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

Abstract

a b s t r a c tIn 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 H 2 uniformly stable in time, depends continuously on initial data, and is unique. Finally, we obtain the limiting behavior of the system as the parameter N is big enough.

Global attractor for the time discretized modified three-dimensional Bénard systems

“…The usual discretization methods include finite difference method, finite element method, weighted residual method and so on (see [1,2,4,6,7,14,16]). In [5], the modified 3D Navier-Stokes equations were discretized on the time by finite difference method, then the existence of the global attractor was proved. In the literature [15], the Benjamin-Bona-Mahony equation was discretized on the time by the Crank-Nicolson scheme.…”

The Three-Dimensional Globally Modified Navier–Stokes Equations: Recent Developments

“…In this spirit Deugoue & Djoko [8] investigated the implicit Euler scheme applied to the GMNSE, specifically…”

Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations

“…The spaces V 1 and V together with u V1 and (u, B) V will be defined later. One notes that when B=0, system (2)-(3) becomes the 3D globally modified Navier-Stokes equations (GMNSE) which have been the object of intense investigations over the last years [4,6,7,8,10,14,15,18,19,21,31]. The 3D GMNSE were introduced and studied in [4].…”