Nonlocal Cahn–Hilliard–Navier–Stokes systems with shear dependent viscosity

We consider a diffuse interface model for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture in three dimensions. The averaged velocity u is governed by a Navier–Stokes system with a shear dependent viscosity controlled by a power p>2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ, that is, the (relative) concentration difference of the two components. The resulting equations are endowed with the no-slip boundary condition for u and the no-flux boundary condition for the chemical potential μ. The latter variable is the functional derivative of a nonlocal and nonconvex Ginzburg–Landau type functional which accounts for the presence of two phases. We first prove the existence of a weak solution in the case p≥11/5. Then we extend some previous results on time regularity and uniqueness if p>11/5.