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In this paper we are concerned with the steady Navier-Stokes and Stokesproblems with mixed boundary conditions involving Tresca slip, leak condition,one-sided leak conditions, velocity, pressure, rotation, stress and normalderivative of velocity together. Relying on the relations among strain,rotation, normal derivative of velocity and shape of boundary surface, we havevariational formulations for the problems, which consist of five formulae withfive unknowns. We get the variational inequalities equivalent to the formulatedvariational problems, which have one unknown. Then, we study the correspondingvariational inequalities and relying the results for variational inequalities,we get existence, uniqueness and estimates of solutions to the Navier-Stokesand Stokes problems with the boundary conditions. Our estimates for solutionsdo not depend on the thresholds for slip and leaks.

The steady Navier-Stokes and Stokes systems with mixed boundary conditions including one-sided leaks and pressure