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

Abstract

The globally modified Navier-Stokes equations (GMNSE) were introduced by Caraballo, Kloeden & Real [1] 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-dimensional Navier-Stokes equations. These results were reviewed by Kloeden et al [11], which was published in 2009, but there have been some important developments since then, which will be reviewed here.Herrn Prof. Dr. Jürgen Scheurle zu seinem sechzigsten Geburtstag gewidmet.

Asymptotic Behavior of the Solutions of the Generalized Globally Modified Navier–Stokes Equations

“…Since the modifying factor F N (‖Λ β u‖) decreases the singularity of the quadratic convection term (u • ∇)u, it allows the authors to derive the existence and uniqueness of global solutions [8]. Following [8], the existence results and the asymptotic behaviors of solutions to problem (1) were extensively studied in different contexts, see e.g., [13][14][15][16][17][18][19][20][21] and the review paper [22].…”

Unique strong and $${ \mathbb {V} }$$ V -attractor of a three-dimensional globally modified two-phase flow model

“…In [30], the authors used the GMNSE to prove that the attainability set of weak solutions of the 3D NS satisfying the energy inequality is weakly compact and weakly connected. Several articles are devoted to the mathematical analysis of the GMNSE, see for instance [5,8,14,24,25,29,30,33], as well as the review paper [4] in which the authors present some recent developments on the GMNSE.…”

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].…”