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Desingularization of nondegenerate rotating vortex patches

Răzvan-Octavian Radu, Noah Stevenson

TL;DR

This work shows that nondegenerate steady rotating vortex patches in the plane or disk can be realized as limits of smooth, compactly supported Euler solutions with dihedral symmetry, effectively desingularizing classical singular vortex patches. The authors introduce a nondegeneracy condition expressed via the triviality of the kernel of a boundary linearization and develop a robust local-invertibility framework based on a boundary-oriented, tubular-neighborhood formulation of the stream function. By regularizing the vorticity nonlinearity and applying a Newton-type method with uniform estimates, they construct smooth desingularizations of admissible patches and demonstrate a splitting and trapping phenomenon for nearby solutions. The paper confirms two canonical nondegenerate families—the Kirchhoff elliptical vortices and Burbea’s Rankine perturbations—as admissible, providing the first general desingularization procedure applicable to broad families of steady rotating vortex patches and enabling the construction of exotic singular patch-like states nearby nondegenerate patches.

Abstract

This paper analyses the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a natural nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, that is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of delicate analysis on thin domains, nonlinear a priori estimates, and a custom version of Newton's method. Our techniques are robust enough to additionally allow us to construct exotic families of singular rotating vortex patch-like solutions nearby a given nondegenerate state. To the best of the authors' knowledge, this work constitutes the first desingularization procedure applicable to general families of steady rotating vortex patches.

Desingularization of nondegenerate rotating vortex patches

TL;DR

This work shows that nondegenerate steady rotating vortex patches in the plane or disk can be realized as limits of smooth, compactly supported Euler solutions with dihedral symmetry, effectively desingularizing classical singular vortex patches. The authors introduce a nondegeneracy condition expressed via the triviality of the kernel of a boundary linearization and develop a robust local-invertibility framework based on a boundary-oriented, tubular-neighborhood formulation of the stream function. By regularizing the vorticity nonlinearity and applying a Newton-type method with uniform estimates, they construct smooth desingularizations of admissible patches and demonstrate a splitting and trapping phenomenon for nearby solutions. The paper confirms two canonical nondegenerate families—the Kirchhoff elliptical vortices and Burbea’s Rankine perturbations—as admissible, providing the first general desingularization procedure applicable to broad families of steady rotating vortex patches and enabling the construction of exotic singular patch-like states nearby nondegenerate patches.

Abstract

This paper analyses the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a natural nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, that is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of delicate analysis on thin domains, nonlinear a priori estimates, and a custom version of Newton's method. Our techniques are robust enough to additionally allow us to construct exotic families of singular rotating vortex patch-like solutions nearby a given nondegenerate state. To the best of the authors' knowledge, this work constitutes the first desingularization procedure applicable to general families of steady rotating vortex patches.

Paper Structure

This paper contains 16 sections, 27 theorems, 286 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Euler equations in the plane and in the disk, steady rotating solutions, and elliptical vortices
  3. Survey of prior work
  4. Main results and discussion
  5. Conventions of notation
  6. Existence of admissible states
  7. Kirchhoff elliptical vortices
  8. Perturbations of the Rankine vortex
  9. Analysis near admissible states
  10. Preliminary estimates and identities
  11. Key consequences of nondegeneracy
  12. Quantitative local invertibility
  13. Synthesis
  14. Tools from analysis
  15. Uniform local level set coordinates
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Psi$, $\Omega$, $c$, $\widetilde{U}$, and $m$ be an admissible steady rotating vortex patch in the sense of Definition defn of admissible patches and let $\gamma\in C^\infty(\mathbb{R})$ be any function that satisfies $\gamma(t) = 0$ for $t\leqslant -1$ and $\gamma(t) = 1$ for $t\geqslant 1$. and the following hold.

Figures (1)

  • Figure 1: Shown here are the sets $U$, $U^{\mathrm{in}}$, and $U^{\mathrm{out}}$ produced by Lemma \ref{['lem on simple consequences of admissibility']} and their relation to $\Sigma$ (the curve in solid black). The region in blue is the topological tubular neighborhood $U$. The set $U^{\mathrm{in}}$ is the bounded connected component of the white region. The set $U^{\mathrm{out}}$ is the complement of $\overline{U\cup U^{\mathrm{in}}}$. The support of the vorticity $\widetilde{U}$ is not emphasized in the diagram, but it is the bounded simply connected region enclosed by $\Sigma$.

Theorems & Definitions (58)

  • Definition 1.1: Steady rotating weak solutions to the Euler system
  • Definition 1.2: Admissible steady rotating vortex patch solutions
  • Theorem 1: Desingularization of admissible states
  • Theorem 2: Splitting of admissible states
  • Theorem 3: Trapping of admissible states
  • Theorem 4: Existence of admissible states
  • Lemma 2.1: Stream functions for Kirchhoff vortices
  • proof
  • Proposition 2.2: Nondegeneracy on Kirchhoff stream functions, I
  • proof
  • ...and 48 more