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

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)
A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 14 / 18