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On a Thermodynamically Consistent Diffuse-Interface Model for Incompressible Two-Phase Flows with Chemotaxis and Mass Transport

Andrea Giorgini, Jingning He, Hao Wu

TL;DR

This work develops and analyzes a thermodynamically consistent diffuse-interface model for two incompressible fluids with unmatched densities, coupled to a soluble chemical via chemotaxis and mass transport. In 2D, it proves the existence of global finite-energy and weak solutions, and, under stronger data, the existence and uniqueness of global strong solutions with propagation of regularity; importantly, the chemical density remains bounded if its initial data are bounded, highlighting a diffusion-driven regularization that prevents Keller–Segel-type blow-up. The analysis combines Onsager-based derivation, a regularized/semi-Galerkin approximation, energy-dissipation methods, and decoupling/bootstrap techniques to handle the coupled NSCH–chemotaxis system, with a detailed study of auxiliary decoupled problems. The results establish a rigorous foundation for the long-time behavior of diffuse-interface two-phase flows with chemotaxis in two dimensions and reveal a fundamental distinction from classical Keller–Segel dynamics by leveraging Cahn–Hilliard diffusion to control the chemical concentration.

Abstract

We investigate a hydrodynamic system of Navier--Stokes/Cahn--Hilliard type, which describes the motion of a two-phase flow of two incompressible fluids with unmatched densities coupled with a soluble chemical species. Derived from Onsager's variational principle, this thermodynamically consistent diffuse-interface model incorporates both the chemotaxis effects induced by the chemical species and the mass transport processes within the mixture. For the two-dimensional initial-boundary value problem, we establish the existence of global finite energy solutions and global weak solutions, using a suitable approximation scheme combined with compactness methods. Next, by carefully analyzing three decoupled subsystems and employing a bootstrap argument, we prove the existence and uniqueness of a global strong solution for sufficiently regular initial data, as well as the propagation of regularity for global weak solutions. In particular, we show that the density of the chemical substance stays bounded for all time if its initial datum is bounded. This implies a significant distinction from the classical Keller--Segel system: diffusion driven by the chemical potential gradient can prevent the formation of concentration singularities.

On a Thermodynamically Consistent Diffuse-Interface Model for Incompressible Two-Phase Flows with Chemotaxis and Mass Transport

TL;DR

This work develops and analyzes a thermodynamically consistent diffuse-interface model for two incompressible fluids with unmatched densities, coupled to a soluble chemical via chemotaxis and mass transport. In 2D, it proves the existence of global finite-energy and weak solutions, and, under stronger data, the existence and uniqueness of global strong solutions with propagation of regularity; importantly, the chemical density remains bounded if its initial data are bounded, highlighting a diffusion-driven regularization that prevents Keller–Segel-type blow-up. The analysis combines Onsager-based derivation, a regularized/semi-Galerkin approximation, energy-dissipation methods, and decoupling/bootstrap techniques to handle the coupled NSCH–chemotaxis system, with a detailed study of auxiliary decoupled problems. The results establish a rigorous foundation for the long-time behavior of diffuse-interface two-phase flows with chemotaxis in two dimensions and reveal a fundamental distinction from classical Keller–Segel dynamics by leveraging Cahn–Hilliard diffusion to control the chemical concentration.

Abstract

We investigate a hydrodynamic system of Navier--Stokes/Cahn--Hilliard type, which describes the motion of a two-phase flow of two incompressible fluids with unmatched densities coupled with a soluble chemical species. Derived from Onsager's variational principle, this thermodynamically consistent diffuse-interface model incorporates both the chemotaxis effects induced by the chemical species and the mass transport processes within the mixture. For the two-dimensional initial-boundary value problem, we establish the existence of global finite energy solutions and global weak solutions, using a suitable approximation scheme combined with compactness methods. Next, by carefully analyzing three decoupled subsystems and employing a bootstrap argument, we prove the existence and uniqueness of a global strong solution for sufficiently regular initial data, as well as the propagation of regularity for global weak solutions. In particular, we show that the density of the chemical substance stays bounded for all time if its initial datum is bounded. This implies a significant distinction from the classical Keller--Segel system: diffusion driven by the chemical potential gradient can prevent the formation of concentration singularities.
Paper Structure (25 sections, 21 theorems, 436 equations)

This paper contains 25 sections, 21 theorems, 436 equations.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Related literature
  4. Summary of results and proof strategies
  5. Main Results
  6. Preliminaries
  7. Statement of results
  8. A Regularized System and Its Semi-Galerkin Approximation
  9. The regularized system
  10. The semi-Galerkin scheme
  11. Existence of Global Finite Energy/Weak Solutions
  12. Uniform estimates
  13. Passage to the limit
  14. Completion of the Proof of Theorem \ref{['WEAK-SOL']}
  15. Proof of Corollary \ref{['finite-to-weak']}
  16. ...and 10 more sections

Key Result

Lemma 2.1

Assume that $m\in C([-1,1])$ and $m_{*} \leq m(r)\leq m^*$ for all $r \in [-1,1]$ with $m_{*}<m^*$ being two given positive constants, $a: \Omega \to [-1,1]$ is a measurable function. For every $f\in V_0^{-1}$, problem epgq admits a unique weak solution $u\in V_0$ satisfying In addition, if $m\in C^1([-1,1])$ and $a\in H^2(\Omega)$, then we have

Theorems & Definitions (44)

  • Lemma 2.1
  • Lemma 2.2: Generalized Young's inequality
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.1
  • Remark 2.4
  • Theorem 2.1: Existence of global finite energy/weak solutions
  • ...and 34 more