Global weak solutions to a Navier–Stokes–Cahn–Hilliard system with chemotaxis and mass transport: Cross diffusion versus logistic degradation

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 [Formula: see text] satisfies a Navier–Stokes system driven by a capillary force, the phase field variable [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], [Formula: see text]. 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.