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 $Σ$ without boundary and flows along $Σ$. 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 $Σ$ 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.

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