Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves

David Lannes[1]

  • [1] École Normale Supérieure DMA et CNRS UMR 8553 45, rue d’Ulm 75005 Paris France

Séminaire Équations aux dérivées partielles (2008-2009)

  • Volume: 2008-2009, page 1-19

Abstract

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This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.

How to cite

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Lannes, David. "Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves." Séminaire Équations aux dérivées partielles 2008-2009 (2008-2009): 1-19. <http://eudml.org/doc/11203>.

@article{Lannes2008-2009,
abstract = {This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.},
affiliation = {École Normale Supérieure DMA et CNRS UMR 8553 45, rue d’Ulm 75005 Paris France},
author = {Lannes, David},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {shallow-water equations; bi-fluid; Euler equations; well-posedness},
language = {eng},
pages = {1-19},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves},
url = {http://eudml.org/doc/11203},
volume = {2008-2009},
year = {2008-2009},
}

TY - JOUR
AU - Lannes, David
TI - Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves
JO - Séminaire Équations aux dérivées partielles
PY - 2008-2009
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2008-2009
SP - 1
EP - 19
AB - This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.
LA - eng
KW - shallow-water equations; bi-fluid; Euler equations; well-posedness
UR - http://eudml.org/doc/11203
ER -

References

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  1. B. Alvarez-Samaniego and D. Lannes, Large time existence for 3 D water-waves and asymptotics, Invent. math. 171 (2008) 485-541. Zbl1131.76012MR2372806
  2. J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water-waves, Arch. Rational Mech. Anal. 178 (2005) 373-410. Zbl1108.76012MR2196497
  3. J. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. 89 (2008) 538-566. Zbl1138.76028MR2424620
  4. W. Craig, P. Guyenne, and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure. Appl. Math. 58 (2005) 1587-1641. Zbl1151.76385MR2177163
  5. W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré, Anal. Non Linéaire 14 (1997) 615-667. Zbl0892.76008MR1470784
  6. W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992) 497-522. Zbl0742.76012MR1158383
  7. V. Duchene, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, arXiv:0906.0839v1, submitted. Zbl1254.76045
  8. P. Grisvard, Quelques propriétés des espaces de Sobolev, utiles dans l’étude des équations de Navier-Stokes, Exposé 4 in “Problèmes d’évolution non linéaires”, Séminaire de Nice 1974-1975. 
  9. P. Guyenne, D. Lannes, J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, submitted. Zbl1191.35166
  10. T. Iguchi, N. Tanaka and A. Tani, On the two-phase free boundary problem for two-dimensional water waves, Math. Ann. 309 (1997) 199-223. Zbl0897.76017MR1474190
  11. D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005) 605-654. Zbl1069.35056MR2138139
  12. D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal. 232 (2006) 495-539. Zbl1099.35191MR2200744
  13. D. Lannes, In preparation 
  14. K. Ohi and T. Iguchi, A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation, Discrete Contin. Dyn. Syst. 23 (2009) 1205-1240. Zbl1155.35416MR2461848
  15. V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968) 190-194. 

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