NAVIER-STOKES EQUATIONS ON THIN 3D DOMAINS. I: GLOBAL ATTRACfORS AND GLOBAL REGULARITY OF SOLUTIONS

We examine the Navier-Stokes equations (NS) on a thin 3-dimen-sional domain n. = Q2 x (0, e), whereQ2 is a suitable bounded domain inlR? and e is a small, positive, real parameter. We consider these equationswith various homogeneous boundary conditions, especially spatially periodicboundary conditions. We show that there are large sets ~(e) in HI (n.) andY(e) in WI,oo«O, 00), L 2(n.)) such that if Uo E ~(e) and FE Y(e) ,then (NS) has a strong solution U(t) that remains in HI (n,) for all t ~ 0 andin H 2(n,) for all t > O. We show that the set of strong solutions of (NS) hasa local attractor 'll, in HI (n,) , which is compact in H 2(n,). Furthermore,this local attractor 'll, turns out to be the global attractor for all the weaksolutions (in the sense of Leray) of (NS). We also show that, under reasonableassumptions, 'll. is upper semicontinuous at e = 0 . LABORATOIRE D'ANALYSE NUMERIQUE, BATIMENT 425, UNIVERSITE DE PARIS SUD (ET CNRS),91405 ORSAY CEDEX, FRANCEE-mail address:[email protected] SCHOOL OF MATHEMATICS AND AHPCRC, UNIVERSITY OF MINNESOTA, MINNEAPOLIS,MINNESOTA 55455E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use