Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T00:59:42.354Z Has data issue: false hasContentIssue false
  • English
  • Français

On mass concentration for the critical generalized Korteweg–de Vries equation

Published online by Cambridge University Press:  17 December 2015

B. Pigott
B. Pigott*
Affiliation:
Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that blow-up solutions of the critical generalized Korteweg–de Vries equation in H1() concentrate at least the mass of the ground state at the blow-up time. The I-method is used to prove a slightly weaker result in Hs() with 16/17 < s < 1. Under an assumption on the precise blow-up rate, we are able to use similar arguments to prove a more precise analogue of the H1() concentration result over the same range of s.

Keywords

Korteweg–de Vries (KdV) equationcritical normmass concentrationblow-upPrimary 35Q53
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 519 - 532
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations, Geom. Funct. Analysis 3 (1993), 107–156.Google Scholar
2. Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: the KdV equation, Geom. Funct. Analysis 3 (1993), 209–262.Google Scholar
3. Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Volume 10 (American Mathematical Society/Courant Institute of Mathematical Sciences, 2003).Google Scholar
4. Colliander, J. and Raphaeäl, P., Rough blowup solutions to the L 2-critical NLS, Math. Annalen 345 (2009), 307–366.Google Scholar
5. Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and . Tao, T., Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eqns 2001(26) (2001), 1–7.Google Scholar
6. Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., Sharp global well-posedness for KdV and modified KdV on and , J. Am. Math. Soc. 16 (2003), 705–749.Google Scholar
7. Colliander, J., Raynor, S., Sulem, C. and Wright, J. D., Ground state mass concentration in the L 2-critical nonlinear Schrödinger equation below H 1 , Math. Res. Lett. 12 (2005), 357–375.CrossRefGoogle Scholar
8. Dodson, B., Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Preprint (arxiv.org/abs/1304.8025; 2013).Google Scholar
9. Farah, L., Global rough solutions to the critical generalized KdV equation, J. Diff. Eqns 249 (2010), 1968–1985.Google Scholar
10. Gérard, P., Description du défaut de compacité de l’injection de Sobolev, ESAIM: Control Optim. Calc. Variations 3 (1998), 213–233.Google Scholar
11. Hmidi, T. and Keraani, S., Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not. 46 (2005), 2815–2828.Google Scholar
12. Hmidi, T. and Keraani, S., Remarks on the blowup for the L 2-critical nonlinear Schrödinger equations, SIAM J. Math. Analysis 38 (2006), 1035–1047.Google Scholar
13. Kenig, C. E., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math. 46 (1993), 527–620.Google Scholar
14. Kenig, C. E., Ponce, G. and Vega, L., On the concentration of blow up solutions for the generalized KdV equation critical in L 2 , in Nonlinear wave equations (Providence, RI, 1998), Contempory Mathematics, Volume 263, pp. 131–156 (American Mathematical Society, Providence, RI, 2000).Google Scholar
15. Martel, Y. and Merle, F., Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation, J. Am. Math. Soc. 15 (2002), 617–664.CrossRefGoogle Scholar
16. Martel, Y., Merle, F. and Raphaël, P., Blow up for the critical gKdV equation I: dynamics near the soliton, Acta Math. 212 (2014), 59–140.CrossRefGoogle Scholar
17. Martel, Y., Merle, F. and Raphaël, P., Blow up for the critical gKdV equation II: minimal mass dynamics, J. Eur. Math. Soc., in press.Google Scholar
18. Martel, Y., Merle, F. and Raphaël, P., Blow up for the critical gKdV equation III: exotic regimes, Annali Scuola Norm. Sup. Pisa V, 8 (2015), 575–631.Google Scholar
19. Merle, F., Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Am. Math. Soc. 14 (2001), 555–578.Google Scholar
20. Miao, C., Shao, S., Wu, Y. and Xu, G., The low regularity global solutions for the critical generalized KdV equation, Dynam. PDEs 7 (2010), 265–288.Google Scholar
21. Tzirakis, N., Mass concentration phenomenon for the quintic nonlinear Schrödinger equation in one dimension, SIAM J. Math. Analysis 37 (2006), 1923–1946.Google Scholar
22. Weinstein, M., Nonlinear Schröodinger equations and sharp interpolation estimates, Commun. Math. Phys. 87(4) (1982), 567–576.Google Scholar