On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations

Journal of the London Mathematical Society

Volumn 53, Issue 3, 1996, Pages 551-559

  Birnir, Björn ^a   Kenig, Carlos E ^c   Ponce, Gustavo ^a   Svanstedt, Nils ^a   Vega, Luis ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

^c UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030158684     PISSN: 00246107     EISSN: None     Source Type: Journal    
DOI: 10.1112/jlms/53.3.551     Document Type: Article

References (16)

1. H. BERESTYCKI and P. L. LIONS, 'Nonlinear scalar field equations', Arch. Rational Mech. Anal. 82 (1983) 313-376.
2. B. BIRNIR, G. PONCE and N. SVANSTEDT, 'On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation', to appear.
3. J. L. BONA, P. E. SOUGANIDIS and W. A. STRAUSS, 'Stability and instability of solitary waves of Korteweg-de Vries type', Proc. Roy. Soc. London Ser. A 411 (1987) 395-412.
4. J. BOURGAIN, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations', Geom. Funct. Anal. 3 (1993) 107-156, 209-262.
5. s, Nonlinear Anal. 14 (1990) 807-836.
6. R. T. GLASSEY, 'On the blowing up solutions to the Cauchy problem for nonlinear Schrödinger equations', J. Math. Phys. 18 (1979) 1794-1797.
7. T. KATO, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Mathematics 8 (1983) 93-128.
8. C. E. KENIG, G. PONCE and L. VEGA, 'Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle', Comm. Pure Appl. Math. 46 (1993) 527-620.
9. C. E. KENIG, G. PONCE and L. VEGA, 'A bilinear estimate with applications to the KdV equation', preprint.
10. H. LINDBLAD, 'A sharp counter example to local existence of low regularity solutions to non-linear wave equations', Duke Math. J. 72 (1993) 503-539.
11. H. LINDBLAD and C. D. SOGGE, 'On existence and scattering with minimal regularity for semilinear wave equations', preprint.
12. G. PONCE and T. C. SIDERIS, 'Local regularity of nonlinear wave equations in three space dimensions', Comm. Partial Differential Equations 18 (1993) 169-177.
13. W. A. STRAUSS, 'Dispersion of low-energy waves for two conservative equations', Arch. Rational Mech. Anal. 55 (1974) 86-92.
14. W. A. STRAUSS, 'Existence of solitary waves in higher dimensions', Comm. Math. Phys. 55 (1977) 149-162.
15. M. I. WEINSTEIN, On the solitary travelling wave of the generalized Korteweg-de Vries equation. Lectures in Applied Mathematics 23 (American Mathematical Society, Providence, 1986) 23-29.
16. V. E. ZAKHAROV and A. B. SHABAT, 'Exact theory of two dimensional self-focussing and one-dimensional self-modulation of waves in non-linear media', Soviet Phys. JETP 34 (1972) 62-69.