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Wellposedness and singularity formation beyond the Yudovich class

Tarek M. Elgindi, Ryan W. Murray, Ayman R. Said

Abstract

We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.

Wellposedness and singularity formation beyond the Yudovich class

Abstract

We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.
Paper Structure (17 sections, 13 theorems, 141 equations)

This paper contains 17 sections, 13 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Main Results
  4. Some remarks on the proof
  5. Organization of the Paper
  6. Wellposedness Theory
  7. Propagation of the analytic regularity
  8. Uniqueness in $C^\omega_{-\alpha}$
  9. Motion of the fluid at the origin
  10. Finite time blow up for $C^\omega_{-\alpha}$ vorticity and some consequences
  11. Proof of Theorem \ref{['thm:finite time 0-hom']}
  12. Nonlinear instability in $L^p_{loc}\left(\mathbb{R}^2\right)$ of the 2d Euler equation
  13. Application: a potential non uniqueness scenario for the unforced 2d Euler equation in $L^p$
  14. Separation of trajectories estimate
  15. The Biot-Savart law in polar coordinates
  16. ...and 2 more sections

Key Result

Theorem 1.1

Consider solutions $\omega(r,\theta,t)$ to eq: 2dEuler1-eq: 2dEuler2 that are odd and $\frac{2\pi}{m}$ periodic in $\theta$ with $m\geq 3$ and singularity not worse than $(r+\theta)^{-\alpha}$ for $\theta\in(0,\frac{\pi}{m})$ for some $0<\alpha<1.$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 14 more