$$H^\infty $$-Calculus for the Surface Stokes Operator and Applications

Abstract

AbstractWe consider a smooth, compact and embedded hypersurface $$\Sigma $$ Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded $$H^\infty $$ H ∞ -calculus with angle smaller than $$\pi /2$$ π / 2 . As an application, we consider critical spaces for the Navier–Stokes equations on the surface $$\Sigma $$ Σ . In case $$\Sigma $$ Σ is two-dimensional, we show that any solution with a divergence-free initial value in $$L_2(\Sigma , \textsf{T}\Sigma )$$ L 2 ( Σ , T Σ ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.