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Small scale creation of the Lagrangian flow in 2d perfect fluids

Ayman Rimah Said

Abstract

In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman.

Small scale creation of the Lagrangian flow in 2d perfect fluids

Abstract

In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like . This initial data dependent norm quantifies the exact decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman.
Paper Structure (19 sections, 25 theorems, 129 equations)

This paper contains 19 sections, 25 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Propagation of exact smoothness in the 2d Euler equation
  3. The forward frequency cascade
  4. Active scalar equations
  5. Organisation of the paper
  6. Exact smoothness propagation in 2d Euler
  7. Analysis of the norm $\left\Vert\cdot \right\Vert_{\omega_0}$
  8. Actions of operators on $\left\Vert \cdot \right\Vert_{\omega_0}$
  9. Convergence of semi-classical measures
  10. Convergence in $L^2(\mathbb{R}^2)$
  11. The rate of convergence
  12. The monotone quantity in 2d Euler
  13. Proof of Theorem \ref{['thm: intro lyap']}
  14. Proof of Corollary \ref{['cor: blw up']}
  15. Notions of mircolocal analysis
  16. ...and 4 more sections

Key Result

Theorem 1.2

Consider the 2d Euler equation on $\mathbb{R}^2.$ Then for $\omega_0\in H^{s}(\mathbb{R}^2)\setminus H^{s+\epsilon}(\mathbb{R}^2)$ for some $s>1$ and all $\epsilon>0$, then roughly the s+1 derivative of the Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$.

Theorems & Definitions (43)

  • Conjecture 1.1: Yudovich (1974), yudovic1974lossyudovich2000loss, quote from morgulis2008loss
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 33 more