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

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 Galerkinʼs 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.