Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations

Abstract

We prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.

Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations

“…It was also proved in [2] that if u 0 ∈ H \ V , and f ∈ L ∞ (0, +∞; H), then there exists a solution u of GMNSE with u(0) = u 0 , but we do not know if it is unique. Nevertheless, in this last case, we know that every solution u of the GMNSE with…”

“…In [2] we proved that if u 0 ∈ V and f ∈ L 2 (0, T ; H), then there exists a unique solution u of the GMNSE with u(0) = u 0 , and…”

“…, φ m }. From the proof of Theorem 7 in [2] and the uniqueness of u, it follows that if u 0 ∈ V and f ∈ L 2 (0, T ; H), then among other things,…”

“…The aim of this paper is to continue with the analysis of the globally modified Navier-Stokes equations, which was initiated recently in the papers [2] and [8]. In fact, we are interested in several aspects related to the statistical analysis of these equations, since statistical solutions have proven to be very useful in the understanding of turbulence in the case of Navier-Stokes equations (see Foias et al [5]).…”

Asymptotic behaviour of the three-dimensional -Navier–Stokes model with locally Lipschitz delay forcing terms

“…It is worth mentioning that, in order to circumvent serious difficulties in analyzing the three dimensional Navier-Stokes equations, there have been many modifications of them starting with Leray and mostly involving the nonlinear term, see the review paper of Constantin [9]. A system, called the globally modified Navier-Stokes equations (GMNSE), was introduced recently by Caraballo, Kloeden and Real [2,3] and a similar analysis to the one carried out in the present paper is being investigated and will be reported elsewhere. In this paper, we will be mainly concerned with the case in which the delay terms are only locally Lipschitz (as is considered in [20] for the 2D Navier-Stokes model with variable delay).…”