1 - Shear flows and their attractors

Summary

Abstract

We consider the problem of the existence and finite dimensionality of attractors for some classes of two-dimensional turbulent boundarydriven flows that naturally appear in lubrication theory. The flows admit mixed, non-standard boundary conditions and time-dependent driving forces. We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions.

Introduction

This work gives a survey of the results obtained in a series of papers by Boukrouche & Łukaszewicz (2004, 2005a,b, 2007) and Boukrouche, Łukaszewicz, & Real (2006) in which we consider the problem of the existence and finite dimensionality of attractors for some classes of twodimensional turbulent boundary-driven flows (Problems I–IV below). The flows admit mixed, non-standard boundary conditions and also time-dependent driving forces (Problems III and IV). We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions. This research is motivated by problems from lubrication theory. Our results generalize some earlier ones devoted to the existence of attractors and estimates of their dimensions for a variety of Navier–Stokes flows. We would like to mention a few results that are particularly relevant to the problems we consider.

Most earlier results on shear flows treated the autonomous Navier–Stokes equations. In Doering & Wang (1998), the domain of the flow is an elongated rectangle ω = (0, L) × (0, h), L ≫ h.