Stability of radially symmetric, monotone vorticities of 2D Euler equations

TL;DR

This paper proves the stability of radially symmetric, monotone vorticity distributions in 2D Euler equations, including vortex patches and Gaussian profiles, using a variational approach based on impulse minimization.

Contribution

It establishes stability results for a broad class of radially symmetric, monotone vorticities without requiring bounds on perturbation size or support.

Findings

Radial monotone vorticities are stable under 2D Euler dynamics.

Stability holds in weighted norms related to angular impulse.

The approach applies to vortex patches and Gaussian distributions.

Abstract

We consider the incompressible Euler equations in $R^2$ when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on $L^\infty$-bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure.