An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations

We prove an existence result for the hydrostatic approximations of Navier-S tokes equations, also called Primitive Equations of the Ocean. We perform a direct analysis of existence of solutions, which is not based upon a singu lar limit process. Our analysis is essentially based upon the existence of a reduced inf-sup condition for (3D) horizontal velocities and (2D) surface pl assures. It applies to general domains, without the need of a boundary talus. Our existence result is, at the same time, a result of approximation by mixed finite elements. (C) 2000 Academie des sciences/Editions scientif iques et medicales Elsevier SAS.