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

Ken Abe, Kyudong Choi, In-Jee Jeong, Young-Jin Sim, Kwan Woo

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.

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

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.

Paper Structure

This paper contains 26 sections, 28 theorems, 234 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sadovskii Vortices
  3. Main Results
  4. Reduction to the Half-Plane and the Vorticity--Stream Function Relation
  5. Key Ideas
  6. Related Topics
  7. Open Problems
  8. Organization of the Paper
  9. Variational Principle
  10. Stream Function Estimates
  11. Kinetic Energy Estimates
  12. Maximization of the Kinetic Energy
  13. Euler--Lagrange Equation
  14. Case $1<p<\infty$
  15. Case $p=\infty$
  16. ...and 11 more sections

Key Result

Theorem 1.2

For each $p \in (1,\infty]$, there exists a nonempty set $\mathcal{S}(p)$ of vorticities defined on $\mathbb R^2_+$ satisfying the following: More precisely, each $\omega \in \mathcal{S}(p)$ maximizes the kinetic energy under unit impulse and unit $L^p$ norm constraints, subject to positivity on $\mathbb R^2_+$ and odd-symmetry, with no restriction on the total mass.

Figures (1)

  • Figure 1: For an odd-symmetric dipolar vortex $\mathbf{1}_{\Omega_+}-\mathbf{1}_{\Omega_-}$, the upper region $\Omega_+$ and the lower region $\Omega_-$ interact to produce the steady flow $(W,0)$ that carries the dipole. Their mutual interaction ($\Omega_+$ pushes $\Omega_-$ and vice versa) overlaps at $(0,0)$, enhancing the central velocity so that $\overline{v}_1(0,0) > 2W$.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1.2: Sadovskii vortex family
  • Theorem 1.3: Stability of the Sadovskii vortex patch
  • Remark 1.4
  • Remark 1.5: Steiner symmetry
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 58 more