Communications in Mathematical Physics

, Volume 330, Issue 2, pp 859–886 | Cite as

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

  • Anca-Voichita MatiocEmail author
  • Bogdan-Vasile Matioc


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.


Vorticity Weak Solution Water Wave Local Bifurcation Water Wave Problem 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WienViennaAustria

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