|
Blow up in finite time and dynamics of blow up solutions for the  -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
Posted:
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  and has decay in  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.
- 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
(96d:65141), http://dx.doi.org/10.1098/rsta.1995.0027
- 2.
Jean
Bourgain, Harmonic analysis and nonlinear partial differential
equations, 2 (Zürich, 1994) Birkhäuser, Basel, 1995,
pp. 31–44. MR 1403913
(97f:35088)
- 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
(90i:35240), http://dx.doi.org/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
(86f:35160)
- 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
(94h:35229), http://dx.doi.org/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
(38 #3620)
- 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
(2001i:37102), http://dx.doi.org/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 (99h:35200)
- 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
(53 #8689)
- 13.
Michael
I. Weinstein, Nonlinear Schrödinger equations and sharp
interpolation estimates, Comm. Math. Phys. 87
(1982/83), no. 4, 567–576. MR 691044
(84d:35140)
- 1.
- J.L. Bona, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Conservative, high order numerical schemes, Phil. Trans. Roy. Soc. London Ser. A. 351 (1995), 107-164. MR 96d:65141
- 2.
- J. Bourgain, Harmonic analysis and nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, 1,2 (Zurich, 1994), 31-44, Birkhäuser, Basel, 1995. MR 97f:35088
- 3.
- J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20, 6 (1989), 1388-1425. MR 90i:35240
- 4.
- T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematical Supplementary Studies, Studies in Applied Math. 8 (1983), 93-128. MR 86f:35160
- 5.
- 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. MR 94h:35229
- 6.
- P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. MR 38:3620
- 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.
- Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. 79 (2000), 339-425. MR 2001i:37102
- 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.
- F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceeding of the International Congress of Mathematicians (Berlin, 1998), Doc. Math. J. DMV. MR 99h:35200
- 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.
- R.M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review 18 (1976), 412-459. MR 53:8689
- 13.
- M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576. MR 84d:35140
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
PII:
S 0894-0347(02)00392-2
Keywords:
Critical KdV equation,
finite time blow up,
blow up rate
Received by editor(s):
March 15, 2001
Posted:
March 8, 2002
Article copyright:
© Copyright 2002 American Mathematical Society
|
