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