Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Global existence for the critical generalized KdV equation


Authors: G. Fonseca, F. Linares and G. Ponce
Journal: Proc. Amer. Math. Soc. 131 (2003), 1847-1855
MSC (2000): Primary 35Q53
DOI: https://doi.org/10.1090/S0002-9939-02-06871-5
Published electronically: November 6, 2002
MathSciNet review: 1955273
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,

\begin{displaymath}u_t+u_{xxx}+u^4\,u_x=0,\quad x,\,t\in\mathbb{R} .\end{displaymath}

The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space $H^1(\mathbb{R} )$, i.e. solutions corresponding to data $u_0\in H^s(\mathbb{R} )$, $s>3/4$, with $\Vert u_0\Vert _{L^2}<\Vert Q\Vert _{L^2}$, where $Q$ is the solitary wave solution of the equation.


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

  • 1. 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
  • 2. J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 5 (1998), 253-283. MR 99f:35184
  • 3. J. Colliander, G. Staffilani and H. Takaoka, Global well-posedness for KdV below ${L}\sp 2$, Math. Res. Lett., 6 (1999), 755-778. MR 2000m:35159
  • 4. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R} $ and $\mathbb{T} $, preprint.
  • 5. G. Fonseca, F. Linares and G. Ponce, Global well-posedness for the modified Korteweg-de Vries equation, Communications in PDE, 24 (3&4) (1999), 683-705. MR 2000a:35210
  • 6. A. Grünrock, A bilinear Airy-estimate with applications to gKdV-3, preprint.
  • 7. C. E. Kenig, G. Ponce and L. Vega, Global well-posedness for semi-linear wave equations, Comm. Partial Differential Equations, 25 (2000), 1741-1752. MR 2001h:35128
  • 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. MR 94h:35229
  • 9. H. Takaoka, Global well-posedness for Schrödinger equations with derivatives in a nonlinear term and data in low-order Sobolev spaces, Electronic J. Differential Equations, 42 (2001). MR 2002f:35033
  • 10. Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123. MR 2002g:35182
  • 11. Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Annals of Math. (2) 155 (2002), 235-280.
  • 12. 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. MR 2002f:35193
  • 13. H. Pecher, Global well-posedness below energy space for the 1-dimensional Zakharov system, Internat. Math. Res. Notices, 19 `(2001), 1027-1056. MR 2002j:35036
  • 14. M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. MR 87f:35023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q53

Retrieve articles in all journals with MSC (2000): 35Q53


Additional Information

G. Fonseca
Affiliation: Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Email: gfonseca@matematicas.unal.edu.co

F. Linares
Affiliation: Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brazil
Email: linares@impa.br

G. Ponce
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: ponce@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06871-5
Received by editor(s): January 30, 2002
Published electronically: November 6, 2002
Additional Notes: The first author was partially supported by DIB-Universidad Nacional de Colombia
The second author was partially supported by CNP-q Brazil
The third author was partially supported by an NSF grant
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society