5. The Steady Navier-Stokes System with Friction Boundary Conditions

The chapter delves into the steady Navier-Stokes system with mixed boundary conditions, including Tresca slip, leak, and one-sided leak conditions. Using strain bilinear forms and theorems from previous sections, it embeds these boundary conditions into variational formulations. The focus is on deriving variational inequalities with one unknown function (velocity) and proving their equivalence to variational formulations with five unknown functions. The chapter also studies the existence, uniqueness, and estimates of solutions for the Navier-Stokes problems with these boundary conditions, highlighting the conditions under which unique solutions exist and providing a detailed analysis of the variational inequalities and their solutions.