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Helical kelvin waves for the 3D Euler equation

Daomin Cao, Boquan Fan, Rui Li, Guolin Qin

Abstract

Helical Kelvin waves were conjectured to exist for the 3D Euler equations in Lucas and Dritschel \cite{LucDri} (as well as in \cite{Chu}) by studying dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these $m$-fold symmetric helical Kelvin waves, in both simply and doubly connected cases, for the 3D Euler equations. The construction is based on linearization of contour dynamics equations and bifurcation theory. Our results rigorously verify the prediction in aforementioned papers and extend $m$-waves of Kelvin from the 2D Euler equations to the 3D helically symmetric Euler equations.

Helical kelvin waves for the 3D Euler equation

Abstract

Helical Kelvin waves were conjectured to exist for the 3D Euler equations in Lucas and Dritschel \cite{LucDri} (as well as in \cite{Chu}) by studying dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these -fold symmetric helical Kelvin waves, in both simply and doubly connected cases, for the 3D Euler equations. The construction is based on linearization of contour dynamics equations and bifurcation theory. Our results rigorously verify the prediction in aforementioned papers and extend -waves of Kelvin from the 2D Euler equations to the 3D helically symmetric Euler equations.

Paper Structure

This paper contains 16 sections, 20 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. The 3D Euler equations under helical symmetry
  3. Kelvin waves for the 2D Euler equations
  4. Main results and ideas of proofs
  5. Notation and organization
  6. Preliminary: the Green function for and trivial solutions
  7. Simply connected time-periodic solutions
  8. Contour dynamics for helical Kelvin waves
  9. Functional setting and regularity analysis
  10. Function spaces
  11. Regularity of the functional $\mathcal{F}$
  12. Kernel of $d_r\mathcal{F}(\Omega,0)$ and Spectral analysis
  13. Proof of Theorem \ref{['thm-simply']}
  14. Doubly connected time-periodic solutions
  15. Crandall-Rabinowitz Theorem
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $a>0, h>0$ be two numbers and $m\geq 2$ be an integer. Then there exist three constants $M(a,h)$, $H_0(a)$ and $H_1(a)$ such that if either $m>M(a,h)$ or $h\in (0, H_0(a))\cup (H_1(a), +\infty)$ , then there exists a curve of simply connected helical $m$-waves of Kelvin $\omega_H$ for eq1.6 bifu at the angular velocity Here $I_m$ and $K_m$ are the modified Bessel functions of the first and se

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 29 more