Well-posedness of a Hydrodynamic Phase-field Model for Functionalized Membrane-Fluid Interaction

TL;DR

This paper analyzes a complex hydrodynamic phase-field model for membrane-fluid interactions, proving existence, uniqueness, and blow-up criteria for solutions in three dimensions with variable properties.

Contribution

It introduces a rigorous mathematical analysis of a coupled Navier-Stokes and sixth-order Cahn-Hilliard system for functionalized membranes, including existence and uniqueness results.

Findings

Existence of global weak solutions with finite initial energy.

Uniqueness of weak solutions under velocity regularity.

Existence and uniqueness of local strong solutions with blow-up criteria.

Abstract

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.