  In this paper, we study a hydrodynamic phase-field system modeling the
deformation of functionalized membranes in incompressible viscous fluids. The
governing PDE system consists of the Navier-Stokes equations coupled with a
convective sixth-order Cahn-Hilliard type equation driven by the functionalized
Cahn-Hilliard free energy, which describes phase separation in mixtures with an
amphiphilic structure. In the three dimensional case, we first prove existence
of global weak solutions provided that the initial total energy is finite. Then
we establish uniqueness of weak solutions under suitable regularity assumptions
only imposed on the velocity field (or its gradient). Finally, we prove the
existence and uniqueness of local strong solutions for arbitrary regular
initial data and derive some blow-up criteria. The results are obtained in the
general setting with variable viscosity and mobility.
