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On the rigidity of uniformly rotating vortex patch near the Rankine vortex

Yupei Huang

Abstract

In this paper, we study the uniformly rotating vortex patch solutions for the 2D incompressible Euler equations. Specifically, we prove that if the patch solution is close to the Rankine vortex in a certain weak topology, it is either the Kirchhoff ellipses or the Rankine vortex.

On the rigidity of uniformly rotating vortex patch near the Rankine vortex

Abstract

In this paper, we study the uniformly rotating vortex patch solutions for the 2D incompressible Euler equations. Specifically, we prove that if the patch solution is close to the Rankine vortex in a certain weak topology, it is either the Kirchhoff ellipses or the Rankine vortex.
Paper Structure (8 sections, 8 theorems, 56 equations)

This paper contains 8 sections, 8 theorems, 56 equations.

Table of Contents

  1. introduction
  2. The V-states and main results
  3. Organization of the paper
  4. Notations
  5. Bifurcation analysis near the Rankine vortex and the proof of Theorem \ref{['Gomez']}
  6. quantitative estimates regarding vortex patch
  7. Appendix
  8. On the proof of Lemma \ref{['bifurcation1']}

Key Result

Theorem 1.1

Let $\mathcal{D}$ be a V-state with angular velocity $\Omega$. Assume that the patch boundary $\partial \mathcal{D}$ can be described as a $C^{1,\frac{1}{2}}$ graph of its argument $\theta$: For sufficiently small $\delta$, if and then $\mathcal{D}$ is a rotating disk or a rotating ellipse.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • proof : Proof of Remark \ref{['Center of vorticity']}
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['Gomez']}
  • Theorem 3.1
  • Theorem 3.2
  • ...and 9 more