3. The Steady Navier-Stokes System

The chapter 'The Steady Navier-Stokes System' focuses on the steady Navier-Stokes equations with mixed boundary conditions, including Dirichlet, pressure, vorticity, stress, and normal derivative of velocity. It begins by discussing the variational formulations for these conditions and the relations among strain, vorticity, normal derivative of velocity, and the shape of boundary surfaces. The second fundamental form and shape operator of surfaces are introduced to describe the shape of surfaces mathematically. The chapter then delves into the properties of vector fields on boundary surfaces and their relations to strain, vorticity, and normal derivatives. It concludes with the variational formulations of the steady problems and the proofs of existence and uniqueness of solutions for the Stokes and Navier-Stokes systems under mixed boundary conditions.