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AN UPPER BOUND ON THE ATTRACTOR DIMENSION OF A 2D TURBULENT SHEAR FLOW WITH A FREE BOUNDARY CONDITION MAHDI BOUKROUCHE Laboratory of Mathemati s, University of Saint-Étienne, EA-3989, 23 rue du Dr Paul Mi helon, 42023 Saint-Étienne, Fran e E-mail: Mahdi.Boukrou he univ-st-etienne.fr GRZEGORZ UKASZEWICZ Mathemati s Department, University of Warsaw, Bana ha 2, 02-957 Warszawa, Poland E-mail: glukasz mimuw.edu.pl Abstra t. We onsider a free boundary problem of a two-dimensional Navier-Stokes shear ow. There exist a unique global in time solution of the onsidered problem as well as the global attra tor for the asso iated semigroup. As in [1℄ and [2℄, we estimate from above the dimension of the attra tor in terms of given data and the geometry of the domain of the ow. This resear h is motivated by a free boundary problem from lubri ation theory where the domain of the ow is usually very thin and the roughness of the boundary strongly a e ts the ow. We show how it an enlarge the dimension of the attra tor. To this end we establish a new version of the Lieb-Thirring inequality with onstants depending on the geometry of the domain. 1. Introdu tion. We onsider the problem of nite dimensionality of a strongly turbulent boundary driven ow onsidered in lubri ation theory. The two-dimensional domain o upied by the ow is assumed to be periodi with a period ell Ω a thin (or elongated) domain along the rst oordinate dire tion (not ne essarily just a re tangle), so that the boundary ∂Ω onsists of three relevant parts: the upper boundary Γ1, the lower boundary Γ2, and the lateral boundary parts ΓL. The velo ity of the uid satis es a free boundary ondition on the upper boundary and non homogeneous boundary ondition on the lower 2000 Mathemati s Subje t Classi ation: 35R35, 76F10.

An upper bound on the attractor dimension of a 2D turbulent shear flow with a free boundary condition