On the time discretization for the globally modified three dimensional Navier–Stokes equations

In this work, we analyze the discrete in time 3D system for the globally modified Navier–Stokes equations introduced by Caraballo (2006) [1]. More precisely, we consider the backward implicit Euler scheme, and prove the existence of a sequence of solutions of the resulting equations by implementing the Galerkin method combined with Brouwer’s fixed point approach. Moreover, with the aid of discrete Gronwall’s lemmas we prove that for the time step small enough, and the initial velocity in the domain of the Stokes operator, the solution is H2H2 uniformly stable in time, depends continuously on initial data, and is unique. Finally, we obtain the limiting behavior of the system as the parameter NN is big enough.