The non-steady Navier–Stokes systems with mixed boundary conditions including friction conditions

. In this paper we are concerned with the non-steady Navier-Stokes and Stokes prob- lems with mixed boundary conditions involving Tresca slip condition, leak condition, one-sided leak conditions, velocity, pressure, rotation, stress and normal derivative of velocity together. We get variational inequalities with one unknown which are equivalent to the original PDE problems for the smooth solutions. Then, we study existence and uniqueness of solutions to the corresponding variational inequalities. Special attention is given to a case that through boundary there is leak, and for such a case 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. Relying the results, we get existence, uniqueness and estimates of solutions to the Navier-Stokes and Stokes problems with the boundary conditions.