Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model

TL;DR

This paper models vesicle membrane dynamics in viscous fluids using coupled Navier-Stokes and Cahn-Hilliard equations, proving global existence of weak solutions and their convergence to equilibrium over time.

Contribution

It introduces a novel vesicle-fluid model coupling Navier-Stokes and Cahn-Hilliard equations and establishes convergence to equilibrium using a modified Lojasiewicz-Simon approach.

Findings

Existence of global weak solutions for the model.

Convergence of solutions to equilibrium as time approaches infinity.

Improved convergence results under additional regularity assumptions.

Abstract

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in $3D$ domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.