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

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, ...
A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 3 / 18