  Strong solutions of the non-stationary Navier-Stokes equations under
non-linearized slip or leak boundary conditions are investigated. We show that
the problems are formulated by a variational inequality of parabolic type, to
which uniqueness is established. Using Galerkin's method and deriving a priori
estimates, we prove global and local existence for 2D and 3D slip problems
respectively. For leak problems, under no-leak assumption at $t=0$ we prove
local existence in 2D and 3D cases. Compatibility conditions for initial states
play a significant role in the estimates.
