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Capillary-gravity water waves with discontinuous vorticity: existence and regularity results. (English) Zbl 1294.35107

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45 Capillarity (surface tension) for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
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