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

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.