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 v satisfies the Navier-Stokes system, the phase field variable phi 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.