Non-stationary Navier-Stokes Equations with Mixed Boundary Conditions

In this paper we are concerned with the initial boundary value problem of the 2, 3-D Navier-Stokes equations with mixed boundary conditions including conditions for velocity, static pressure, stress, rotation and Navier slip condition together. Under a compatibility condition at the initial instance it is proved that for the small data there exists a unique solution on the given interval of time. Also, it is proved that if a solution is given, then there exists a unique solution for small perturbed data satisfying the compatibility condition. Our smoothness condition for initial functions in the compatibility condition is weaker than one in such a previous result.