### Abstract

It is shown that the existence of a smooth solution to a nonlinear pseudo-differential 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.

Original language | English |
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Pages (from-to) | 179-189 |

Number of pages | 11 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 162 |

DOIs | |

Publication status | Published - 30 Apr 2002 |