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

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 $ \mathit{\boldsymbol{v}} $ satisfies the Navier–Stokes system, the phase field variable $ \varphi $ satisfies a convective Cahn–Hilliard equation with a singular potential (e.g., the Flory–Huggins type), the nutrient density $ \sigma $ 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 we can thus deduce that the global attractor is of finite fractal dimension.