A pseudo-differential equations for Stokes waves

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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.
Original languageEnglish
Pages (from-to)179-189
Number of pages11
JournalArchive for Rational Mechanics and Analysis
Volume162
DOIs
Publication statusPublished - 30 Apr 2002

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