Stochastic phase field α-Navier-Stokes vesicle-fluid interaction model

We consider a stochastic perturbation of the phase field alpha-Navier-Stokes model with vesicle-fluid interaction. It consists in system nonlinear evolution partial differential equations modeling fluid-structure interaction associated to dynamics an elastic vesicle immersed moving incompressible viscous fluid. This couples phase-field equation -for interface between fluid and vesicle- fluid- extra term, namely bending energy. The is additive space-time noise trace class on each system. prove existence uniqueness solution classical spaces L2 functions estimates non-linear terms based priori estimate about regularity solutions finite dimensional systems, tightness approximated solution.