In this talk, I discuss a class of coupled partial differential equation systems describing the interaction between an incompressible viscous fluid and a deformable membrane modeled through a phase-field variable. The analysis is carried out within the Navier-Stokes framework, allowing for the description of physically relevant regimes arising in biological membranes and amphiphilic materials. Particular attention is devoted to an associated optimal control problem in two spatial dimensions. The results highlight the interplay between fluid flow and membrane curvature in driving phase evolution and shaping membrane morphology.
