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An Expanding Self-Similar Vortex Configuration for the 2D Euler Equations

Juan Dávila, Manuel del Pino, Monica Musso, Shrish Parmeshwar

Abstract

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $ω_{\varepsilon}(x,t)$ concentrated around points $ξ_{j}(t)$ that converge to a sum of Dirac delta masses as $\varepsilon\to0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $ξ_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $ξ_{j}$.

An Expanding Self-Similar Vortex Configuration for the 2D Euler Equations

Abstract

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity concentrated around points that converge to a sum of Dirac delta masses as . These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius around each .

Paper Structure

This paper contains 20 sections, 16 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Scheme of the proof
  3. Power Nonlinearity Vortex
  4. Vortex trajectories
  5. Approximate solution on $[T_0,T]$
  6. Construction of a solution
  7. Linearisation Around Point Vortices
  8. First Approximation on $\left[T_{0},T\right]$
  9. Constructing Solutions on $\left[T_{0},T\right]$
  10. Homotopic Operators
  11. Weighted $L^{2}$ Function Space
  12. The Poisson Equation
  13. Lower Bound on a Quadratic Form
  14. Weighted $L^{2}$ A Priori Estimates
  15. A Priori Estimates on a Projected Linear Problem
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\gamma\geq19$. There exists a constant $C>0$ such that for all $\varepsilon>0$ small enough, and all $T_0>0$ large enough, there exists a solution to 2d-euler-vorticity-stream, $(\omega_\varepsilon , \Psi_\varepsilon)$, on the whole interval $[T_0,\infty)$, with where we have the decomposition of the trajectories $\xi_{j}=\xi_{*j}+\tilde{\xi}_{j}$ with $\xi_{*j}$ defined in complex variables

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 4.1
  • Remark 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • proof
  • Remark 4.6
  • Remark 5.1
  • ...and 20 more