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On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications

Tarek. M. Elgindi, Ryan. W. Murray, Ayman. R. Said

Abstract

We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise constant with finitely many jumps. All continuous scale-invariant solutions become singular and homogenize in infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin. This monotonicity also applies locally to solutions of the 2d Euler equation that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of symmetry. Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis then gives generic spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry.

On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications

Abstract

We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on relax to states that are piece-wise constant with finitely many jumps. All continuous scale-invariant solutions become singular and homogenize in infinite time. On , this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin. This monotonicity also applies locally to solutions of the 2d Euler equation that are -fold symmetric () and have radial limits at the point of symmetry. Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like and ). Our analysis then gives generic spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry.
Paper Structure (25 sections, 27 theorems, 170 equations, 4 figures)

This paper contains 25 sections, 27 theorems, 170 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Scale-Invariant Solutions
  3. Summary of results
  4. The zero-homogeneous Euler System: the theorems
  5. Proof of Theorem \ref{['0-hom Euler Cauchy thm']}
  6. The Riccati structure
  7. $\bullet$ The equation on $\partial_\theta G$.
  8. An Entropy: The expanding and contracting sets
  9. Detailed analysis of the expanding and contracting sets
  10. Integrating the Riccati equation:
  11. On the finiteness of the expanding set
  12. $\bullet$ Scenario 1.
  13. $\bullet$ Scenario 2.
  14. Relaxation to jump states when $g_0$ admits limits on $E$
  15. Relaxation to the average when $E$ contains a continuity point
  16. ...and 10 more sections

Key Result

Theorem 1

Consider the 2d Euler equation on $\mathbb{R}^2.$

Figures (4)

  • Figure 1: The expanding set associated to the evolution in Figure \ref{['fig:homoclinic']}
  • Figure 2: An orbit of the system which approaches zero at $t = \pm \infty$.
  • Figure 3: An orbit of the system connecting two different steady states towards $t = \pm \infty$.
  • Figure 4: Evolution on the Sphere. On the left an initial vorticity is displayed, and on the right a caricature of how the vorticity will look at large time.

Theorems & Definitions (51)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1
  • Definition 1.3
  • Definition 1.4
  • Theorem : Main theorem of EJ-symmetries
  • Corollary 1.5
  • Theorem 2
  • Corollary 1.6
  • Remark 1.7
  • ...and 41 more