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Cusp Formation in Vortex Patches

Tarek M. Elgindi, Min Jun Jo

TL;DR

This work resolves the conjecture that an acute corner of a vortex patch for the 2D Euler equations cusp-forms instantaneously. By exploiting a two-fold symmetry and a moving-frame reformulation, the authors derive an effective angular-transport system for corner geometry, reduce it to an autonomous ODE for the corner parameters A(τ), B(τ), and then justify this reduction via a perturbative analysis around an approximate solution ω_g. A key ingredient is the average-perturbation functional F(t,r), for which recursive, log-scale estimates show the perturbation vanishes near the origin on the cusp time scale, forcing the cusp and yielding a bound indicating logarithmic sharpness of the cusp. The results rigorously confirm numerical predictions and lay a framework to study cusp formation in 2D Euler patches, including potential extensions to other corner geometries and times beyond the immediate cusp onset.

Abstract

We prove instantaneous cusp formation for any initial vortex patch with acute corners. This was conjectured to occur in the numerical literature.

Cusp Formation in Vortex Patches

TL;DR

This work resolves the conjecture that an acute corner of a vortex patch for the 2D Euler equations cusp-forms instantaneously. By exploiting a two-fold symmetry and a moving-frame reformulation, the authors derive an effective angular-transport system for corner geometry, reduce it to an autonomous ODE for the corner parameters A(τ), B(τ), and then justify this reduction via a perturbative analysis around an approximate solution ω_g. A key ingredient is the average-perturbation functional F(t,r), for which recursive, log-scale estimates show the perturbation vanishes near the origin on the cusp time scale, forcing the cusp and yielding a bound indicating logarithmic sharpness of the cusp. The results rigorously confirm numerical predictions and lay a framework to study cusp formation in 2D Euler patches, including potential extensions to other corner geometries and times beyond the immediate cusp onset.

Abstract

We prove instantaneous cusp formation for any initial vortex patch with acute corners. This was conjectured to occur in the numerical literature.

Paper Structure

This paper contains 25 sections, 13 theorems, 149 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Simplificiation via two-fold symmetry
  4. Initial vorticity configuration
  5. Reformulation under two-fold symmetry
  6. Some related works
  7. Heuristics of proof
  8. Outline of the paper
  9. The Effective System
  10. Formal derivation of the effective system Lg
  11. Local well-posedness of Lg
  12. Asymptotic equivalence of Lg and Lg
  13. Global existence, the decay of Lg, and the convergence of Lg
  14. Perturbation Estimates
  15. Set-up of the perturbative regime
  16. ...and 10 more sections

Key Result

Theorem A

Assume $\omega_0=\chi_{\Omega_0}$ for some simply connected set $\Omega_0\subset \mathbb{R}^2$. Assume that $\Omega_0$ has an acute corner at $x=0$. Then the corresponding unique vortex patch solution $\omega(\cdot,t)=\chi_{\Omega(t)}$ develops a cusp instantaneously at the image of $x=0$ under the where we are evaluating at $t=t(r)\rightarrow 0$ as $r\rightarrow 0.$ Here, $\Phi_t(0)$ denotes the

Figures (8)

  • Figure 1: Bahouri-Chemin solution on a periodic square
  • Figure 2: Danchin2, p.497: Fig. 10, evolution of an initial acute-angle of a vortex patch.
  • Figure 3: Danchin2, p.494: Fig. 7(a), angle-in-time graph computed for different resolutions in the case of an initial patch with an acute corner. The resolution increases from top to bottom.
  • Figure 4: A prototype of our initial patch $\Omega_0$
  • Figure 5: Instantaneous cusping near $r=0$ at $t\to 0^{+}$
  • ...and 3 more figures

Theorems & Definitions (32)

  • Conjecture 1.1: Danchin2, p.496
  • Definition 1.2
  • Theorem A: Instantaneous cusp formation
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: EJ, Chapter 5
  • Remark 2.2
  • Lemma 2.3
  • ...and 22 more