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

On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type