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 $\mathrm{\Sigma }$ without boundary and flows along $\mathrm{\Sigma }$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $\mathrm{\Sigma }$ and we show that each equilibrium on $\mathrm{\Sigma }$ 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.