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Capillary–Gravity Water Waves with Discontinuous Vorticity: Existence and Regularity Results

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Abstract

In this paper we construct periodic capillarity–gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by re-expressing, in the height function formulation of the water wave problem, the boundary condition obtained from Bernoulli’s principle as a nonlocal differential equation. This enables us to establish the existence of weak solutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we investigate the a priori regularity of these weak solutions and prove that they are in fact strong solutions of the problem, describing waves with a real-analytic free surface. Moreover, assuming merely integrability of the vorticity function, we show that any weak solution corresponds to flows having real-analytic streamlines.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Nordbergstrasse 15, 1090, Vienna, Austria

    Anca-Voichita Matioc & Bogdan-Vasile Matioc

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  1. Anca-Voichita Matioc
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  2. Bogdan-Vasile Matioc
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Correspondence to Anca-Voichita Matioc.

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Communicated by W. Schlag

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Matioc, AV., Matioc, BV. Capillary–Gravity Water Waves with Discontinuous Vorticity: Existence and Regularity Results. Commun. Math. Phys. 330, 859–886 (2014). https://doi.org/10.1007/s00220-014-1918-z

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  • DOI: https://doi.org/10.1007/s00220-014-1918-z

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