Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction

Abstract

<p style='text-indent:20px;'>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 the phase separation process in mixtures with an amphiphilic structure. In the three dimensional case, we 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 that are only imposed on the velocity field or its gradient. Next, we prove existence and uniqueness of local strong solutions for arbitrary regular initial data and derive some blow-up criteria. Finally, we show the eventual regularity of global weak solutions for large time. The results are obtained in a general setting with variable fluid viscosity and diffusion mobility.</p>