On the Navier-Stokes equations on surfaces

Abstract

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface Σ without boundary and flows along Σ. 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 on Σ, and we show that each equilibrium on Σ 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.