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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

On the Long-Time Stability of the Implicit Euler Scheme for the Two-Dimensional Navier--Stokes Equations