In this talk, we deal with a class of stochastic diffuse interface models driven by conservative noise. More precisely, we introduce the Cahn–Hilliard and the conserved Allen–Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure on one hand that the order parameter takes its values in the physical range and, on the other, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. Existence and uniqueness of probabilistically-strong solutions is discussed, highlighting the key technical points arising from the structure of the noise. Further directions of research will also be presented.

The mentioned activities are carried out within the project: NoisyFluid "Noise in Fluids", Grant Agreement 101053472, CUP E53C22001720006.