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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Well-posedness of the water-waves equations
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by David Lannes
J. Amer. Math. Soc. 18 (2005), 605-654
Published electronically: April 7, 2005


We prove that the water-waves equations (i.e., the inviscid Euler equations with free surface) are well-posed locally in time in Sobolev spaces for a fluid layer of finite depth, either in dimension $2$ or $3$ under a stability condition on the linearized equations. This condition appears naturally as the Lévy condition one has to impose on these nonstricly hyperbolic equations to insure well-posedness; it coincides with the generalized Taylor criterion exhibited in earlier works. Similarly to what happens in infinite depth, we show that this condition always holds for flat bottoms. For uneven bottoms, we prove that it is satisfied provided that a smallness condition on the second fundamental form of the bottom surface evaluated on the initial velocity field is satisfied. We work here with a formulation of the water-waves equations in terms of the velocity potential at the free surface and of the elevation of the free surface, and in Eulerian variables. This formulation involves a Dirichlet-Neumann operator which we study in detail: sharp tame estimates, symbol, commutators and shape derivatives. This allows us to give a tame estimate on the linearized water-waves equations and to conclude with a Nash-Moser iterative scheme.
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Bibliographic Information
  • David Lannes
  • Affiliation: MAB, Université Bordeaux 1 et CNRS UMR 5466, 351 Cours de la Libération, 33405 Talence Cedex, France
  • Email:
  • Received by editor(s): November 25, 2003
  • Published electronically: April 7, 2005
  • Additional Notes: This work was partly supported by the ‘ACI jeunes chercheurs du Ministère de la Recherche “solutions oscillantes d’EDP” et “Dispersion et non-linéarités ”, GDR 2103 EAPQ CNRS and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 605-654
  • MSC (2000): Primary 35Q35, 76B03, 76B15; Secondary 35J67, 35L80
  • DOI:
  • MathSciNet review: 2138139