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Small scale creation for solutions of the incompressible two dimensional Euler equation

Alexander Kiselev, Vladimir Sverak

TL;DR

The paper investigates whether the known double exponential upper bound for the growth of the vorticity gradient in the 2D Euler equations is sharp. It constructs smooth, symmetry-preserving initial data on the unit disk that induce persistent hyperbolic dynamics near the boundary, isolating a nonlocal stretching mechanism that drives the gradient growth. A Key Lemma decouples the main hyperbolic term from bounded error terms, enabling precise tracking of boundary trajectories and a Gronwall-type analysis that yields double exponential growth for ∥∇ω∥ and ∥∇u∥ at all times. The result confirms that the double exponential bound is optimal in this setting and highlights boundary effects as a robust conduit for rapid small-scale creation in incompressible flows.

Abstract

We construct an initial data for two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp - the double exponential growth is the fastest possible growth rate.

Small scale creation for solutions of the incompressible two dimensional Euler equation

TL;DR

The paper investigates whether the known double exponential upper bound for the growth of the vorticity gradient in the 2D Euler equations is sharp. It constructs smooth, symmetry-preserving initial data on the unit disk that induce persistent hyperbolic dynamics near the boundary, isolating a nonlocal stretching mechanism that drives the gradient growth. A Key Lemma decouples the main hyperbolic term from bounded error terms, enabling precise tracking of boundary trajectories and a Gronwall-type analysis that yields double exponential growth for ∥∇ω∥ and ∥∇u∥ at all times. The result confirms that the double exponential bound is optimal in this setting and highlights boundary effects as a robust conduit for rapid small-scale creation in incompressible flows.

Abstract

We construct an initial data for two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp - the double exponential growth is the fastest possible growth rate.

Paper Structure

This paper contains 5 sections, 4 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Key Lemma
  4. The Proof of the Main Theorem
  5. Discussion

Key Result

Theorem 1.1

Consider two-dimensional Euler equation on a unit disk $D.$ There exists a smooth initial data $\omega_0$ with $\|\nabla \omega_0\|_{L^\infty}/\|\omega_0\|_{L^\infty} >1$ such that the corresponding solution $\omega(x,t)$ satisfies for some $c>0$ and for all $t \geq 0.$

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['upperdexp']}
  • proof : Proof of Proposition \ref{['kato']}
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['mainthm']}