Attractors for the Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Singular Potential in 2D

TL;DR

This paper studies the long-term behavior of a complex fluid model combining Navier-Stokes, Cahn-Hilliard, and chemotaxis effects with singular potentials, proving the existence of finite-dimensional attractors in 2D.

Contribution

It establishes the existence of global and exponential attractors for a coupled Navier-Stokes-Cahn-Hilliard system with chemotaxis and singular potential in two dimensions.

Findings

Existence of a global attractor in 2D

Existence of an exponential attractor

Global attractor has finite fractal dimension

Abstract

We analyze the long-time behavior of solutions to a Navier-Stokes-Cahn-Hilliard system with chemotaxis effects and a solution-dependent mass source term. The fluid velocity satisfies the Navier-Stokes system, the phase field variable satisfies a convective Cahn-Hilliard equation with a singular potential (e.g., the Flory-Huggins type), the nutrient density satisfies an advection-diffusion-reaction. For the initial boundary value problem in 2D, we prove the existence of the global attractor in a suitable phase space. Furthermore, we obtain the existence of an exponential attractor, and it can be implied that the global attractor is of finite fractal dimension.