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Multifractal scaling and the Euler equations on R^3/Z^3

Henrik Ueberschaer

Abstract

We study the Euler equations describing the motion of an incompressible fluid on the cubic torus with real initial data. We construct solutions on the Fourier side which display a sudden loss of regularity within finite time even for highly regular initial data. Moreover, the solution may regain its initial regularity within finite time. This loss of regularity may coincide with the appearance of a certain type of multifractal scaling of the solutions.

Multifractal scaling and the Euler equations on R^3/Z^3

Abstract

We study the Euler equations describing the motion of an incompressible fluid on the cubic torus with real initial data. We construct solutions on the Fourier side which display a sudden loss of regularity within finite time even for highly regular initial data. Moreover, the solution may regain its initial regularity within finite time. This loss of regularity may coincide with the appearance of a certain type of multifractal scaling of the solutions.
Paper Structure (12 sections, 2 theorems, 52 equations)

This paper contains 12 sections, 2 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Set-up.
  3. Main result
  4. Multifractal scaling
  5. Real data on the $3$-torus
  6. Derivation
  7. Construction of the solution
  8. Proof of theorem \ref{['3d-blowup']}
  9. Blow-up of Sobolev norms
  10. Regaining regularity
  11. Proof of Corollary \ref{['mult']}
  12. Loss of regularity for complex data on the $2$-torus

Key Result

Theorem 1.1

Fix two functions $f_1,f_2:{\mathbb Z}^3\to{\mathbb R}_+$. There exists a solution to Fourier-system such that the initial data satisfy $|u_0(\xi)|\leq f_1(\xi)$ for $|\xi|>1$, the $\hat{u}(\xi,\cdot)$ are smooth in the time variable and there exists a time $T_0>0$ and a vector $\xi_1\in{\mathbb Z}^

Theorems & Definitions (5)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Corollary 2.1
  • Remark 3