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Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Authors: H. A. Biagioni and F. Linares
Journal: Trans. Amer. Math. Soc. 353 (2001), 3649-3659
MSC (1991): Primary 35Q55, 35Q51
Published electronically: May 3, 2001
MathSciNet review: 1837253
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Abstract | References | Similar Articles | Additional Information


Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in $H^s(\mathbb R)$, $s<1/2$. This result implies that best result concerning local well-posedness for the IVP is in $H^s(\mathbb R),\, s\ge1/2$. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.

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  • 1. T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
  • 2. B. Birnir, G. Ponce and N. Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. Henri Poincaré 13 No. 4 (1996), 529-535. MR 97e:35152
  • 3. B. Birnir, C. Kenig, G. Ponce, N. Svanstedt and L. Vega, On the ill-posedness for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), 551-559. MR 97d:35233
  • 4. Guo Boling and Wu Yaping, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. Differential Equation 123 (1995), 35-55. MR 96k:35166
  • 5. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, Geom. Funct. Anal. 3 (1993), 107-156, 209-262. MR 95d:35160a; MR 95d:35160b
  • 6. N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D 55(1992), 14-36. MR 93h:35190
  • 7. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math. 8(1983), 93-128. MR 86f:35160
  • 8. C.E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1-21. MR 94g:35196
  • 9. -, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 342 (1994), 155-172. MR 94e:35121
  • 10. -, A bilinear estimate with applications to the KdV equations, J. Amer. Math. Soc. 9 (1996), 573-603. MR 96k:35139
  • 11. -, On ill-posedness of some canonical dispersive equations, preprint 1999.
  • 12. R. J. Iorio, Jr, The Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), 1031-1081. MR 88b:35034
  • 13. K. Mio, T. Ogino, K. Minami, S. Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), 265-271. MR 57:2116
  • 14. H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), 1082-1091. MR 53:2129
  • 15. T. Ozawa,On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137-163. MR 98b:35186
  • 16. T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11(1998), 201-222. MR 2000m:35167
  • 17. G. Ponce, On the well posedness of some nonlinear evolution equations, Nonlinear waves (Sapporo, 1995), Gakuto International Series 10, (1997), 393-424. MR 99k:35140
  • 18. -, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), 527-542. MR 92e:35137
  • 19. H. Takaoka,Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), 561-580. MR 2000e:35221
  • 20. -, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, preprint (2000).
  • 21. W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized Ginzburg-Landau equations, Phys. D 56 (1992), 303-367. MR 93k:35195
  • 22. M. Wadadi, H. Sanuki, K. Konno, Y.-H. Ichikawa, Circular polarized nonlinear Alfvén waves - a new type of nonlinear evolution equation in plasma physics. Conference on the Theory and Application of Solitons (Tucson, Ariz., 1976), Rocky Mountain J. Math. 8 (1978), 323-331.
  • 23. M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations 12 No.10 (1987), 1133-1173. MR 88h:35107

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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, Schr\"odinger equation, Benjamin-Ono equation
Received by editor(s): April 5, 2000
Received by editor(s) in revised form: July 24, 2000
Published electronically: May 3, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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