Abstract. In this paper we study the stability for all positive time of the fully implicit Euler scheme for the two-dimensional Navier-Stokes equations. More precisely, we consider the time discretization scheme and with the aid of the discrete Gronwall lemma and the discrete uniform Gronwall lemma we prove that the numerical scheme is stable.

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier-Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the L 2 and H 1 norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.

In this article we study the limit, as the Rossby number e goes to zero, of the primitive equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales.

The aim of this article is to present a qualitative study of the Primitive Equations in a threedimensional domain, with periodical boundary conditions. We start by recalling some already existing results regarding the existence locally in time of weak solutions and existence and uniqueness of strong solutions, and we prove the existence of very regular solutions, up to C 1 -regularity. In the second part of the article we prove that the solution of the Primitive Equations belongs to a certain Gevrey class of functions.