Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport

We analyze a diffuse interface model that describes the dynamics of incompressible viscous two-phase flows, incorporating mechanisms such as chemotaxis, active transport, and long-range interactions of Oono’s type. The evolution system couples the Navier–Stokes equations for the volume-averaged fluid velocity $\mathit{v}$, a convective Cahn–Hilliard equation for the phase-field variable $\phi$, and an advection-diffusion equation for the density of a chemical substance $\sigma$. For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\mathit{v},\phi ,\sigma )$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable $\phi$ transitions into a strong solution, whereas the chemical density $\sigma$ only partially regularizes. Subsequently, the velocity field $\mathit{v}$ becomes regular after a sufficiently large time, followed by a further regularization of the chemical density $\sigma$, which in turn enhances the spatial regularity of $\phi$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\to +\mathrm{\infty }$. Our analysis uncovers the influence of chemotaxis, active transport, and long-range interactions on the propagation of regularity at different stages of time. The proof relies on several key points, including a novel regularity result for a convective Cahn–Hilliard–diffusion system with a velocity field $\mathit{v}$ of Leray type, the strict separation property of $\phi$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(\phi ,\sigma )$ with a given velocity, respectively.