On degenerate circular and shear flows: the point vortex and power law circular flows

TL;DR

This paper investigates the stability and damping of perturbations in degenerate circular flows, including point vortices and flows with power law singularities, addressing challenges posed by non-monotonicity and weighted Sobolev spaces.

Contribution

It develops a framework for analyzing asymptotic stability and linear inviscid damping in degenerate circular flows with singularities, extending previous results to more complex flow profiles.

Findings

Established stability criteria for degenerate circular flows.

Analyzed the effects of power law singularities on flow stability.

Provided mathematical tools for weighted Sobolev space analysis.

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 $r \uparrow \infty$.