On the Convergence of the Uniform Attractor for the 2D Leray-α Model

Gabriel Deugoué

doi:10.1155/2017/1681857

Abstract

We consider a nonautonomous 2D Leray-α model of fluid turbulence. We prove the existence of the uniform attractor Aα. We also study the convergence of Aα as α goes to zero. More precisely, we prove that the uniform attractor Aα converges to the uniform attractor of the 2D Navier-Stokes system as α tends to zero.

Highlights

In the past decades, the study of nonautonomous dynamical systems has been paid much attention as evidenced by the references cited in [1,2,3,4,5,6,7,8]
In [9], the author considers some special classes of nonautonomous dynamical systems and studies the existence and uniqueness of uniform attractors
Instead of considering a single process associated with the dynamical system, the authors consider a family of processes depending on a parameter σ in some Banach space

Summary

Introduction

The study of nonautonomous dynamical systems has been paid much attention as evidenced by the references cited in [1,2,3,4,5,6,7,8]. In [17], the authors studied the convergence of the solution of the 2D stochastic Lerayα model to the solution of the stochastic 2D Navier-Stokes equations as α approaches 0. They proved the convergence in probability with the rate of convergence at most O(α). In [6], the authors studied the convergence of the uniform attractors of the 2D Navier-Stokes-α model when α tends to zero.

The 2D Navier-Stokes System and Its Uniform Attractor

The 2D Leray-α Model and Its Uniform Attractor

Convergence of the Uniform Attractors of the 2D Leray-α Model