Stochastic homogenization of interfaces moving by oscillatory normal velocity Adina CIOMAGA joint work with P.E. Souganidis and H.V. Tran Universit´ e Paris Diderot Laboratoire Jacques Louis-Lions LJLL Seminar October 17, 2014 Adina Ciomaga (Paris 7) Stochastic homogenization of interfaces l LJLL Seminar 1 / 24
24
Embed
Stochastic homogenization of interfaces moving by ... · PDF fileCoercive Hamiltonians:Lions-Papanicolau-Varadhan ’88, Evans ’89, Ishii ’99 Stochastic media:Souganidis,...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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).
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
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