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On Finiteness of Stationary Configurations of the Planar Five-vortex Problem

Xiang Yu, Shuqiang Zhu

TL;DR

The paper advances the finiteness theory for the planar five-vortex problem by combining Albouy–Kaloshin singular-sequence methods with a detailed diagrammatic analysis. It proves sharp universal upper bounds for equilibria and rigidly translating configurations and shows finiteness of relative equilibria and collapse configurations outside codimension-2 vorticity subvarieties, with all-same-sign vorticities guaranteeing finiteness. The approach hinges on translating singular-sequence limits into two-colored diagrams, constructing and excluding 39 candidate diagrams (ultimately retaining 31), and deriving explicit constraints on the vorticities for the surviving cases. The results provide a rigorous framework linking algebraic geometry, central configurations, and fluid-mechanical vortex dynamics, with implications for the qualitative understanding of multi-vortex motion and its generic finiteness properties.

Abstract

The finiteness problem of stationary configurations for the planar five-vortex problem is considered in this paper. The numbers of equilibria and rigidly translating configurations are shown to be at most 6 and 24 respectively. The numbers of relative equilibria and collapse configurations are shown to be finite, except perhaps if the 5-tuple of vorticities belongs to a given codimension 2 subvariety of the vorticity space. In particular, if the vorticities are of the same sign, the number of stationary configurations is finite.

On Finiteness of Stationary Configurations of the Planar Five-vortex Problem

TL;DR

The paper advances the finiteness theory for the planar five-vortex problem by combining Albouy–Kaloshin singular-sequence methods with a detailed diagrammatic analysis. It proves sharp universal upper bounds for equilibria and rigidly translating configurations and shows finiteness of relative equilibria and collapse configurations outside codimension-2 vorticity subvarieties, with all-same-sign vorticities guaranteeing finiteness. The approach hinges on translating singular-sequence limits into two-colored diagrams, constructing and excluding 39 candidate diagrams (ultimately retaining 31), and deriving explicit constraints on the vorticities for the surviving cases. The results provide a rigorous framework linking algebraic geometry, central configurations, and fluid-mechanical vortex dynamics, with implications for the qualitative understanding of multi-vortex motion and its generic finiteness properties.

Abstract

The finiteness problem of stationary configurations for the planar five-vortex problem is considered in this paper. The numbers of equilibria and rigidly translating configurations are shown to be at most 6 and 24 respectively. The numbers of relative equilibria and collapse configurations are shown to be finite, except perhaps if the 5-tuple of vorticities belongs to a given codimension 2 subvariety of the vorticity space. In particular, if the vorticities are of the same sign, the number of stationary configurations is finite.

Paper Structure

This paper contains 23 sections, 22 theorems, 83 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Basic notations
  3. Equilibria and rigidly translating configurations
  4. Equilibria
  5. Rigidly translating configurations
  6. Singular sequences for central configurations and coloring rules
  7. Central configurations of the planar N-vortex problem
  8. Singular sequences
  9. The two-colored diagrams
  10. Constraints when some sub-diagrams appear
  11. Construction of the 5-vortex diagrams
  12. No two-colored vertex
  13. $C=2$
  14. $C=3$
  15. $C=4$
  16. ...and 8 more sections

Key Result

Theorem 1.1

For the planar five-vortex problem with nonzero vorticities $\Gamma_n$$(n\in\{1,2,3,4, 5\})$,

Figures (16)

  • Figure 1: On the left, vertices 1, 2 are $z$-circled, and a $z$-edge is between them; In the middle, vertices 1, 2 are $z$- and $w$-circled, and a $zw$-edge is between them; On the right, vertices 1, 2 are $w$-circled, and a $w$-edge is between them.
  • Figure 2: Four possible diagram for no two-colored vertex. They correspond to Diagram 15 of Figure \ref{['fig:list2']}, Diagram 1 of Figure \ref{['fig:list1']}, Diagram 6 of Figure \ref{['fig:list1']}, and Diagram 16 of Figure \ref{['fig:list2']} respectively.
  • Figure 3: Two diagrams for $C=2$. The first one has been excluded. The second one corresponds to Diagram 18 of Figure \ref{['fig:list2']}.
  • Figure 4: Two diagrams for $C=3$. Both have been excluded.
  • Figure 5: Five diagrams for $C=4$, two $zw$-edges. The second and fourth one has been excluded. The first, third and fifth one correspond to Diagram 1 of Figure \ref{['fig:list2']}, Diagram 2 of Figure \ref{['fig:list1']}, and Diagram 20 of Figure \ref{['fig:list2']} respectively.
  • ...and 11 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Definition 2.4
  • Proposition 3.1
  • ...and 37 more