On existence of Sadovskii vortex patch: A touching pair of symmetric counter-rotating uniform vortex

The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has ga...

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Bibliographic Details
Main Authors Choi, Kyudong, Jeong, In-Jee, Sim, Young-Jin
Format Journal Article
LanguageEnglish
Published 17.06.2024
Subjects
Mathematics - Analysis of PDEs
Mathematics - Mathematical Physics
Physics - Mathematical Physics
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Summary:The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in inviscid limit of planar flows via Prandtl--Batchelor theory and as the asymptotic state for vortex ring dynamics. In this work, we prove the existence of a Sadovskii vortex patch, by solving the energy maximization problem under the exact impulse condition and an upper bound on the circulation.
DOI:10.48550/arxiv.2406.11379