*Navier-Stokes Equations with Navier Boundary Condition in Nearly Flat
Domains*

**Luan T. Hoang**

University of Minnesota

email: `lthoang@math.umn.edu`

**Abstract**: 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
H1. 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.