On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...

On the Stochastic Cahn-Hilliard/Allen-Cahn equation 

Annie Millet 

SAMM, Paris 1 and PMA 

Workshop Stochastic PDEs 

Isaac Newton Institute, Cambridge -September 11, 2012 

(joint work with D. Antonopoulou & G. Karali) 

A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 1 / 18

Outline 

1 Introduction 

2 Well-posedeness result in L q (O) 

3 Regularity of the solution 

4 Regularity of the distribution 


Outline 






Outline 






Outline 






The stochastic model 

O is a rectangular domain in Rd , d = 1, 2, 3, 

or O is a bounded piecewise smooth convex or a smooth Lipschitz domain 

(with ”smooth” boundary) 

ν outward normal, ϱ > 0 is a ”diffusion constant”, 

f is a polynomial of degree 3 with a positive leading coefficient, such as 

f = F ′ where F (u) = (1 − u2 ) 2 is a double equal-well potential, 

W is space-time white noise 

� � � � 

ut = −ϱ∆ ∆u − f (u) + ∆u − f (u) + σ(u) ˙W in O × [0, T ], 

u(x, 0) = u0(x) in O, 

∂u 

∂ν 

= ∂∆u 

∂ν 

= 0 on ∂O × [0, T ], 

Deterministic forcing for a physical model studied by Cahn, Ertl, Halperin, 

Hilliard, Hohenberg, Karali, Katsoutalis, Imbihl, Novick-Cohen, Segel, 

Vlachos, ... 


Physical interpretation 

In the Cahn-Hilliard equation u scaled concentration of an alloy; 

−∆u + f (u) models the chemical potential, Neumann BC: 

conservation of mass and insulation from outside, F free energy. 

The general equation models surface diffusion/adsorption vapor 

deposition, catalysis, surface processes involving transport and 

chemistry of precursors in a gas phase (unconsumed reactants and 

radical absorb so that surface diffusion, or reaction and desorption 

back to the gas) 

Statistical mechanics Arrhenius adsorption/desorption dynamics, 

Metropolis surface diffusion and simple unimolecular reaction; 

Simplified mean field mathematical model : Combination of the 

Cahn-Hiliard eq. (surface diffusion, particle/particle interaction) and 

Allen-Cahn eq. (adsorption to and desorption to surface): mean field 

PDE (0 < ɛ















PDE (0 < ɛ















PDE (0 < ɛ















PDE (0 < ɛ

Previous results with stochastic forcing 

Stochastic Cahn-Hilliard equation on a rectangle (only the bi-Laplacian): 

f = F ′ where F is some ”double well potential” polynomial 

∂tu(x, t) = −∆[∆u(x, t) − f (u(x, t))] + σ(u(x, t)) ˙W (x, t) 

dimension 1 with additive space-time white noise distribution 

solutions (dimension d ≥ 2, L 2 (O) solution for addive colored noise) 

Da Prato, Debussche, 1996) 

dimension 1 to 3: Cardon-Weber with multiplicative space time white 

noise on a rectangle 2001 (dimension 4 and 5, Cardon-Weber & M. 

with colored noise, 2004); the diffusion coefficient σ is Lipschitz and 

bounded. 

O = (0, 1) and f (u) = θ/2 ln( 1+u 

1−u ) − θcu with θ < θc with additive 

noise and reflection measures Debussche & Goudenège (2011) 












bounded. 

O = (0, 1) and f (u) = θ/2 ln( 1+u 














bounded. 

O = (0, 1) and f (u) = θ/2 ln( 1+u 




Weak formulation 

O ”regular” bounded domain in Rd , d = 1, 2, 3 

space-time white noise written as W (dx, ds) � for a one-dimensional � 

(d + 1)-parameter Wiener process W := 

� 

� 

W (x, t) : t ∈ [0, T ], x ∈ O 

and Ft = σ W (x, s) : s ≤ t, x ∈ O 

ϱ = 1, f polynomial of degree 3 with positive leading coefficient 

� � 

� 

u(x, t) − u0(x) φ(x) dx = 

O 

� t � � 

− ∆ 

0 O 

2 � 

φ(x)u(x, s) + ∆φ(x)[f (u(x, s)) + u(x, s)] dxds 

� t � 

� t � 

− φ(x)f (u(x, s)dxds + φ(x)σ(u(x, s)) W (dx, ds) 

0 O 

0 O 

for φ ∈ C 4 (O) with ∂φ 

∂ν 

= ∂∆φ 

∂ν 

= 0 on ∂O. 


Global existence 

Theorem 

Let u0 ∈ L q (O) with q ∈ [6, +∞) and σ : R → R be globally Lipschitz 

with the following sublinear growth: 

There exists constants α ∈ [0, 1/9) and C > 0 such that for all u ∈ R, 

|σ(u)| ≤ C(1 + |u| α ) 

Then for every T > 0, the stochastic Cahn-Hilliard/Allen-Cahn equation 

has a unique weak solution (strong in the probabilistic sense) 

u(x, t) : (x, t) ∈ O × [0, T ]. Furthermore a.s. u ∈ L ∞� 0, T ; L q (O) � 

Remark: The above result also holds true for the solution to the stochastic 

Cahn-Hilliard equation. This improves previous results of Cardon-Weber. 


Sketch of the proof: Integral representation 

A = −∆ on D(A) := 

� 

u ∈ H 2 (O) : ∂u 

∂ν 

� 

= 0 with ∂O 

S(t) semi-group generated by L = −∆2 + ∆ 

Green function G fundamental solution to δtu − Lu = 0 with B.C. 

Represent the weak solution of the stochastic Cahn-Hilliard/Allen-Cahn 

equation in an integral form for any x ∈ O and t ∈ [0, T ]: 

� 

u(x, t) = u0(y)G(x, y, t) dy 

O 

� t � 

� � 

+ ∆G(x, y, t − s) − G(x, y, t − s) f (u(y, s)) dyds 

0 O 

� t � 

+ G(x, y, t − s)σ(u(y, s)) W (dy, ds) 

0 

O 


Sketch of proof (continued) 

Then there exist positive constants C, c such that 

d 

− 

|G(x, y, t)| ≤ Ct 4 exp � − c |x − y| 4 

3 t 

d+2 

− 

|∆G(x, y, t)| ≤ Ct 4 exp � − c |x − y| 4 

3 t 

d+4 

− 

|∂tG(x, y, t)| ≤ Ct 4 exp � − c |x − y| 4 

3 t 

Furthermore, given x, z ∈ O and t ≥ s ≥ 0, 

� 

� t 

0 

� s 

0 

� 

1 � − 3 , 

− 1 

3 

� , 

1 � − 3 . 

|G(x, y, t − r) − G(z, y, t − r)| 

O 

2 dydr ≤ C|x − z| γ 

|G(x, y, t − r) − G(x, y, s − r)| 

O 

2 dydr ≤ C|t − s| γ′ 

� t � 

|G(x, y, t − r)| 

s O 

2 dr ≤ C|t − s| γ′ 

for γ < 4 − d, γ ≤ 2 and γ ′ < 1 − d 

4 . 



Study the truncated equation: cut-off function χn of class C1 such that 

|χn| ≤ 1, |χ ′ n| ≤ 2, χn(x) = 1 if x ≤ n and χn(x) = 0 if x ≥ n + 1. 

� 

un(x, t) = u0(y)G(x, y, t) dy 

O 

� t � 

� � 

+ ∆G(x, y, t − s) − G(x, y, t − s) χn(�un(·, s)�q) f (un(y, s)) dyds 

0 O 

� t � 

+ G(x, y, t − s) χn(�un(·, s)�q) σ(un(y, s)) W (dy, ds) 

0 

O 



The truncated SPDE has a unique solution on every time interval [0, T ]. 

For ”small” T and β ∈ [q, ∞), fixed point theorem in the space 

� 

HT = u adapted : �u� β 

� 

= sup E �u(., t)� HT 

t≤T 

β � � 

q < ∞ 

Extend the unique solution on any time interval 

L(un)(x, t) = 

� t 

0 

� 

O 

G(x, y, t − s)κn(�un(y, s)�q)σ(un(y, s))W (dy, ds) 

Then for any p ∈ [1, ∞): 

� 

E sup |L(un)(x, t)| 

t,x 

2p� 

≤ Cp(T )(1 + n 2αp ) 

and vn = un − L(un). 



Using vn = un − L(un) which solves 

∂tvn + (−∆ + 1)(−∆)vn + (−∆ + 1)χn(�vn + Lun�q)f (vn + L(un)) = 0 

• Apply (−∆ + Id) −1 yields the existence of constants C1 and C2 such that 

� t 

0 

χn(�un(., s)�q)�un(., s)� 4 4ds ≤C1(1.B(u0)) 

+ C2 

� t 

0 

χn(�un(., s)�q)�Lun(., s)� 4 4ds 

• energie estimates on the Galerkin approximation, 

|f (x + y) − f (x)| ≤ C|y|(1 + x 2 + y 2 ) and Hölder’s inequality yield 

�� 

E 

� T 

�un(·, t)� a �β� � 

q dt ≤ C(T ) 

0 

�u0� aβ 

2 +1+n3aαβ +(1+n aαβ )B(u0) aβ 

2 

for a ∈ [q, ∞) and β ∈ [1, ∞), where for some ONB (ei, i ≥ 0) of L2 (O) 

B(u0) := 1 

� ∞� 

� 

� 

1 

− � 

� [λi + 1] 2 (u0, ei) L2 (O)ei � 

2 

2 

2 . 

i=0 

A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 12 / 18 

�


Let Tn = inf{t > 0 : �un(., t)�q ≥ n} ∧ T the above bounds yield that 

lim n P(Tn < T ) ≤ C(T )n −2β(1−9α) 

if the sublinear growth of the ”diffusion” coefficient σ is such that 

|σ(u) ≤ C(1 + |u| α ) with α ∈ [0, 1 

9 ) 

Borel Cantelli’s lemma yields a.s. the existence of a global solution. 


Hölder regulatity 

Theorem 

Suppose that O = (0, π) d , d = 1, 2, 3 

there exists constants α ∈ � 0, 1 

� 

36 and C > 0 such that for all u ∈ R, 

|σ(u)| ≤ C(1 + |u| α ) 

(i) If u0 is continuous, then the solution u has almost surely continuous 

trajectories. 

(ii) If u0 is γ-Hölder continuous for 0 < γ < 1, then the trajectories of the 

solution u are almost surely: 

β1 Hölder-continuous in space with β1 ≤ γ and β1 < (2 − d 

2 ) 

β2 Hölder-continuous in time, with β2 ≤ γ 

4 and β2 < 1 d 

4 (2 − 2 ). 

Remark: Theorem true for the stochastic Cahn-Hilliard equation 

(improves Cardon-Weber’s result) 


Idea of the proof 

(i) Use the explicit form of the Green function for the integral involving 

the initial condition (requires that O is a rectangle) 

(ii) Use the Kolmogorov criterion for the stochastic integral, the identity 

u = un on {Tn = T } and the above upper estimates of un and P(Tn < T ) 

(iii) Use upper bounds of the space and time partial derivatives of the 

Green function. 

(iv) For the deterministic integrals use the factorization method. 

Remark : Unlike S. Cerrai 2003 (and M. Kuntze & J. Van Neerven 2012) 

we do not see how to use subdifferentials and have linear growth of σ, 

even for ”regular” initial conditions. 

This is due to the Laplace operator in front of the non-linearity 


Regularity of the distribution 

Theorem 

Suppose that O = (0, π) d , d = 1, 2, 3, u0 is continuous, 

|σ(u)| ≤ C(1 + |u| α ) for α ∈ � 0, 1 

� 

36 and σ does not vanish. 

Then for every t > 0 and x ∈ D, the distribution of u(x, t) has a density. 

Idea of the proof: 

(i) use the continuity of u to have P(Ωn) → 1 where 

Ωn = {ω : sup 

t 

sup |u(x, t, ω)| ≤ n} 

x 

(ii) for χn previous cut-off function, let ũn = u on Ωn satisfy 

� 

ũn(x, t) = u0(y)G(x, y, t) dy 

O 

� t � 

� � 

+ ∆G(x, y, t − s) − G(x, y, t − s) χn(|ũn(y, s)|)f (ũn(y, s)) dyds 

0 O 

� t � 

+ G(x, y, t − s)σ(ũn(y, s)) W (dy, ds) 

0 O 


Regularity of the distribution (continued) 

Prove that for every (t, x), ũn(x, t) belongs to D1,2 � t � 

and 

sup E 

t,x 

|Dz,r ũn(x, t)| 2 dzdr = Cn < ∞ 

0 

O 

� t � 

sup sup E 

s∈[t−ɛ,t] y t−ɛ 

|Dz,r ũn(y, s)| 

O 

2 dzdr ≤ Cnɛ 2−d/2 

(iii) Suppose that σ = 1; then Dz,r ũn(x, t) = G(t − r, x, z) + Qz,r (x, t) 

and for t > 0 and 0 < ɛ < t: 

� t � 

|Dz,r ũn(x, t)| 2 dzdr ≥ 1 

2 I1(t, x, ɛ) − I2(x, t, ɛ) 

0 

O 

for I1(x, t, ɛ) = � t 

t−ɛ 

� 

O G 2 (t − r, x, z)dzdr≥ Cɛ 1−d/4 

I2(x, t, ɛ) = 

� t 

t−ɛ 

� 

O 

Qz,r (x, t) 2 dzdr 


Regularity of the distribution (continued) 

For σ = 1, the ”main” term is ɛ1−d/4 since 2 − d/2 > 1 − d/4. 

Thus the truncated sequence ũn is such that for t > 0, 

� t � 

|Dz,r ũn(x, t)| 2 dzdr > 0 a.s. 

(iv) For σ which does not vanish, write 

0 

O 

Dz,r ũn(x, t) = σ(ũn(x, t))vn(z, r) 

where vn(z, r) behaves as the previous Malliavin derivative (when σ = 1) 

for t > 0, 

� t 

0 

� 

O 

|Dz,r ũn(x, t)| 2 dzdr > 0 a.s. 

ũn(x, t) has a density (Bouleau-Hirsch criterion) 

u = ũn on Ωn and P(Ωn) → 1 implies that u(x, t) has a density.

On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...

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