamiltonians: Bhattacharya-Craciun ’03,Cardaliaguet-Lions-Souganidis ’09.



Homogenization of oscillating fronts

Theorem (Main result)

There exists an event of full probability Ω ⊆ Ω such that, for each ω ∈ Ω, theunique solution uε = uε(·, ω) of (HJε), satisfies that, for each R,T > 0,

uε(·, ω)∗ u as ε→ 0 in L∞(BR × (0,T )) a.s. in Ω, (2)

where the weak limit u is given by the convex combination

u = θ0u0 +∑i∈I

θi ui . (3)

with ui,t + H i (Dui ) = 0 in Rn × (0,∞),

ui (·, 0) = u0 in Rn.(HJi )



Average behavior of moving interfaces

Figure 4: Oscillating interface (red) and its average behavior (black).

Ansatz

uε(x , t, ω) = u(x , t)︸︷︷︸averaged profile

+ εw(x ,

x

ε, ω)

︸︷︷︸corrector at scale ε

+o(ε2).



Formal Asymptotic Expansion

Ansatz

uε(x , t) = u(x , t) + εw(x ,

x

ε, ω)

+ o(ε2).

Plugging the expansion in (HJε) and identifying the terms in front of powers of ε

ut(x , t) + a (x/ε, ω) |Dxu(x , t) + Dyw(x/ε, t, ω)| = 0.

Problem (Step 1 - Effective Hamiltonian / Macroscopic problem)

For each p ∈ Rn there exists a constant µi = H i (p) such that

a(y , ω)|p + Dwi | = µi in Ui = cca > 0.

Problem (Step 2 - Convergence / Sublinear decay at infinity)

lim|y |→∞

wi (y , ω; p)

|y |= lim|y |→∞

wi (y , ω; 0)− p · y|y |

= 0.


Homogenization of oscillating fronts The macroscopic metric problem

The metric problem

Fix ω ∈ Ω. Let U(ω) be an unbounded, connected component of a(·, ω) > 0.For each z ∈ U(ω), we consider the metric problema(y , ω)|Dm| = 1 in U(ω) \ z,

m(·, z , ω) = 0 at z.(4)

Proposition (Well-posedness)

The metric problem has a maximal subsolution

m(y , z , ω) := supw(y , ω) : w(·, ω) ∈ L, w(z , ω) = 0, (5)

a(y , ω)|Dw | ≤ 1 in U(ω),

where L is the class of globally Lipschitz functions.



A subadditive quantity

Lemma (Pseudo-Metric)

m(·, ·, ω) : U(ω)× U(ω)→ [0,∞) is symmetric, and subadditive, i.e. for allx , y , z ∈ U(ω)

m(x , z , ω) ≤ m(x , y , ω) + m(y , z , ω),

and for each fixed z ∈ U(ω),

limd(y ,∂U(ω))→0

m(y , z , ω) = +∞.

Proof. Define the barrier

bδ(y , ω) = −1/L log a(y , ω) + c

Then bδ is an admissible test function for m, with ||Dbδ(·, ω)||L∞(Uδ(ω)) ≤ 1/δ:

∣∣Dbδ(·, ω)∣∣ =

∣∣∣∣1L Da(·, ω)

a(·, ω)

∣∣∣∣ ≤ 1

δ.



Behavior of the maximal subsolution

Lemma (Lipschitz Continuity)

There exists a constant C (ω) s.t. for each δ ∈ (0, δ0), for all y1, y2 ∈ Uδ(ω),

|m(y1, z , ω)−m(y2, z , ω)| ≤ C (ω)

δ|y1 − y2|, (6)

Proof. Let w be an admissible function. Observe that

δ|Dw | ≤ a(y , ω)|Dw | ≤ 1 in Uδ(ω) = y ∈ U(ω) : a(y , ω) > δ.

For all y1, y2 ∈ Uδ(ω),

w(y1, ω)− w(y2, ω) ≤ infγ

∫ 1

0

|Dw(γ(s))||γ′(s)|ds

≤ 1

δdUδ(ω)(y1, y2) ≤ C (ω)

δ|y1 − y2|.



Behavior of the maximal subsolution

Lemma (Lipschitz Continuity)

There exists a constant C (ω) s.t. for each δ ∈ (0, δ0), for all y1, y2 ∈ Uδ(ω),

|m(y1, z , ω)−m(y2, z , ω)| ≤ C (ω)

δ|y1 − y2|, (7)

and, there exists a deterministic constant C > 0 such that, a.s. in ω, for ally1 ∈ Uδ(ω),

lim supy2∈Uδ(ω)|y2−y1|→∞

|m(y1, z , ω)−m(y2, z , ω)||y1 − y2|

≤ C

δ. (8)

Idea.

lim supy2∈Uδ(ω)|y2−y1|→∞

dUδ(ω)(y1, y2, )

|y1 − y2|≤ C

δ. (9)


Homogenization of oscillating fronts Homogenization of the metric problem

Homogenization of the metric problem

Theorem

There exist an event of full probability Ω ⊆ Ω and m : Rn → R, such that, forevery ω ∈ Ω, y , z ∈ Rn and every δ ∈ (0, δ0),

lim supt→∞

ty ,tz∈Uδ(ω)

m(ty , tz , ω)

t= lim inf

t→∞ty ,tz∈U(ω)

m(ty , tz , ω)

t= m(y − z).



Homogenization around the origin I

Lemma

Suppose that 0 ∈ U(ω) a.s. in ω. There exists a set of full probability Ωµ,δy ∈ F

and a Lipschitz extension of m, mδ(·, ·ω) : R+y × R+y → R s.t., for ω ∈ Ωµ,δy ,

limt→∞

1

tmδ(ty , 0, ω) =: mδ(y , ω).

Proof. For each t ≥ 0, consider the entrance and exit times in a hole

t∗ := sup

0 ≤ s ≤ t : sy ∈ Uδ(ω)

and t∗ := infs ≥ t : sy ∈ U

δ(ω).

Let, for t = (1− α) t∗ + αt∗, s = (1− β)s∗ + βs∗,

mδ(ty , sy , ω) = (1− α)((1− β) m(t∗y , s∗y , ω) + β m(t∗y , s

∗y , ω))

+

α((1− β) m(t∗y , s∗y , ω) + β m(t∗y , s∗y , ω)

). (10)



Homogenization around the origin II

The random process Q : I → L1(Ω,P) defined given by

Q([s, t))(ω) = mδ(ty , sy , ω).

is continuous and subadditive on (Ω,F ,P) w.r. to the semigroup(τty)t≥0

.

In view of the subadditive ergodic theorem, there exists an event Ωµ,δy of full

probability such that, for all ω ∈ Ωµ,δy , there exists

mδ(y , ω) := limt→∞

1

tmδ(ty , 0, ω) = lim

t→∞ty ,0∈Uδ(ω)

1

tm(ty , 0, ω).



Deterministic average I

Lemma

There exists a set of full probability Ωµ,δ ∈ F and mδ : Rn → R such that, forevery ω ∈ Ωδ and y ∈ Rn,

limt→∞

ty∈Uδ(ω)

1

tm(ty , 0, ω) = mδ(y). (11)

mδ is 1−positively homogeneous, C-Lipschitz continuous and subadditive.

Proof. To show that mδ(y , ·) is deterministic, we check that, for all z ∈ Rn,

mδ(y , ω) = mδ(y , τzω). (12)


Homogenization of oscillating fronts Effective Hamiltonian

The effective Hamiltonian

The effective Hamiltonian H(p, ω) corresponding to U(ω) is given by

H(p, ω) = infµ > 0 : ∃ w(·, ω) ∈ S+ s.t. a(y , ω)|p + Dw | ≤ µ in U(ω)

.

= infµ > 0 : ∃ w(·, ω) ∈ S+ s.t. a(y , ω)

∣∣∣∣pµ + Dw

∣∣∣∣ ≤ 1 in U(ω).

where S+ :=w ∈ Lloc : lim inf|y |→∞

w(y)|y | ≥ 0

.

Proposition

There exists a set of full probability Ω ⊆ Ω such that, for every ω ∈ Ω and p ∈ Rn,

H(p) = H(p, ω) = infw∈S+

[ess supy∈U(ω)

(a(y , ω)|p + Dw |

)].



The sublinear decay

Proposition (The deterministic effective Hamiltonian)

For each p ∈ Rn and ω ∈ Ω,

H(p) = inf

µ > 0 : lim inf

|y |→∞,y∈U(ω)

m(y , z , ω)− p/µ · (y − z)

|y |≥ 0

. (13)



Main Homogenization Result

Theorem (Convergence)

limε→0

supx∈Uδi,ε(ω)∩BR

|uε(x , ω; p) + ui (x)| = 0.

where ui solves (HJi ), and uε u := θ0u0 +∑

i∈I θiui

Step 1. The unique solution wε = wε(·, ω; p) of

wε + a(xε, ω)|p + Dwε| = 0 in Rn × Ω (14)

satisfies, for all δ > 0 sufficiently small and R > 0,

limε→0

supx∈Uδi,ε(ω)∩BR

|wε(x , ω; p) + H i (p)| = 0.

Conclude using perturbed test function argument.Step 2.

wε w :=∑i∈I

θiH i (p).


Open questions and future work

Open questions and future work

State constraint HJs - perforated domains (in progress - with P.Cardaliaguet and P.E. Souganidis)

uε + H(x ,Du, ω) = 0 in U(ω)

uε = g on ∂U(ω).

Viscous case (in progress - with I. Kim)uεt − ε∆u + a

(xε , ω

)|Duε| = 0 in Rn × (0,∞)

uε = u0 in Rn.


Documents

Stochastic homogenization of interfaces moving by ... · PDF fileCoercive Hamiltonians:Lions-Papanicolau-Varadhan ’88, Evans ’89, Ishii ’99 Stochastic media:Souganidis, Rezakhanlou-Tarver