Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches

Yuchen Wang, Xin Xu, Maolin Zhou

TL;DR

The paper addresses degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches in the 2D Euler equations. It develops a real-analytic Lyapunov-Schmidt framework with a degenerate two-dimensional kernel, introducing a reduced two-parameter function F_2(λ,t;a) and analyzing it via higher-order expansions to overcome the loss of transversality. By computing Jacobian, Hessian, and explicit higher-order derivatives of F_2 around the degenerate points λ_{2p} (p = 2,3,4) and exploiting a blow-up in a, the authors establish the existence of transcritical bifurcations emanating from annuli A_{b_{2p}} with Ω = (1 − b_{2p}^2)/4, where b_{2p} solves b^{2p} = p − 1 − p b^2. This work extends the landscape of rotating vortex patches under strong degeneracy and provides a systematic methodology potentially applicable to other degenerate bifurcation problems in fluid dynamics.

Abstract

In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem, neither the kernel of linearization is one-dimensional nor the transeversallity condition holds. To this end, we make a detailed analysis on the nonlinear functional and the bifurcation curves are obtained by perturbing real algebraic varieties defined by truncated polynomials. In addition, our result partially answers an problem proposed by Hmidi and Mateu in \cite{Hmidi2016a} (\emph{Adv.Math.302 (2016), 799-850}).

Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches

TL;DR

The paper addresses degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches in the 2D Euler equations. It develops a real-analytic Lyapunov-Schmidt framework with a degenerate two-dimensional kernel, introducing a reduced two-parameter function F_2(λ,t;a) and analyzing it via higher-order expansions to overcome the loss of transversality. By computing Jacobian, Hessian, and explicit higher-order derivatives of F_2 around the degenerate points λ_{2p} (p = 2,3,4) and exploiting a blow-up in a, the authors establish the existence of transcritical bifurcations emanating from annuli A_{b_{2p}} with Ω = (1 − b_{2p}^2)/4, where b_{2p} solves b^{2p} = p − 1 − p b^2. This work extends the landscape of rotating vortex patches under strong degeneracy and provides a systematic methodology potentially applicable to other degenerate bifurcation problems in fluid dynamics.

Abstract

In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem, neither the kernel of linearization is one-dimensional nor the transeversallity condition holds. To this end, we make a detailed analysis on the nonlinear functional and the bifurcation curves are obtained by perturbing real algebraic varieties defined by truncated polynomials. In addition, our result partially answers an problem proposed by Hmidi and Mateu in \cite{Hmidi2016a} (\emph{Adv.Math.302 (2016), 799-850}).
Paper Structure (24 sections, 21 theorems, 312 equations)

This paper contains 24 sections, 21 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Set-up
  3. Conformal mappings and regularity
  4. Linearization on annulus
  5. Lyapunov-Schmidt reduction
  6. Heuristic discussion and Proof of the main result
  7. Heuristic discussion on the scheme of bifurcations
  8. Proof of main result Theorem \ref{['T:main']}
  9. Jacobian and Hessian of $F_2(\lambda,t;a)$
  10. First-order derivatives
  11. Compute $\frac{\mathrm{d}}{\mathrm{d} t}I_i(\phi_j(t,w))_{|t=0}$
  12. Compute $\frac{\mathrm{d}^2}{\mathrm{d} t^2}I_i(\phi_j(t,w))_{|t=0}$
  13. Compute $\overline{\mu}_{ij}$
  14. Second-order derivatives
  15. admissible parameters for non-isolated intersections
  16. ...and 9 more sections

Key Result

Theorem 1.1

There are non-trivial two-fold uniformly rotating vortex patches emanating from the annulus $A_{b_{2p}}$ with angular velocity $\Omega =\frac{1-b_{2p}^2}{4}$ for $p =2, 3, 4$, where $0<b_{2p}<1$ satisfies the algebraic equation Moreover, these bifurcation are transcritical.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['L:Real-analyticity']}
  • Lemma 2.3
  • proof
  • Remark 3.1
  • proof : Proof of Theorem \ref{['T:main']}
  • ...and 27 more