Dynamics and stochastic limit for phase transition problems. problems with noise. Dimitra Antonopoulou IACM-FORTH

Transcription

1 Dynamics and stochastic limit for phase transition problems with noise IACM-FORTH International Conference on Applied Mathematics, September 16-20, 2013, ACMAC, Heraklion, Greece

2 Table of contents Space-Time noise definition Stochastic dynamics Stochastic limit Stochastic dynamics and stability in probability

3 Phase separation 1.1 Space-Time noise definition A binary alloy (two phases mixture) forced to homogenization is in a bounded vessel Ω R n, n = 1, 2, 3. So, since the alloy is not in an equilibrium state evolution starts Evolution Stages (ε > 0 will be a measure for the inner interfaces width): (a) Homogenization (Constant concentration) (b) Spinodal (ε > 0 - ε 0 + ) (c) Coarsening (Sharp Interface ε = 0) (d) Equilibrium The system is not always closed and it may coexist: Thermal fluctuations, external fields, external mass supply deterministic forcing or additive noise.

4 Fourier Brownian, smooth in space 1.1 Space-Time noise definition Ẇ : the formal derivative of a Wiener process W. W : is a Q-Wiener process in H = L 2 (Ω), Q a symmetric operator and (e k ) k N is an orthonormal basis with eigenvalues αk 2, such that Qe k = α 2 k e k and W (t) = α k β k (t)e k, k=1 for a sequence of independent real-valued standard Brownian motions {β k (t)} t 0, Da Prato, Zabczyk 92. Assume mass conservation Ω Ẇ = 0.

5 The problem 2.1. Stochastic dynamics u t = ( ε 2 u + f (u)) + G(ε; x, t), t > 0: time, x Ω: space, u: concentration, f (u)=f (u): F a double equal-well potential, a typical example is: F (u) := 1 4 (u2 1) 2, f (u) := u 3 u, G(ε; x, t): deterministic forcing (smooth) or G = Ẇ additive noise (non-smooth), Cahn-Hilliard J.C.P. 58, Cook A.Metal. 70, Hohemberg and Halperin J.R.Mod.P D Deterministic Problem Dynamics (Ẇ = 0): Fusco-Hale JDDE 89, Carr-Pego CPAM 89, P.R.S.Ed. 90, Bates-Xun JDE 94, 95. Deterministic solution is approximated to a manifold Dynamics.

6 2.1. Stochastic dynamics The interfaces (fronts) stochastic motion A, Blomker, Karali SIMA 12 Integrated Stochastic Cahn-Hilliard Equation: ũ t = ε 2 ũ xxxx + (f (ũ x )) x + Ẇ, 0 < x < 1, t > 0, (ISC-H) ũ(0, t) = 0, ũ(1, t) = M = mass, ũ xx (0, t) = ũ xx (1, t) = 0. Existence: Cardon-Weber Bernoulli 01 and A., Karali DCDS B 11. Noise smooth in space: can use-define an approximate invariant manifold M by piecing together steady state solutions parametrized by the position of the fronts.

7 2.1. Stochastic dynamics Dependency on time is stochastic: Itô calculus higher order derivatives related to the stationary problem, that we estimate. Let ξ := (ξ 1,..., ξ N ) fronts positions and ũ ξ j := ũξ ξ j. ũ (ξ, ṽ), ũ(t) := ũ ξ(t) + ṽ(t) ũ: sum of stochastic processes and ũ ξ manifold. ṽ the projection ṽ, E ξ j = 0 for j = 1,..., N, where E ξ j approximate the first eigenfunctions of the linearized integrated Cahn-Hilliard operator. Define higher derivatives: il := E ξ i, E ξ ξ ilk := 2 E ξ i, ũ ξ l ξ l ξ kl := 2 ũ ξ. k ξ k ξ l E ξ

8 2.1. Stochastic dynamics Apply Itô-formula and differentiate in t the orthogonality condition ṽ, E ξ i = 0 to get: a system in dξ 1,, dξ N : [ ] ũ ξ j, E ξ i ṽ, E ξ ij dξ j = ε 2 (ũxxxx ξ + ṽ xxxx ) + (f (ũx ξ + ṽ x )) x, E ξ i dt j + l,k [ 1 2 ṽ, E ξ ilk 1 2 ũξ kl, E ξ i ũ ξ k, E ξ il ]dξ l dξ k + j dw, E ξ ij dξ j + E ξ i, dw. - The last three higher order additive terms give the difference from the deterministic Cahn-Hilliard system - A polynomially strong noise (in ε) dominates as the deterministic dynamics are exponentially small. - Stability for polynomially strong noise holds for a very long time with high probability.

9 The problem 3.1. Stochastic limit A., Blomker, Karali 13. We consider the stochastic Cahn-Hilliard equation on D in R n n = 2, 3 du ε = ( ε u ε + ε 1 f (u ε ))dt + ε σ Ẇ (x, t), with Neumann boundary conditions. Introduce chemical potential v ε and stochastic system: du ε = v ε dt + ε σ Ẇ v ε = 1 ε f (uε ) + ε u ε. - For Ẇ = 0 deterministic: for the sharp interface limit cf. Alikakos, Bates and X. Chen ARMA The Alikakos, Bates and X. Chen approach (modified) is applicable in the stochastic case too!!! for Ẇ smooth in space.

10 Deterministic regime 3.1. Stochastic limit Our first result is for sufficiently small noise σ large. Let σ > σ 0 > 1. The limit of u ε and v ε as ε 0 + solves on a given time interval [0, T ] the deterministic Hele-Shaw problem v = 0 in D\Γ(t), t > 0 n v = 0 on D v = λh on Γ(t) V = 1 2 ( nv + n v ) on Γ(t) Γ(0) = Γ 0, where H and V are the mean curvature and velocity respectively of the zero level surface Γ(t).

11 Stochastic regime 3.1. Stochastic limit Let σ = 1: larger noise. Use the change of variables: û ε = u ε εw. û ε and v ε solve the following system: t û ε = v ε v ε = 1 ε f (ûε + εw ) + ε û ε + ε 2 W. Advantages: - The above contains now a random PDE without stochastic differentials. - We can treat the equation path-wisely for every fixed realization of W.

12 3.1. Stochastic limit The limit of u ε and v ε solves the stochastic Hele-Shaw problem v = 0 in D\Γ(t), t > 0 n v = 0 on D v = λh + W on Γ(t) V = 1 2 ( nv + n v ) on Γ(t) Γ(0) = Γ 0.

13 The problem 4.1. Stochastic dynamics and stability in probability Let ε > 0 and δ > 0 rescale initial Ω Ω δ := δ 1 Ω in R 2 ut ε (x, t) = ε 2 u ε (x, t) f (u ε (x, t)) + 1 f (u ε (x, t))dx + Ω δ Ẇ (x, t; ε), x Ω δ, t > 0, Ω δ n u ε (x, t) = 0, x Ω δ, t > 0, u ε (x, 0) = u ε 0(x), x Ω δ. When Ẇ = 0 deterministic problem. - Alikakos-X.Chen-Fusco Cal. Var 00, X.Chen-Kowalczyk Comm. PDE 96, Bates-Jin 12. Mass conservation phase separation - total interface perimeter shortening - the layer maintains a semicircular shape (droplet) placed on the boundary enclosing a point of locally maximum curvature.

14 4.1. Stochastic dynamics and stability in probability A., Bates, Karali 13. We apply the Alikakos-X.Chen-Fusco manifold construction. Let L ε (v) := ε 2 v f (v)+ 1 Ω δ Ω δ f (v)dx in Ω δ, n v = 0 on Ω δ. There exist a droplet like state u = u(x, ξ, ε) and a scalar (velocity) field c = c(ξ, ε) = O(δ 2 ) such that where ξ L ε (u) = ε 2 c(ξ, ε) ξ (u) + O(ε k ) in Ω δ, n u = 0 on Ω δ, u = Ω δ π, Ω δ ( ) 0, Ω δ (1D) is the arc-length parameter of Ω δ. (1)

15 4.1. Stochastic dynamics and stability in probability We consider the manifold { } M := u(, ξ, ε) : ξ [0, Ω δ ], consisting of these u satisfying (1) (droplets). Let w := u ε approximated by some u in M and written as w(t) = u(, ξ(t), ε) + v(t), for v(t) L 2 (Ω δ ) ξ (u(, ξ(t), ε)), v small and ξ is a 1D-diffusion process dξ = b(ξ)dt + (σ(ξ), dw ), for some scalar field b : R R and some variance σ : R H.

16 Main results 4.1. Stochastic dynamics and stability in probability - Apply Itô calculus. - Assume that the initial condition is sufficiently smooth to prove dξ = ε 2 c(ξ(t), ε)dt + O(ε k 1/2 δ 2 )dt + da s = 4 [ ] 3π K Ω δ (ξ)ε 2 δ 1 + O(δ) dt + O(δ 4 ε 2 )dt + da s, locally in time where A s is the stochastic diffusion process da s =A 1[ 1 2 (v, 3 ξ (u)) 3 ] 2 ( 2 ξ (u), ξ(u)) (Qσ, σ)dt + A 1 (σ, Q ξ 2 (u))dt + A 1 ( ξ (u), dw ), A := ξ (u) 2 (v, 2 ξ (u)) and K Ω δ (ξ) (in arc-length) the derivative of the curvature of Ω δ.

17 4.1. Stochastic dynamics and stability in probability - The deterministic dynamics are of polynomial order in ε a polynomially strong noise is not always dominant (main difference to the one-dimensional stochastic Cahn-Hilliard). - For sufficiently smooth and bounded noise we prove L 2 - and H 1 ε (ε-weighted norm) - stability in probability for time T ε = O(ε 1 ) + as ε +0 with high probability P 1. - Stochastic stability in the H 1 ε -norm the local in time stochastic dynamics are eventually global.