H∞-calculus for the surface Stokes operator and applications

Abstract

We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded H∞-calculus with angle smaller than π/2. As an application, we consider critical spaces for the Navier–Stokes equations on the surface Σ. In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L2(Σ,TΣ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.