6. The Non-steady Navier-Stokes System with Friction Boundary Conditions

The chapter delves into the non-steady Navier-Stokes system with friction boundary conditions, embedding all boundary conditions into variational formulations. It introduces variational inequalities equivalent to the original PDE problems for smooth solutions and studies the existence, uniqueness, and estimates of solutions for Navier-Stokes and Stokes problems. The chapter also addresses the case of total pressure and static pressure, proving existence and uniqueness under specific conditions. Additionally, it presents a comprehensive analysis of the Stokes problem with mixed boundary conditions, highlighting the unique challenges and solutions in the context of non-steady fluid dynamics.