About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 404781, 8 pages
http://dx.doi.org/10.1155/2014/404781
Research Article

The Cauchy Problem for a Fifth-Order Dispersive Equation

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received 21 January 2014; Accepted 8 February 2014; Published 27 March 2014

Academic Editor: Changsen Yang

Copyright © 2014 Hongjun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 30, no. 4-6, pp. 761–781, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. X. F. Liu and Y. Y. Jin, “The Cauchy problem of a shallow water equation,” Acta Mathematica Sinica, vol. 21, no. 2, pp. 393–408, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. Li, W. Yan, and X. Yang, “Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity,” Journal of Evolution Equations, vol. 10, no. 2, pp. 465–486, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. A. Olson, “Well posedness for a higher order modified Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 10, pp. 4154–4172, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Wang and S. Cui, “Global well-posedness of the Cauchy problem of the fifth-order shallow water equation,” Journal of Differential Equations, vol. 230, no. 2, pp. 600–613, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Grunrock, New applications of the fourier restriction norm method to well-posedness problems for nonlinear evolution equations, [Doctoral Dissertation], University of Wuppertal, 2002.
  7. A. Grünrock, “An improved local well-posedness result for the modified KdV equation,” International Mathematics Research Notices, no. 61, pp. 3287–3308, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. E. Kenig, G. Ponce, and L. Vega, “On the ill-posedness of some canonical dispersive equations,” Duke Mathematical Journal, vol. 106, no. 3, pp. 617–633, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. E. Kenig, G. Ponce, and L. Vega, “A bilinear estimate with applications to the KdV equation,” Journal of the American Mathematical Society, vol. 9, no. 2, pp. 573–603, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Sharp global well-posedness for KdV and modified KdV on R and T,” Journal of the American Mathematical Society, vol. 16, no. 3, pp. 705–749, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness for Schrödinger equations with derivative,” SIAM Journal on Mathematical Analysis, vol. 33, no. 3, pp. 649–669, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness for KdV in Sobolev spaces of negative index,” Electronic Journal of Differential Equations, vol. 2001, no. 26, pp. 1–7, 2001. View at Zentralblatt MATH · View at MathSciNet
  14. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation,” Mathematical Research Letters, vol. 9, no. 5-6, pp. 659–682, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. M. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Review, vol. 18, no. 3, pp. 412–459, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. Guo, “Global well-posedness of Korteweg-de Vries equation in H-3/4(R),” Journal de Mathématiques Pures et Appliquées, vol. 91, no. 6, pp. 583–597, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. D. Ionescu and C. E. Kenig, “Global well-posedness of the Benjamin-Ono equation in low-regularity spaces,” Journal of the American Mathematical Society, vol. 20, no. 3, p. 753–798 (electronic), 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. D. Ionescu, C. E. Kenig, and D. Tataru, “Global well-posedness of the KP-I initial-value problem in the energy space,” Inventiones Mathematicae, vol. 173, no. 2, pp. 265–304, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,” Geometric and Functional Analysis, vol. 3, no. 3, pp. 209–262, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. I. Bejenaru and T. Tao, “Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation,” Journal of Functional Analysis, vol. 233, no. 1, pp. 228–259, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. E. Kenig, G. Ponce, and L. Vega, “The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,” Duke Mathematical Journal, vol. 71, no. 1, pp. 1–21, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. E. Kenig, G. Ponce, and L. Vega, “Oscillatory integrals and regularity of dispersive equations,” Indiana University Mathematics Journal, vol. 40, no. 1, pp. 33–69, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. Ginibre, “Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain),” Astérisque, no. 237, pp. 163–187, 1996. View at Zentralblatt MATH · View at MathSciNet
  24. N. Tzvetkov, “Remark on the local ill-posedness for KdV equation,” Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 329, no. 12, pp. 1043–1047, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet