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

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.