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

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
A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 16 / 18