On the solvability of models for two-phase flows of viscous incompressible fluid with shear-dependent viscosity

In this paper we consider a model describing the motion of a mixture of two incompressible and (partially) immiscible fluids in dd-dimension (d≤4d≤4) space domain whose viscosities have a pp-structure depending on the shear rate (or called with shear-dependent viscosity). The existence of weak solutions is established for the exponent p≥p1≐d+3d2+4dd+2 under the no-slip boundary condition to velocity and non-flux boundary condition to relative density. This improves the known results proved for p≥3d+2d+2 in Kim et al. (2006).