1. An Approach to the Primitive Equations for Oceanic and Atmospheric Dynamics by Evolution Equations

The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the theory of evolution equations. In particular, we discuss the hydrostatic Stokes operator, well-posedness results in critical spaces within the L ^p(L ^q)-scale, within the scaling invariant space L ^∞(L ¹) for Neumann boundary conditions, and within the L ^∞(L ^p) space for mixed boundary conditions and p > 3. In addition, we investigate real analyticity of solutions, convergence of the scaled Navier-Stokes equations to the primitive equations, and the existence of periodic solutions for large forces.