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Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation

Woohyu Jeon

Abstract

In this paper, we investigate the persistency of loglog singular structure for 2D Euler equation under the perturbation of continuous function. First, we prove that when loglog vortex is perturbed by the function with modulus of continuity $μ(r)=\left( \log \frac{1}{r}\right)^{-α} ~~ (0<α<1)$, the loglog vortex is propagated while the perturbation term maintaining same modulus of continuity. We then show that this result is sharp: There exists a smooth function such that when the loglog vortex is perturbed by the smooth function, the norm of perturbation term corresponding to modulus of continuity $μ(r)=\left( \log \frac{1}{r}\right)^{-α} $ instantly blows up for all $α>1$.

Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation

Abstract

In this paper, we investigate the persistency of loglog singular structure for 2D Euler equation under the perturbation of continuous function. First, we prove that when loglog vortex is perturbed by the function with modulus of continuity , the loglog vortex is propagated while the perturbation term maintaining same modulus of continuity. We then show that this result is sharp: There exists a smooth function such that when the loglog vortex is perturbed by the smooth function, the norm of perturbation term corresponding to modulus of continuity instantly blows up for all .
Paper Structure (13 sections, 18 theorems, 158 equations, 1 table)

This paper contains 13 sections, 18 theorems, 158 equations, 1 table.

Table of Contents

  1. Introduction
  2. Persistence of singular structure
  3. Main Result
  4. Notations
  5. Idea of the proof
  6. Organization of the paper
  7. Conservation of Modulus of Continuity
  8. Elementary Lemmas
  9. Proof of the Main Theorem: Conservation Part
  10. Breakdown of Modulus of Continuity
  11. Construction of Initial Data and Basic Properties of the Solution
  12. The Key Lemma
  13. Proof of the Main Theorem: Breakdown Part

Key Result

Proposition 1.1

Let $g$ be any bounded function with compact support. Then, there exists a unique global weak solution $($in the standard distribution sense$)$ for EE-Biot-Savart starting from initial vorticity In addition, the solution $w$ is Lagrangian; The solution $w(x,t)$ could be written by where $\Phi_t^{-1}(x)$ is the inverse flow map of $u~(\coloneqq \nabla^\perp \Delta^{-1} w)$.

Theorems & Definitions (37)

  • Proposition 1.1
  • Definition 1.3
  • Theorem 1.4: Conservation of modulus of continuity in the propagation of loglog vortex
  • Theorem 1.5: Breakdown of modulus of continuity in the propagation of loglog vortex
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1
  • Lemma 2.2
  • ...and 27 more