An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory

We consider a two-dimensional Navier-Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup. Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type bound on the time-averaged energy dissipation rate, independent of viscosity at large Reynolds numbers. Then we establish a version of the Lieb-Thirring inequality for a class of functions defined on the considered non-rectangular flow domain. This research is motivated by a problem from lubrication theory. (C) 2004 Elsevier Ltd. All rights reserved.