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Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results

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Abstract

In this paper we consider the steady water wave problem for waves that possess a merely \(L_r\)-integrable vorticity, with \(r\in (1,\infty )\) being arbitrary. We first establish the equivalence of the three formulations – the velocity formulation, the stream function formulation, and the height function formulation – in the setting of strong solutions, regardless of the value of \(r\). Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small-amplitude capillary and capillary–gravity water waves with an \(L_r\)-integrable vorticity.

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Acknowledgments

The authors thank the anonymous referees for their valuable suggestions, which have improved the quality of the article.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 , Wien, Austria

    Calin Iulian Martin & Bogdan-Vasile Matioc

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  1. Calin Iulian Martin
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  2. Bogdan-Vasile Matioc
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Correspondence to Calin Iulian Martin.

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Communicated by P. Newton.

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Martin, C.I., Matioc, BV. Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results. J Nonlinear Sci 24, 633–659 (2014). https://doi.org/10.1007/s00332-014-9201-1

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  • DOI: https://doi.org/10.1007/s00332-014-9201-1

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