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Bounds on Growth and Impossibility of Collapse for Point Vortex Systems

Samuel Zbarsky

Abstract

We consider 2D point vortex systems and, under certain conditions on the masses of the point vortices, prove that collapse is impossible and provide bounds on the growth of the system. The bounds are typically of the form $O(t^a)$ for some $a<1/2$, but we obtain various results for various assumptions on the masses.

Bounds on Growth and Impossibility of Collapse for Point Vortex Systems

Abstract

We consider 2D point vortex systems and, under certain conditions on the masses of the point vortices, prove that collapse is impossible and provide bounds on the growth of the system. The bounds are typically of the form for some , but we obtain various results for various assumptions on the masses.
Paper Structure (9 sections, 9 theorems, 62 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bounds on the kernel and notation
  3. Acknowledgements
  4. Examples
  5. Movement bound
  6. One big negative
  7. Clusters
  8. No collisions and growth slower than $T^{1/2}$
  9. Improvements, corollaries, and comments

Key Result

Theorem 3

Figures (2)

  • Figure 1: This is an illustration of the clusters in the statement of Lemma \ref{['lem:nocollisionsandslowgrowth']}
  • Figure 2: This is an illustration of the clusters $U_i$

Theorems & Definitions (17)

  • Theorem 3
  • Conjecture 4
  • Lemma 5
  • proof
  • Theorem 6
  • Theorem 7
  • proof
  • Definition 8
  • Lemma 9
  • proof
  • ...and 7 more