8. The Non-steady Boussinesq System

The chapter discusses the non-steady Boussinesq system, focusing on the mathematical formulations and proofs of existence and uniqueness of solutions under different boundary conditions. It distinguishes between problems based on static and total pressure, presenting variational formulations and assumptions for each case. For the static pressure case, the chapter proves the existence of a unique solution under small data conditions and specific compatibility conditions. The total pressure case is also addressed, demonstrating the existence of a solution without restrictions on the data. The chapter offers a comprehensive exploration of the non-steady Boussinesq system, providing valuable insights into the behavior of fluid dynamics under varying conditions.