Navier-stokes Equations with Navier Boundary Conditions in Nearly Flat Domains

We consider the Navier–Stokes equations in a thin domain of which the top and bottom are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides on the domain. The model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H. The long time dynamics of the solutions are also discussed. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affects the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a simple proof of the existence of globally-defined strong solutions. Furthermore we show the existence of a global attractor for the class of all globally-defined strong solutions. It is shown that this attractor is also the global attractor for the weak solutions of the Navier–Stokes equations. This is a joint work with George Sell. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, U.S.A. E-mail address: [email protected]