Existence and stability of Sadovskii vortices: from patch to smooth vortices

TL;DR

This paper develops a scaling-invariant variational framework for Sadovskii vortices in 2D Euler flows, unifying patch and smooth vortex solutions, and proves their existence, stability, and properties through energy inequalities and concentration-compactness.

Contribution

It introduces a new variational principle without mass constraints, encompassing classical Sadovskii vortices and smooth vortices, and establishes their stability and propagation characteristics.

Findings

Existence of a family of Sadovskii vortices interpolating between patch and smooth vortices.

Proved stability of axis-touching vortex configurations under small perturbations.

Derived bounds on vortex center of mass movement, showing near-constant propagation speed.

Abstract

We establish a scaling-invariant variational framework for steadily translating dipoles of the two-dimensional incompressible Euler equations. Specifically, we consider the maximization of the kinetic energy subject to constraints on the impulse and the Lp-norm (1<p\leq\infty) of vorticity without imposing any restriction on the total mass. In contrast to variational constructions with a mass constraint, the shape of the vortices in our framework is determined directly by the scaling-invariant structure of the energy. We prove that every vortex arising from this variational principle necessarily touches the symmetry axis and is therefore what is known in the physics literature as a Sadovskii vortex, a configuration known to emerge as the endpoint of steady vortex-dipole branches. The construction is based on a sharp energy inequality under the two constraints of fixed impulse and Lp-norm, combined with an application of the concentration-compactness principle. This yields a family of axis-touching solutions parameterized by 1<p\leq\infty, interpolating between the classical Sadovskii vortex patch (corresponding to p=\infty) and vortices of arbitrarily high regularity as p\to1. In particular, the classical Chaplygin-Lamb dipole (corresponding to p=2) is recovered within this family. In previous variational constructions with mass constraints, the natural scaling leads to the restriction p>4/3. By removing the mass constraint, we obtain a unified scaling-invariant variational principle valid for all 1<p\leq\infty. As a consequence of the variational structure, we establish a Lyapunov-type stability result, demonstrating that the axis-touching geometry persists under small perturbations. Finally, we derive a quantitative bound on the horizontal center of mass of perturbed solutions, showing that they propagate at nearly the same speed as the underlying Sadovskii vortex.