Note on the stability of planar stationary flows in an exterior domain without symmetry

TL;DR

This paper investigates the asymptotic stability of small stationary flows in a non-symmetric exterior domain, demonstrating stability under small perturbations and establishing key $L^p$-$L^q$ estimates for the linearized flow semigroup.

Contribution

It provides new stability results for non-symmetric exterior flows and derives $L^p$-$L^q$ estimates for the associated linearized semigroup.

Findings

Stability of small stationary flows with decay rate $O(|x|^{-1})$ at infinity.

Establishment of $L^p$-$L^q$ estimates for the linearized flow semigroup.

Validation of stability under small initial perturbations.

Abstract

The asymptotic stability of two-dimensional stationary flows in a non-symmetric exterior domain is considered. Under the smallness condition on initial perturbations, we show the stability of the small stationary flow whose leading profile at spatial infinity is given by the rotating flow decaying in the scale-critical order $O(|x|^{-1})$. Especially, we prove the $L^p$-$L^q$ estimates to the semigroup associated with the linearized equations.