Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results

Calin Iulian Martin; Bogdan-Vasile Matioc

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arXiv · 1311.6935

Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results

Calin Iulian Martin·Bogdan-Vasile Matioc

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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 a $L_r-$integrable vorticity.

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Calin Iulian Martin, Bogdan-Vasile Matioc. 2013-11-27. Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results. https://arxiv.org/abs/1311.6935

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