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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.

MSC:
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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