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

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.