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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
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 2 / 24
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 .
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 3 / 24
Average behavior of moving interfaces Moving interfaces and Hamilton Jacobi equations
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 4 / 24
Average behavior of moving interfaces Moving interfaces and Hamilton Jacobi equations
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 5 / 24
Average behavior of moving interfaces Moving interfaces and Hamilton Jacobi equations
Periodic environments
Figure 2: Periodic configurations of environments: the velocity a(·) is positiveinside white regions and negative on black regions.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 6 / 24
Average behavior of moving interfaces Moving interfaces and Hamilton Jacobi equations
Random environments
Figure 3: Random configurations of environments: the velocity a(·) is positiveinside white regions and negative on black regions.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 7 / 24
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)
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 8 / 24
Average behavior of moving interfaces Known results and main contribution
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 9 / 24
Average behavior of moving interfaces Known results and main contribution
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 )
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 10 / 24
Average behavior of moving interfaces Known results and main contribution
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).
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 11 / 24
Average behavior of moving interfaces Known results and main contribution
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 12 / 24
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 13 / 24
Homogenization of oscillating fronts The macroscopic metric problem
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 14 / 24
Homogenization of oscillating fronts The macroscopic metric problem
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|.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 15 / 24
Homogenization of oscillating fronts The macroscopic metric problem
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)
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 16 / 24
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).
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 17 / 24
Homogenization of oscillating fronts Homogenization of the metric problem
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)
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 18 / 24
Homogenization of oscillating fronts Homogenization of the metric problem
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).
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 19 / 24
Homogenization of oscillating fronts Homogenization of the metric problem
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)
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 20 / 24
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 |
)].
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 21 / 24
Homogenization of oscillating fronts Effective Hamiltonian
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)
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 22 / 24
Homogenization of oscillating fronts Effective Hamiltonian
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).
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 23 / 24
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.
Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 24 / 24