- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 404781, 8 pages
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. Grunrock, New applications of the fourier restriction norm method to well-posedness problems for nonlinear evolution equations, [Doctoral Dissertation], University of Wuppertal, 2002.
- A. Grünrock, “An improved local well-posedness result for the modified KdV equation,” International Mathematics Research Notices, no. 61, pp. 3287–3308, 2004.
- 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.
- 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.
- 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.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Sharp global well-posedness for KdV and modified KdV on and ,” Journal of the American Mathematical Society, vol. 16, no. 3, pp. 705–749, 2003.
- 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.
- 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.
- 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.
- R. M. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Review, vol. 18, no. 3, pp. 412–459, 1976.
- Z. Guo, “Global well-posedness of Korteweg-de Vries equation in ,” Journal de Mathématiques Pures et Appliquées, vol. 91, no. 6, pp. 583–597, 2009.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.