  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.
