The three dimensional globally modified Navier-Stokes equations: Recent developments

The globally modified Navier-Stokes equations (GMNSE) were introduced by Caraballo, Kloeden & Real in 2006 and have been investigated in a number of papers since then, both for their own sake and as a means of obtaining results about the 3-dimensionalNavier-Stokes equations. These results were reviewed by Kloeden et al, which was published in 2009, but there have been some important developments since then, which will be reviewed here

On the convergence of solutions of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms

Existence and uniqueness of strong solutions for the three dimensional system of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms are established in this article. Galerkin's method and Aubin Lions compactness theorem are the main mathematical tools we use to prove the existence result. Moreover, we prove that, from a sequence of weak solutions of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms, we can extract a subsequence which converges in an adequate sense to a weak solution of three dimensional magnetohydrodynamics equations with locally Lipschitz delays terms

Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays

Existence and uniqueness of strong solutions for three dimensional system of globally modified magnetohydrodynamics equations containing infinite delays terms are established together with some qualitative properties of the solution in this work. The existence is proved by making use of; Galerkin's method, Cauchy-Lipshitz's theorem, a priori estimates, the Aubin-Lions compactness theorem. Moreover, we study the asymptotic behavior of the solution

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.The University of Pretoria and the National Research Foundation South Africa.http://www.tandfonline.com/loi/gapa20hb2016Mathematics and Applied Mathematic

Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-α Model

We propose and analyze a splitting-up scheme for the numerical approximation of the 3D stochastic Navier-Stokes- model. We prove the convergence of the scheme to the unique variational solution of the 3D stochastic Navier-Stokes- a model when the time step tends to zeroClaude Leon Foundation Postdoctoral Fellowship and the University of Pretoriahttp://www.tandfonline.com/loi/lsaa20hb201

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 H2 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. 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 H2 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

Approximation of the trajectory attractor of the 3D MHD System

We study the connection between the long-time dynamics of the 3D magnetohydrodynamic- model and the exact 3D magnetohydrodynamic system. We prove that the trajectory attractor U of the 3D magnetohydrodynamic- model converges to the trajectory attractor U0 of the 3D magnetohydrodynamic system (in an appropriate topology) when approaches zero.Claude Leon Foundation Postdoctoral Fellowship and the University of Pretoriahttp://www.aimsciences.org/journals/home.jsp?journalID=3hb201