Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel

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Abstract

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary.
Original languageEnglish
Pages (from-to)376-422
Number of pages47
JournalJournal of Differential Equations
Volume400
Early online date3 May 2024
DOIs
Publication statusE-pub ahead of print - 3 May 2024

Data Availability Statement

No data was used for the research described in the article.

Acknowledgements

We also thank the anonymous referee for their careful reading and useful comments.

Keywords

  • math.AP

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