Growth of perimeter for vortex patches in a bulk

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on T2 and R2 whose perimeter grows with time. More precisely, for any constant M>0, we construct a vortex patch in T2 whose smooth boundary has length of order 1 at the initial time su...

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Bibliographic Details
Published in:Applied mathematics letters 2021-03, Vol.113, p.106857, Article 106857
Main Authors: Choi, Kyudong, Jeong, In-Jee
Format: Article
Language:English
Subjects:
2D Euler
Large time behavior
Particle trajectory
Perimeter
Stability
Vortex patch
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Summary:We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on T2 and R2 whose perimeter grows with time. More precisely, for any constant M>0, we construct a vortex patch in T2 whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant M within finite time. The construction is done by cutting a thin slit out of an almost square patch. A similar result holds for an almost round patch with a thin handle in R2.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2020.106857