  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.

Analysis of PDEs