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Ill-posedness for the Zakharov system with generalized nonlinearity

Authors: H. A. Biagioni and F. Linares
Journal: Proc. Amer. Math. Soc. 131 (2003), 3113-3121
MSC (2000): Primary 35Q55, 35Q51
Published electronically: February 6, 2003
MathSciNet review: 1993221
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Abstract: We study the ill-posedness question for the one-dimensional Zakharov system and a generalization of it in one and higher dimensions. Our point of reference is the criticality criteria introduced by Ginibre, Tsutsumi and Velo (1997) to establish local well-posedness.

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  • 1. H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir em dimension 2, C. R. Acad. Sci. Paris 299 (1984), 551-554. MR 86g:35163
  • 2. H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79 (1988), 183-210. MR 89h:35273
  • 3. H. Berestycki and P. L. Lyons, Nonlinear scalar field equations, Arch. Rational Mech. Anal. 82 (1983), 313-376. MR 84h:35054a; MR 84h:35054b
  • 4. H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 353 (2001), 3649-3659. MR 2002e:35215
  • 5. B. Birnir, C. E. Kenig, G. Ponce, N. Svanstedt and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), 551-559. MR 97d:35233
  • 6. B. Birnir, G. Ponce and N. Svanstedt, The local ill-posedness of the modified KdV equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 4, 529-535. MR 97e:35152
  • 7. J. Bourgain and J. Colliander, On well-posedness of the Zakharov system, Int. Math. Res. Not. 11 (1996), 515-546. MR 97h:35206
  • 8. J. Colliander, Wellposedness for Zakharov systems with generalized nonlinearity, J. Differential Equations 148 (2), (1998), 351-363. MR 99h:35196
  • 9. J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal 151 (1997), 384-436. MR 2000c:35220
  • 10. C.E. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127 (1995), 204-234. MR 96a:35197
  • 11. C.E. Kenig, G. Ponce and L. Vega, On ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. MR 2002c:35265
  • 12. T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, RIMS Kyoto Univ. 28 (1992), 329-361. MR 93k:35246
  • 13. W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162. MR 56:12616
  • 14. C. Sulem and P. L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris 289 (1979), 173-176. MR 80i:35165
  • 15. Y. Wu, Orbital stability of solitary waves of Zakharov system, J. Math. Phys. 35 (1994), 2413-2422. MR 95e:35170
  • 16. V.E. Zakharov, The collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), 908-914.

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Additional Information

H. A. Biagioni
Affiliation: Departamento de Matemática, IMECC-UNICAMP, 13081-970, Campinas, SP, Brasil

F. Linares
Affiliation: Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brasil

Keywords: Ill-posedness, Zakharov system
Received by editor(s): June 15, 2001
Received by editor(s) in revised form: April 28, 2002
Published electronically: February 6, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

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