Some estimates for the anisotropic navier-strokes equations and for the hydrostatic approximation

This paper is devoted to the study of the Navier-Stokes equations describing the flow of an incompressible fluid in a shallow domain and to the hydrostatic approximation of these equations. We first study the behaviour of solutions of the Navier-Stokes equations when the depth of the domain tends to zero. We then derive the existence of solutions for the hydrostatic approximation.