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On a pseudo-differential equation for Stokes waves. (English) Zbl 1028.35126
Author’s summary: It is shown that the existence of a smooth solution to a nonlinear pseudodifferential equation on the unit circle is equivalent to the existence of a globally injective conformal mapping in the complex plane which gives a smooth solution to the nonlinear elliptic free-boundary problem for Stokes waves in hydrodynamics.
A dual formulation is used to show that the equation has no non-trivial smooth solutions, stable or otherwise, that would correspond to a Stokes wave with gravity acting in a direction opposite to that which is physically realistic.

35Q35 PDEs in connection with fluid mechanics
35S05 Pseudodifferential operators as generalizations of partial differential operators
35R35 Free boundary problems for PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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