We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples the Navier-Stokes equations for the fluid velocity, a convective Cahn-Hilliard equation for the phase field variable with an advection-diffusion-reaction equation for the nutrient density. In the analysis, we consider a singular (e.g., logarithmic type) potential in the Cahn-Hilliard equation and prove the existence of global weak solutions in both two and three dimensions. Besides, in the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. The singular potential guarantees that the phase field variable always stays in the physically relevant interval [−1, 1] during time evolution. This property enables us to obtain the well-posedness result without any extra assumption on the coefficients that has been made in the previous literature

No abstract

We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono’s type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when t>0. In particular, with the aid of the viscous term, we prove the so-called instantaneous separation property of the phase variable such that it stays away from the pure states ±1 as long as t>0. Furthermore, we study long-time behavior of the system, by proving the existence of a global attractor and characterizing its ω-limit set.

No abstract