Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. S. ALINHAC, P. GÉRARD, Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels. InterEditions, Paris; Editions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991. 190 pp. MR 1172111 (93g:35001)
  • 2. J. BONA, T. COLIN, D. LANNES, Long wave approximations for water-waves, Arch. Ration. Mech. Anal., to appear.
  • 3. T. BEALE, T. HOU, J. LOWENGRUB, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math. 46 (1993), no. 9, 1269-1301. MR 1231428 (95c:76016)
  • 4. G. BIRKHOFF, Helmholtz and Taylor instability, 1962 Proc. Sympos. Appl. Math., Vol. XIII pp. 55-76, American Mathematical Society, Providence, R.I. MR 0137423 (25:875)
  • 5. G. CARBOU, Penalization method for viscous incompressible flow around a porous thin layer, Nonlinear Anal. Real World Appl. 5 (2004), no. 5, 815-855. MR 2085697 (2005d:76034)
  • 6. M. CHRIST, J.-L. JOURNÉ, Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159 (1987), no. 1-2, 51-80. MR 0906525 (89a:42024)
  • 7. R. R. COIFMAN, G. DAVID, Y. MEYER, La solution des conjecture de Calderón Adv. in Math. 48 (1983), no. 2, 144-148. MR 0700980 (84i:42025)
  • 8. R. R. COIFMAN, A. MCINTOSH, Y. MEYER, L'intégrale de Cauchy définit un opérateur borné sur $L\sp{2}$pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361-387.MR 0672839 (84m:42027)
  • 9. R. COIFMAN, Y. MEYER, Nonlinear harmonic analysis and analytic dependence, Pseudodifferential operators and applications (Notre Dame, Indiana, 1984), 71-78, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, 1985. MR 0812284
  • 10. W. CRAIG, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787-1003. MR 0795808 (87f:35210)
  • 11. W. CRAIG, Nonstrictly hyperbolic nonlinear systems, Math. Ann. 277 (1987), no. 2, 213-232. MR 0886420 (88d:35134)
  • 12. W. CRAIG, D. NICHOLLS, Traveling gravity water waves in two and three dimensions. Eur. J. Mech. B Fluids 21 (2002), no. 6, 615-641. MR 1947187 (2004b:76018)
  • 13. W. CRAIG, U. SCHANZ, C. SULEM, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 5, 615-667.MR 1470784 (99i:35144)
  • 14. W. CRAIG, C. SULEM, P.-L. SULEM, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497-522. MR 1158383 (93k:76012)
  • 15. M. DAMBRINE, M. PIERRE, About stability of equilibrium shapes, M2AN Math. Model. Numer. Anal. 34 (2000), no. 4, 811-834.MR 1784487 (2001h:49037)
  • 16. H. FLASCHKA, G. STRANG, The correctness of the Cauchy problem, Advances in Math. 6 (1971), 347-379.MR 0279425 (43:5147)
  • 17. P. R. GARABEDIAN, M. SCHIFFER, Convexity of domain functionals J. Analyse Math. 2, (1953), 281-368. MR 0060117 (15:627a)
  • 18. J. E. GILBERT, M. A. MURRAY, Clifford algebras and Dirac operators in harmonic analysis. Cambridge Studies in Advanced Mathematics, 26. Cambridge University Press, Cambridge, 1991. MR 1130821 (93e:42027)
  • 19. P. GRISVARD, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 0775683 (86m:35044)
  • 20. J. HADAMARD, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées, [B] Mém. Sav. étrang. (2) 33, Nr. 4, 128 S. (1908).
  • 21. R. HAMILTON, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222. MR 0656198 (83j:58014)
  • 22. T. HOU, Z. TENG, P. ZHANG, Well-posedness of linearized motion for $3$-D water waves far from equilibrium, Comm. Partial Differential Equations 21 (1996), no. 9-10, 1551-1585.MR 1410841 (98c:76013)
  • 23. I. L. HWANG, The $L\sp 2$-boundedness of pseudodifferential operators, Trans. Amer. Math. Soc. 302 (1987), no. 1, 55-76.MR 0887496 (88e:47096)
  • 24. T. KATO, G. PONCE, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907.MR 0951744 (90f:35162)
  • 25. D. LANNES, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, Preprint Université Bordeaux 1.
  • 26. J.L. LIONS, E. MAGENES, Problèmes aux limites non homogènes applications. Vol. 1. (French), Travaux et Recherches Mathématiques, No. 17 Dunod, Paris 1968. MR 0247243 (40:512)
  • 27. V. I. NALIMOV, The Cauchy-Poisson problem. (Russian) Dinamika Splosn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, (1974), 104-210, 254. MR 0609882 (58:29458)
  • 28. D. NICHOLLS, F. REITICH, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 6, 1411-1433. MR 1869643 (2003b:35216)
  • 29. L. V. OVSJANNIKOV, Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification. Applications of methods of functional analysis to problems in mechanics (Joint Sympos., IUTAM/IMU, Marseille, 1975), pp. 426-437. Lecture Notes in Math., 503. Springer, Berlin, 1976. MR 0670760 (58:32358)
  • 30. M. SABLÉ-TOUGERON, Régularité microlocale pour des problèmes aux limites non linéaires, Ann. Inst. Fourier 36, No.1, 39-82 (1986). MR 0840713 (88b:35021)
  • 31. X. SAINT RAYMOND, A simple Nash-Moser implicit function theorem, Enseign. Math. (2) 35 (1989), no. 3-4, 217-226.MR 1039945 (91g:58018)
  • 32. G. SCHNEIDER, C. E. WAYNE, The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53 (2000), no. 12, 1475-1535. MR 1780702 (2002c:76025a)
  • 33. G. TAYLOR, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I., Proc. Roy. Soc. London. Ser. A. 201, (1950). 192-196. MR 0036104 (12:58f)
  • 34. M. TAYLOR, Partial differential equations. II. Qualitative studies of linear equations, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. MR 1395149 (98b:35003)
  • 35. F. TRÈVES Introduction to pseudodifferential and Fourier integral operators. Vol. 1. Pseudodifferential operators, The University Series in Mathematics. Plenum Press, New York-London, 1980. MR 0597144 (82i:35173)
  • 36. S. WU, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math. 130 (1997), no. 1, 39-72. MR 1471885 (98m:35167)
  • 37. S. WU, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), no. 2, 445-495.MR 1641609 (2001m:76019)
  • 38. H. YOSIHARA, Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 49-96. MR 0660822 (83k:76017)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35Q35, 76B03, 76B15, 35J67, 35L80

Retrieve articles in all journals with MSC (2000): 35Q35, 76B03, 76B15, 35J67, 35L80

Additional Information

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

PII: S 0894-0347(05)00484-4
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.