---
abstract: 'We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. Prototypical examples are given by circular flows with power law singularities or zeros as $r\downarrow 0$ or $r \uparrow \infty$.'
address:
- 'Department of Mathematics, Imperial College London, London, SW7 2AZ, UK'
- 'Department of Mathematics, University of Southern California, Los Angeles, California 90089, USA'
author:
- Michele Coti Zelati and Christian Zillinger
title: 'On degenerate circular and shear flows: the point vortex and power law circular flows'
---

Introduction
============

When considering the 2D Euler equations close to Couette flow on $\T \times \R$, it is a classical result by Orr that the linearized problem for the vorticity reduces to free transport. Using the explicit solution, it can then be shown that the vorticity weakly, but not strongly, converges to its $x$ average and that the associated (perturbation to the) velocity field strongly converges to a shear flow with sharp algebraic decay rates provided the initial 