$H^{\infty }$-calculus for the surface Stokes operator and applications

Abstract

We consider a smooth, compact and embedded hypersurface $\Sigma$ without boundary and show that the corresponding (shifted) surface Stokes operator $\omega +A_{S,\Sigma }$ admits a bounded $H^{\infty }$-calculus with angle smaller than $\pi /2$, provided $\omega >0$. As an application, we consider critical spaces for the Navier-Stokes equations on the surface $\Sigma$. In case $\Sigma$ is two-dimensional, we show that any solution with a divergence-free initial value in $L_2(\Sigma ,\mathsf{T}\Sigma )$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.

arXiv comment25 pages

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Cited by in corpus (1)

• A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface