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 $$\Sigma $$ without boundary and flows along $$\Sigma $$ . 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 $$\Sigma $$ , and we show that each equilibrium on $$\Sigma $$ 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.