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

TL;DR

This paper proves the bounded $H^$-calculus for the surface Stokes operator on smooth hypersurfaces and applies it to establish global existence and exponential convergence of solutions to surface Navier-Stokes equations in 2D.

Contribution

It establishes the $H^$-calculus for the surface Stokes operator and applies this to analyze the global behavior of surface Navier-Stokes solutions.

Findings

Bounded $H^$-calculus for the surface Stokes operator with angle < 2

Global existence of divergence-free solutions in 2D surface Navier-Stokes

Exponential convergence to equilibrium (Killing fields) in 2D case

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.