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

Jingning He, Hao Wu

TL;DR

This work analyzes a diffuse-interface Navier–Stokes–Cahn–Hilliard model for incompressible two-phase flows that includes chemotaxis, active transport, and nonlocal Oono-type interactions, with a physically relevant singular potential in 3D. The authors establish propagation of regularity for global weak solutions: after any fixed time, the phase field $\varphi$ becomes a strong solution, followed by staged regularity improvements in $\sigma$ and the velocity $\bm{v}$, and then a refinement of $\varphi$; they also prove convergence to a single equilibrium when $\varPsi$ is analytic, aided by a Łojasiewicz–Simon type framework. The analysis hinges on a decoupled convective Cahn–Hilliard–diffusion subsystem with divergence-free drift, uniform energy estimates, an eventual separation property for $\varphi$, and a bootstrap argument that yields strong solutions and uniqueness at large times. The results illuminate how chemotaxis, active transport, and long-range interactions influence regularity and long-time behavior, and they provide a rigorous roadmap for understanding complex multiphase flows with biological transport effects. The findings have potential implications for modelling phase separation in reactive, diffusive, and chemotactic environments and for validating long-time simulations of such coupled systems.

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 $σ$. For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\bm{v}, \varphi, σ)$ 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 $σ$ 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 $σ$, 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 $(\varphi, σ)$ with a given velocity, respectively.

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 work analyzes a diffuse-interface Navier–Stokes–Cahn–Hilliard model for incompressible two-phase flows that includes chemotaxis, active transport, and nonlocal Oono-type interactions, with a physically relevant singular potential in 3D. The authors establish propagation of regularity for global weak solutions: after any fixed time, the phase field becomes a strong solution, followed by staged regularity improvements in and the velocity , and then a refinement of ; they also prove convergence to a single equilibrium when is analytic, aided by a Łojasiewicz–Simon type framework. The analysis hinges on a decoupled convective Cahn–Hilliard–diffusion subsystem with divergence-free drift, uniform energy estimates, an eventual separation property for , and a bootstrap argument that yields strong solutions and uniqueness at large times. The results illuminate how chemotaxis, active transport, and long-range interactions influence regularity and long-time behavior, and they provide a rigorous roadmap for understanding complex multiphase flows with biological transport effects. The findings have potential implications for modelling phase separation in reactive, diffusive, and chemotactic environments and for validating long-time simulations of such coupled systems.

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 , a convective Cahn--Hilliard equation for the phase-field variable , and an advection-diffusion equation for the density of a chemical substance . For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable transitions into a strong solution, whereas the chemical density only partially regularizes. Subsequently, the velocity field becomes regular after a sufficiently large time, followed by a further regularization of the chemical density , which in turn enhances the spatial regularity of . Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as . 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 of Leray type, the strict separation property of for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for with a given velocity, respectively.
Paper Structure (23 sections, 14 theorems, 318 equations)

This paper contains 23 sections, 14 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Main results
  5. Proof strategies
  6. Existence of global weak solutions in $[0,+\infty)$
  7. Global regularity
  8. Convergence to a single equilibrium
  9. A Cahn--Hilliard--Diffusion System with Divergence-free Drift
  10. Existence of weak solutions and conditional uniqueness
  11. The approximate problem
  12. Uniform estimates
  13. Existence
  14. Conditional uniqueness
  15. Solutions with higher-order regularity
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $\Omega$ be a bounded domain of class $C^2$ in $\mathbb{R}^3$. For any $\bm{f} \in \bm{L}^2_{0,\mathrm{div}}(\Omega)$, there exists a unique pair $(\bm{u},p)\in D(\bm{S})\times V_0$ such that $-\Delta \bm{u}+\nabla p=\bm{f}$ a.e. in $\Omega$, that is, $\bm{u}=\bm{S}^{-1}\bm{f}$. Moreover, it hol where the constant $C>0$ only depends on $\Omega$.

Theorems & Definitions (31)

  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1: Existence of global weak solutions
  • Remark 2.2
  • Theorem 2.1: Global regularity
  • Theorem 2.2: Convergence to a single equilibrium
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.2
  • ...and 21 more