Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T08:19:07.806Z Has data issue: false hasContentIssue false
  • English
  • Français

On a model system for the oblique interaction of internal gravity waves

Published online by Cambridge University Press:  15 April 2002

Jean-Claude Saut  and
Nikolay Tzvetkov
Jean-Claude Saut
Affiliation:
Analyse numérique et EDP, Université de Paris-Sud, Bt. 425, 91405 Orsay Cedex, France. ([email protected])
Nikolay Tzvetkov
Affiliation:
Analyse numérique et EDP, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France. ([email protected] )
Get access

Abstract

We give local and global well-posedness results for a system of twoKadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhuto model the oblique interaction of weakly nonlinear, two dimensional,long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves.We use the Fourier transform restriction norms introduced by J. Bourgain and the Strichartz estimates for the linear KP group. Finallywe extend the result of [3] for lower order perturbationof the system in the absence of transverse effects.

Keywords

Internal gravity wavesKadomtsev-Petviashvili equationsCauchy problem.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 501 - 523
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albert, J., Bona, J. and Saut, J.C., Model equations for waves in stratified fluids. Proc. Roy. Soc. Lond. A 453 (1997) 1213-1260. CrossRef
S. Alinhac and P. Gérard, Opérateurs pseudo-différentiel et théorème de Nash-Moser. Éditions du CNRS, EDP Sciences (1991).
Ash, J.M., Cohen, J. and Wang, G., On strongly interacting internal solitary waves. J. Fourier Anal. and Appl. 5 (1996) 507-517.
Bona, J., Ponce, G., Saut, J.C. and Tom, M., A model system for strong interaction between internal solitary waves. Comm. Math. Phys. 143 (1992) 287-313. CrossRef
Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales de l'ENS 14 (1981) 209-246.
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. GAFA 3 (1993) 107-156.
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation. GAFA 3 (1993) 209-262.
Bourgain, J., On the Cauchy problem for the Kadomtsev-Petviashvili equation. GAFA 3 (1993) 315-341.
J.-Y. Chemin, Fluid parfaits incompressibles. Astérisque 230 (1995).
R. Coifman and Y. Meyer, Au delà des operateurs pseudodifférentiels. Astérisque 57 (1978).
Gallagher, I., Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054. CrossRef
Gear, J.A. and Grimshaw, R., Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 65 (1984) 235-258. CrossRef
J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain). Séminaire Bourbaki 796, Astérique 237 (1995) 163-187.
Ginibre, J., Tsutsumi, Y. and Velo, G., On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 (1997) 384-436. CrossRef
R. Grimshaw, Y. Zhu, Oblique interactions between internal solitary waves. Stud. Appl. Math. 92 (1994) 249-270.
Iftimie, D., The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matematica Ibero-Americana 15 (1999) 1-36.
Iório Jr, R.J., Nunes, W.V.L., On equations of KP-type. Proc. Roy. Soc. Edinburgh A 128 (1998) 725-743. CrossRef
P. Isaza, J. Mejia and V. Stallbohm, El problema de Cauchy para la ecuacion de Kadomtsev-Petviashvili (KP-II) en espacios de Sobolev H s , s>0, preprint (1997).
Linares, F., L 2 global well-posedness of the initial value problem associated to the Benjamin equation. J. Differential Equations 152 (1999) 377-393. CrossRef
Kenig, C., Ponce, G. and Vega, L., A bilinear estimate with applications to the KdV equations. J. AMS 9 (1996) 573-603.
Kenig, C., Ponce, G. and Vega, L., Quadratic forms for 1-D semilinear Schrödinger equation. Trans. Amer. Math. Soc. 348 (1996) 3323-3353. CrossRef
Saut, J.C., Remarks on the generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J. 42 (1993) 1017-1029. CrossRef
Strichartz, R., Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977) 705-714. CrossRef
H. Takaoka, Well-posedness for the Kadomtsev-Petviashvili II equation, preprint (1998).
N. Tzvetkov, Global low regularity solutions for Kadomtsev-Petviashvili equation. Diff. Int. Eq. (to appear).