Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results |
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Authors: | Calin Iulian Martin Bogdan-Vasile Matioc |
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Affiliation: | 1. Institut für Mathematik, Universit?t Wien, Nordbergstra?e 15, 1090?, Wien, Austria
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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 (rin (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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