On a pseudo-differential equation for Stokes waves.

*(English)*Zbl 1028.35126Author’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.

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.

Reviewer: P.Godin (Bruxelles)

##### 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 |