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

In this paper we consider the steady water wave problem for waves that possess a merely Lr−integrable vorticity, with r ∈ (1,∞) being arbitrary. We rst 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 Lr−integrable vorticity.