On a Navier-Stokes-Cahn-Hilliard System for Viscous Incompressible Two-phase Flows with Chemotaxis, Active Transport and Reaction

TL;DR

This paper studies a complex Navier-Stokes-Cahn-Hilliard model incorporating chemotaxis, active transport, and reactions for viscous incompressible two-phase flows, proving well-posedness and analyzing long-term behavior.

Contribution

It introduces a thermodynamically consistent model generalizing Model H and establishes local and global well-posedness, along with asymptotic behavior analysis.

Findings

Existence and uniqueness of local strong solutions.

Global strong solutions under small initial data.

Asymptotic convergence to equilibrium with rate estimates.

Abstract

We analyze a Navier-Stokes-Cahn-Hilliard model for viscous incompressible two-phase flows where the mechanisms of chemotaxis, active transport and reaction are taken into account. The evolution system couples the Navier-Stokes equations for the volume-averaged fluid velocity, a convective Cahn-Hilliard equation for the phase-field variable, and an advection-diffusion equation for the density of certain chemical substance. This system is thermodynamically consistent and generalizes the well-known ``Model H'' for viscous incompressible binary fluids. For the initial-boundary value problem with a physically relevant singular potential in a general bounded smooth domain $\Omega\subset \mathbb{R}^3$, we first prove the existence and uniqueness of a local strong solution. When the initial velocity is small and the initial phase-field function as well as the initial chemical density are small perturbations of a local minimizer of the free energy, we establish the existence of a unique global strong solution. Afterwards, we show the uniqueness of asymptotic limit for any global strong solution as time goes to infinity and provide an estimate on the convergence rate. The proofs for global well-posedness and long-time behavior are based on the dissipative structure of the system and the Lojasiewicz-Simon approach. Our analysis reveals the effects of chemotaxis, active transport and a long-range interaction of Oono's type on the global dynamics of the coupled system.