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

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.
A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 6 / 18