On the large time $L^{\infty}$-estimates of the Stokes semigroup in two-dimensional exterior domains

TL;DR

This paper establishes that the Stokes semigroup in two-dimensional exterior domains is bounded and analytic on $L^{ obreak ext{infty}}_{ obreak ext{sigma}}$, extending previous $L^{p}$ results and addressing the endpoint case.

Contribution

It proves the boundedness and analyticity of the Stokes semigroup on $L^{ obreak ext{infty}}_{ obreak ext{sigma}}$ in 2D exterior domains, an endpoint case of earlier $L^{p}$-boundedness results.

Findings

The Stokes semigroup is bounded and analytic on $L^{ obreak ext{infty}}_{ obreak ext{sigma}}$ for 2D exterior domains.

The proof uses the non-existence of bounded steady flows (Stokes paradox).

Asymptotic formulas for the net force of the Stokes resolvent are employed.

Abstract

We prove that the Stokes semigroup is a bounded analytic semigroup on $L^{\infty}_{\sigma}$ of angle $\pi/2$ for two-dimensional exterior domains. This result is an end point case of the $L^{p}$-boundedness of the semigroup for $p\in (1,\infty)$, established by Borchers and Varnhorn (1993) and an extension of finite time $L^{\infty}$-estimates studied by the author and Giga (2014). The proof is based on the non-existence result of bounded steady flows (the Stokes paradox) and some asymptotic formula for the net force of the Stokes resolvent.