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

TL;DR

This paper develops a thermodynamically consistent diffuse-interface model for incompressible two-phase flows with chemotaxis and mass transport, establishing existence, uniqueness, and regularity of solutions, and highlighting differences from classical Keller--Segel systems.

Contribution

It introduces a novel model derived from Onsager's principle that couples fluid dynamics, chemotaxis, and mass transport, and provides rigorous mathematical analysis of its solutions.

Findings

Existence of global finite energy solutions in 2D.

Existence and uniqueness of global strong solutions for regular initial data.

Chemical density remains bounded, preventing singularities.

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.