Some Mathematical Problems in Geophysical Fluid Dynamics

Abstract This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions 2 and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 1990s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction ( Section 1.4 ).

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