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Well-posedness of the water-waves equations

Author: David Lannes
Journal: J. Amer. Math. Soc. 18 (2005), 605-654
MSC (2000): Primary 35Q35, 76B03, 76B15; Secondary 35J67, 35L80
Published electronically: April 7, 2005
MathSciNet review: 2138139
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Abstract: 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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Additional Information

David Lannes
Affiliation: MAB, Université Bordeaux 1 et CNRS UMR 5466, 351 Cours de la Libération, 33405 Talence Cedex, France

Keywords: Water-waves, Dirichlet-Neumann operator, free surface
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.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.