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

Jingning He, Hao Wu

Published 2022-12-13Version 1

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 logarithmic type 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 system's dissipative structure and the {\L}ojasiewicz--Simon approach. Our analysis reveals the role of chemotaxis, active transport and a long-range interaction of Oono's type in the global dynamics of the coupled system.