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On Singular Vortex Patches, I: Well-posedness Issues
About this Title
Tarek M. Elgindi and In-Jee Jeong
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 283, Number 1400
ISBNs: 978-1-4704-5682-5 (print); 978-1-4704-7401-0 (online)
DOI: https://doi.org/10.1090/memo/1400
Published electronically: January 20, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Background Material
- 3. Global well-posedness for symmetric patches in an intermediate space
- 4. Global well-posedness for symmetric $C^{1,\alpha }$-patches with corners
- 5. Ill-posedness results for vortex patches with corners
- 6. Effective system for the boundary evolution near the corner
- A. Appendix
Abstract
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally $m$-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as $m\geq 3.$ In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle $\frac {\pi }{2}$ for all time. Even in the case of vortex patches with corners of angle $\frac {\pi }{2}$ or with corners which are only locally $m$-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on $\mathbb {R}^2$ with interesting dynamical behavior such as cusping and spiral formation in infinite time.- R. C. Alexander. Family of similarity flows with vortex sheets. The Physics of Fluids, 14(2):231–239, 1971.
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