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.
@article{SEDP_2008-2009____A7_0,
author = {Lannes, David},
title = {Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:7},
pages = {1--19},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2008-2009},
language = {en},
url = {http://archive.numdam.org/item/SEDP_2008-2009____A7_0/}
}
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 (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:7 PY - 2008-2009 SP - 1 EP - 19 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2008-2009____A7_0/ LA - en ID - SEDP_2008-2009____A7_0 ER -
%0 Journal Article %A Lannes, David %T Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:7 %D 2008-2009 %P 1-19 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2008-2009____A7_0/ %G en %F SEDP_2008-2009____A7_0
Lannes, David. Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 7, 19 p. http://archive.numdam.org/item/SEDP_2008-2009____A7_0/
[1] B. Alvarez-Samaniego and D. Lannes, Large time existence for water-waves and asymptotics, Invent. math. 171 (2008) 485-541. | MR | Zbl
[2] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water-waves, Arch. Rational Mech. Anal. 178 (2005) 373-410. | MR | Zbl
[3] J. Bona, D. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl. 89 (2008) 538-566. | MR | Zbl
[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. | MR | Zbl
[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. | Numdam | MR | Zbl
[6] W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992) 497-522. | MR | Zbl
[7] V. Duchene, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, arXiv:0906.0839v1, submitted.
[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.
[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. | MR | Zbl
[11] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005) 605-654. | MR | Zbl
[12] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal. 232 (2006) 495-539. | MR | Zbl
[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. | MR | Zbl
[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.
