On the Navier-Stokes equations on surfaces

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface without boundary and flows along that surface. Local-in-time well-posedness is established in the framework of Lp-Lq-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields and we show that each equilibrium is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.