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Generic small-scale creation in the two-dimensional Euler equation

Thomas Alazard, Ayman Rimah Said

Abstract

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_δ$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.

Generic small-scale creation in the two-dimensional Euler equation

Abstract

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.
Paper Structure (40 sections, 42 theorems, 367 equations)

This paper contains 40 sections, 42 theorems, 367 equations.

Table of Contents

  1. Introduction
  2. The equations
  3. The Yudovich Conjecture
  4. Strategy of the proof and plan of the paper
  5. Preliminaries
  6. Notations and conventions
  7. Well-posedness and Flow Map
  8. Alinhac's Paracomposition Operators
  9. The Paracomposition Operator
  10. Main Identities
  11. Paracomposition of the Laplacian
  12. Parametrix Construction
  13. Preliminaries
  14. The Elliptic Parametrix
  15. The vector field behind Shnirelman's unknown
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $s> 4$ be a real number. The set of initial data $\omega_0\in H^{s}_0(\mathbb{T}^2)$ such that the corresponding unique solution of the two-dimensional incompressible Euler equations satisfies is a dense $G_\delta$ set in $H^s_0(\mathbb{T}^2)$.

Theorems & Definitions (90)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Conjecture 1.6: Yudovich (1974), yudovic1974losszbMATH01620965, quote from zbMATH05308906
  • Remark 1.7
  • Theorem 2.1: Global Well-posedness
  • Definition 3.1
  • Remark 3.1
  • ...and 80 more