Stochastic Cahn-Hilliard and conserved Allen-Cahn equations with logarithmic potential and conservative noise

Abstract

We investigate the Cahn-Hilliard and the conserved Allen-Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn-Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen-Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one. The analysis is carried out by studying the Cahn-Hilliard/conserved Allen-Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.