The Navier-Stokes equations on manifolds with boundary

Abstract

We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold Mwith boundary. The motion on Mis modeled by the incompressible Navier-Stokes equations, and the fluid is sub-ject to pure or partial slip boundary conditions of Navier type on ∂M. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on Mthat satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case Mis two-dimensional we show that solutions with divergence free initial condition in L2(M; TM)exist globally and converge to an equilibrium exponen-tially fast.