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Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. (English) Zbl 1361.35137

Summary: In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Hölder spaces.

MSC:

35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35D30 Weak solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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