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In this paper we are concerned with a flow of inhomogeneous incompressible fluid-like bodies (IIFB). The concept of IIFB is arised from the analysis of a certain type of granular flows. It is esssentially important to assign the so-called ‘slip’ boundary condition due to its behaviour at the surface, thus we take into account the Navier’s slip condition. Here, the theorem on the unique solvability, local in time, is proved.

The initial-boundary value problem for the Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established in Holder spaces locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.