The Navier-Stokes equations on manifolds with boundary

We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold $\text{sM}$ with boundary. The motion on $\text{sM}$ is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on $\mathrm{\partial }\text{sM}$. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on $\text{sM}$ that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case $\text{sM}$ is two-dimensional we show that solutions with divergence free initial condition in $L_2(\text{sM};T\text{sM})$ exist globally and converge to an equilibrium exponentially fast.