Global weak solutions to a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential

Abstract

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