Mathematical Analysis of Axisymmetrization and Enhanced Inviscid Damping in 2D Linearized Euler Flow

TL;DR

This paper rigorously analyzes the decay and stability of axisymmetric vortices in 2D Euler flows, demonstrating enhanced inviscid damping and vorticity depletion effects with new decay rates and spectral methods.

Contribution

It provides the first rigorous proof of enhanced inviscid damping rates for axisymmetric vortices, advancing the mathematical understanding of vortex stability in inviscid flows.

Findings

Established new optimal decay rates for velocity components

Demonstrated vorticity depletion enhances damping effects

Developed spectral techniques and Greens functions for asymptotic analysis

Abstract

This paper extends the mathematical theory of axisymmetrization and vorticity depletion within the two-dimensional (2D) Euler equations, with an emphasis on the dynamics of radially symmetric, monotonic vorticity profiles. By analyzing inviscid damping, we establish new optimal decay rates for radial and angular velocity components in weighted Sobolev spaces, showing that vorticity depletion enhances damping effects beyond those observed in passive scalar dynamics. Our methodology involves the construction of advanced Greens functions and the use of spectral techniques to achieve precise asymptotic expansions, providing a comprehensive framework for analyzing long-term stability. These results mark the first rigorous confirmation of enhanced inviscid damping rates in axisymmetric vortices, substantially advancing the theoretical understanding of coherent vortex structures in high Reynolds number flows. This work has potential applications in fluid stability theory, turbulence modeling, and other fields that involve the study of inviscid flows.