Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation


Authors: Yvan Martel and Frank Merle
Journal: J. Amer. Math. Soc. 15 (2002), 617-664
MSC (1991): Primary 35Q53; Secondary 35B05, 35Q51
Published electronically: March 8, 2002
MathSciNet review: 1896235
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we describe the dynamics of blow up solutions for the critical generalized KdV equation such that the initial data is close to the soliton in $L^2$ and has decay in $L^2$ at the right. In particular, we prove that blow up occurs in finite time, and we obtain an upper bound on the blow up rate.


References [Enhancements On Off] (What's this?)

  • 1. J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 351 (1995), no. 1695, 107–164. MR 1336983, 10.1098/rsta.1995.0027
  • 2. Jean Bourgain, Harmonic analysis and nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 31–44. MR 1403913
  • 3. J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388–1425. MR 1019307, 10.1137/0520091
  • 4. Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
  • 5. Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, 10.1002/cpa.3160460405
  • 6. Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 0235310
  • 7. Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), 74-123.
  • 8. Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. (9) 79 (2000), no. 4, 339–425. MR 1753061, 10.1016/S0021-7824(00)00159-8
  • 9. Y. Martel and F. Merle, Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, to appear in Ann. of Math.
  • 10. Frank Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 57–66 (electronic). MR 1648140
  • 11. F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555-578.
  • 12. Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 0404890
  • 13. Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 35Q53, 35B05, 35Q51

Retrieve articles in all journals with MSC (1991): 35Q53, 35B05, 35Q51


Additional Information

Yvan Martel
Affiliation: Département de Mathématiques, Université de Cergy–Pontoise, 2, av. A. Chauvin, 95302 Cergy Pontoise, France
Email: Yvan.Martel@math.u-cergy.fr

Frank Merle
Affiliation: Département de Mathématiques, Université de Cergy–Pontoise, 2, av. A. Chauvin, 95302 Cergy Pontoise, France – and – Institut Universitaire de France
Email: Frank.Merle@math.u-cergy.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-02-00392-2
Keywords: Critical KdV equation, finite time blow up, blow up rate
Received by editor(s): March 15, 2001
Published electronically: March 8, 2002
Article copyright: © Copyright 2002 American Mathematical Society