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.

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Abstract. This article is devoted to the study of multivalued semigroups and their asymptotic behavior, with particular attention to iterations of set-valued mappings. After developing a general abstract framework, we present an application to a time discretization of the two-dimensional Navier-Stokes equations. More precisely, we prove that the fully implicit Euler scheme generates a family of discrete multivalued dynamical systems, whose global attractors converge to the global attractor of the continuous system as the time-step parameter approaches zero.