Well-posedness of the Cauchy problem for models of large amplitude internal waves

Nonlinearity

Volumn 23, Issue 2, 2010, Pages 237-275

  Guyenne, Philippe ^a   Lannes, David ^b   Saut, Jean Claude ^c  

^a UNIVERSITY OF DELAWARE   (United States)

^b ÉCOLE NORMALE SUPÉRIEURE   (France)

^c UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 74849088012     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/23/2/003     Document Type: Article

References (29)

1. Alinhac S 1995 Blowup for nonlinear hyperbolic equations Progress in Nonlinear Equations and Their Applications vol 17 (Boston: Birkhäuser)
2. Alvarez-Samaniego B and Lannes D 2008 A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations Indiana Univ. Math. J. 57 97-131
3. Benjamin T B 1966 Internal waves of finite amplitude and permanent form J. Fluid Mech. 25 241-70
4. Benjamin T B and Bridges T J 1997 Reappraisal of the Kelvin-Helmholtz problem: I. Hamiltonian structure J. Fluid Mech. 333 301-25
5. Bona J, Lannes D and Saut J-C 2008 Asymptotic models for internal waves J. Math. Pures Appl. 89 538-66
6. Bouchut F and Zeitlin V 2009 A robust well-balanced scheme for multi-layer shallow water equations DCDC-S to appear
7. Camassa R, Holm D D and Levermore C D 1997 Long-time shallow-water equations with a varying bottom J. Fluid Mech. 349 173-89
8. Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic Stability (Oxford: Clarendon)
9. Choi W and Camassa R 1999 Fully nonlinear internal waves in a two-fluid system J. Fluid Mech. 396 1-36
10. Chumakova L, Menzaque F E, Milewski P A, Tabak E G and Turner C V 2009 Stability properties and nonlinear mappings of two and three-layer stratified flows Stud. Appl. Math. 122 123-37
11. Craig W and Groves M D 2000 Normal forms for wave motion in fluid interfaces Wave Motion 31 21-41
12. Craig W, Guyenne P and Kalisch H 2005 Hamiltonian long-wave expansions for free surfaces and interfaces Commun. Pure. Appl. Math. 58 1587-641 (Pubitemid 43082564)
13. Craig W, Sulem C and Sulem P-L 1992 Nonlinear modulation of gravity waves: a rigorous approach Nonlinearity 5 497-522
14. Dafermos C 2005 Hyperbolic Conservation laws in Continuum Physics 2nd edn (Heidelberg: Springer)
15. Duchne V 2009 Asymptotic shallow water models for internal waves in a two-fluid system with a free surface arXiv:0906.0839
16. Grisvard P 1974-1975 Quelques propriétés des espaces de Sobolev, utiles dans l'étude des équations de Navier-Stokes Exposé 4 Problèmes d'volution Non Linéaires, Séminaire de Nice 1974-1975
17. Guyenne P 2006 Large-amplitude internal solitary waves in a two-fluid model C. R. Mecanique 334 341-6
18. Hörmander L 1997 Lectures on Nonlinear Hyperbolic Differential Equations (Mathématiques Applications vol 26) (Berlin: Springer)
19. Hou T Y and Zhang P 1998 Growth rates for the linearized motion of 3-D fluid interfaces with surface tension far from equilibrium Asian J. Math. 2 263-88
20. Israwi S 2009 Large time existence for 1D Green-Naghdi equations (arXiv:0909.2232) submitted
21. John F 1974 Formation of singularities in one-dimensional nonlinear wave propagation Commun. Pure Appl. Math. 27 377-405
22. Lannes D 2006 Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators J. Funct. Anal. 232 495-539
23. Li Y A 2006 A shallow-water approximation to the full water wave problem Commun. Pure Appl. Math. 59 1225-85
24. Liu T-P 1979 Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations J. Diff. Eqns 33 92-111
25. Majda A 1984 Incompressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Heidelberg: Springer)
26. Milewski P, Tabak E, Turner C, Rosales R and Menzaque F 2004 Nonlinear stability of two-layer flows Commun. Math. Sci. 2 427-42
27. Smoller J 1983 Shock Waves and Reaction-Diffusion Equations (Heidelberg: Springer)
28. Taylor M E 1997 Partial Differential Equations: III. Nonlinear Equations (Applied Mathematical Sciences vol 117) (New York: Springer) Corrected reprint of the 1996 original
29. Zakharov V E 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid J. Appl. Mech. Tech. Phys. 2 190-4