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

TL;DR

This paper proves that solutions to a complex Navier-Stokes-Cahn-Hilliard system with chemotaxis and active transport become more regular over time and eventually stabilize, revealing how these mechanisms influence solution behavior.

Contribution

It introduces a novel regularity propagation result for a coupled Navier-Stokes-Cahn-Hilliard system with chemotaxis, active transport, and long-range interactions in three dimensions.

Findings

Global weak solutions become regular after positive time.

Solutions stabilize towards a single equilibrium as time approaches infinity.

The analysis highlights the impact of chemotaxis and active transport on regularity propagation.

Abstract

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 $\bm{v}$, a convective Cahn--Hilliard equation for the phase-field variable $\varphi$, 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 $(\bm{v}, \varphi, \sigma)$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable $\varphi$ transitions into a strong solution, whereas the chemical density $\sigma$ only partially regularizes. Subsequently, the velocity field $\bm{v}$ 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 $\varphi$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\to +\infty$. 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 $\bm{v}$ of Leray type, the strict separation property of $\varphi$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(\varphi, \sigma)$ with a given velocity, respectively.