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

Abstract

We prove that the Stokes semigroup is a bounded analytic semigroup on $L_{\sigma }^{\infty }$ 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.

20 pages