On degenerate circular and shear flows: the point vortex and power law circular flows
Michele Coti Zelati, Christian Zillinger
TL;DR
The paper tackles linear stability and inviscid damping for perturbations of degenerate circular and point-vortex flows in 2D Euler, where degeneracy occurs as $r\to0$ or $r\to\infty$. It develops a localized, weighted-energy approach that combines a scattering formulation with a dyadic annulus partition and frequency-by-frequency analysis to prove $L^2$, $H^1$, and weighted $H^2$ stability and damping for mildly degenerate profiles. Central contributions include the construction of a time-dependent Lyapunov functional via local operators $A_j(t)$, a careful treatment of boundary layers through augmented elliptic operators and a splitting of differentiated fields into well-behaved and boundary-layer components, and rigorous bounds for the evolution maps and commutator terms. The results extend the inviscid damping theory to non-monotone and power-law singular profiles, enabling precise decay and scattering statements for circular flows and the point-vortex setting with potential applications to stability analyses in fluid dynamics and vortex dynamics.
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 $r \uparrow \infty$.