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2D point vortex dynamics in bounded domains: global existence for almost every initial data

Martin Donati

Abstract

In this paper, we prove that in bounded planar domains with $C^{2,α}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [13] for the unit disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. In this paper, we make extensive use of the estimates given in [7]. We establish our relevant inequalities first in simply connected domains using conformal maps, then in multiply connected domains.

2D point vortex dynamics in bounded domains: global existence for almost every initial data

Abstract

In this paper, we prove that in bounded planar domains with boundary, for almost every initial condition in the sense of the Lebesgue measure, the point vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [13] for the unit disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. In this paper, we make extensive use of the estimates given in [7]. We establish our relevant inequalities first in simply connected domains using conformal maps, then in multiply connected domains.

Paper Structure

This paper contains 14 sections, 29 theorems, 242 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simply connected and exterior domains
  3. Holomorphic maps
  4. The Riemann Mapping Theorem
  5. Green's Function
  6. Intermediate lemmas
  7. Inequalities for simply connected bounded domains and exterior domains
  8. Multiply connected domain
  9. Biot-Savart law
  10. Inequalities for multiply connected domains
  11. Completion of the proof of Theorem \ref{['mainresult']}
  12. Construction of a regularized dynamic
  13. End of the proof of Theorem \ref{['mainresult']}
  14. Acknowledgments.

Key Result

Theorem 1.1

If $\Omega = D(0,1)$ the open unit disk, then the point vortex dynamics ptvortexdynamic for any fixed number of points $N \ge 1$ and masses $(a_i)_i \in \mathbb{R}^N$ is globally well defined except maybe for a set of initial conditions in $\Omega^N$ which has vanishing Lebesgue measure.

Figures (2)

  • Figure 1: An example for $m=1$. The domains $\Omega$, $\Omega_0$ and $\Omega_1$ and their boundaries.
  • Figure 2: Decomposition \ref{['decompOmegaVj']}.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2: Riemann Mapping Theorem
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 36 more