Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Mass Transport: Cross Diffusion versus Logistic Degradation

TL;DR

This paper proves the existence of global weak solutions for a complex fluid-chemotaxis model involving Navier-Stokes, Cahn-Hilliard, and reaction-diffusion equations, highlighting the effects of cross diffusion and logistic degradation.

Contribution

It introduces a novel semi-Galerkin scheme to establish global weak solutions for a coupled Navier-Stokes-Cahn-Hilliard-chemotaxis system with singular potentials.

Findings

Existence of global weak solutions in 2D and 3D domains.

Regularity and uniqueness of solutions in 2D under certain conditions.

Insights into phase separation influenced by chemotaxis and cross diffusion.

Abstract

We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows influenced by interactions with a soluble chemical substance, encompassing the chemotaxis effect, mass transport, and reactions. In the resulting coupled evolutionary system, the macroscopic fluid velocity field $\boldsymbol{v}$ satisfies a Navier--Stokes system driven by a capillary force, the phase field variable $\varphi$ is governed by a convective Cahn--Hilliard equation incorporating a mass source and a singular potential (e.g., the Flory--Huggins type), and the chemical concentration $\sigma$ obeys an advection-reaction-diffusion equation with logistic degradation, exhibiting a cross-diffusion structure akin to the Keller--Segel model for chemotaxis. Under general structural assumptions, we establish the existence of global weak solutions to the initial boundary value problem within a bounded smooth domain $\Omega\subset \mathbb{R}^d$, $d=2,3$. The proof hinges on a novel semi-Galerkin scheme for a suitably regularized system, featuring a non-standard approximation of the singular potential. Moreover, with more restrictive assumptions on coefficients and data, we establish regularity properties and uniqueness of global weak solutions in the two-dimensional case. Our analysis contributes to a further understanding of phase separation processes under the interplay of fluid dynamics and chemotaxis, in particular, the influence of cross diffusion and logistic degradation.