This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space with . We also establish the ill-posedness for the initial data in with . Thus, the regularity requirement for the fifth-order dispersive equations is sharp.
References
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. MR2153514 1076.35098 10.1081/PDE-200059284 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. MR2153514 1076.35098 10.1081/PDE-200059284
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. MR2135300 1095.35041 10.1007/s10114-004-0420-5 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. MR2135300 1095.35041 10.1007/s10114-004-0420-5
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. MR2643805 1239.35141 10.1007/s00028-010-0057-z 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. MR2643805 1239.35141 10.1007/s00028-010-0057-z
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. MR2514739 1170.35087 10.1016/j.jde.2008.12.001 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. MR2514739 1170.35087 10.1016/j.jde.2008.12.001
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. MR2269935 1106.35068 10.1016/j.jde.2006.04.008 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. MR2269935 1106.35068 10.1016/j.jde.2006.04.008
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. 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. MR2096258 1072.35161 A. Grünrock, “An improved local well-posedness result for the modified KdV equation,” International Mathematics Research Notices, no. 61, pp. 3287–3308, 2004. MR2096258 1072.35161
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. MR1211741 0808.35128 10.1002/cpa.3160460405 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. MR1211741 0808.35128 10.1002/cpa.3160460405
C. E. Kenig, G. Ponce, and L. Vega, “On the ill-posedness of some canonical dispersive equations,” Duke Mathematical Jour-nal, vol. 106, no. 3, pp. 617–633, 2001. MR1813239 1034.35145 10.1215/S0012-7094-01-10638-8 euclid.dmj/1092403945
C. E. Kenig, G. Ponce, and L. Vega, “On the ill-posedness of some canonical dispersive equations,” Duke Mathematical Jour-nal, vol. 106, no. 3, pp. 617–633, 2001. MR1813239 1034.35145 10.1215/S0012-7094-01-10638-8 euclid.dmj/1092403945
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. MR1329387 0848.35114 10.1090/S0894-0347-96-00200-7 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. MR1329387 0848.35114 10.1090/S0894-0347-96-00200-7
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. MR1969209 1025.35025 10.1090/S0894-0347-03-00421-1 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. MR1969209 1025.35025 10.1090/S0894-0347-03-00421-1
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. MR1871414 1002.35113 10.1137/S0036141001384387 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. MR1871414 1002.35113 10.1137/S0036141001384387
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. MR1824796 0967.35119 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. MR1824796 0967.35119
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. MR1906069 1152.35491 10.4310/MRL.2002.v9.n5.a9 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. MR1906069 1152.35491 10.4310/MRL.2002.v9.n5.a9
R. M. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Review, vol. 18, no. 3, pp. 412–459, 1976. MR0404890 0333.35021 10.1137/1018076 R. M. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Review, vol. 18, no. 3, pp. 412–459, 1976. MR0404890 0333.35021 10.1137/1018076
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. MR2531556 1173.35110 10.1016/j.matpur.2009.01.012 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. MR2531556 1173.35110 10.1016/j.matpur.2009.01.012
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. MR2291918 1123.35055 10.1090/S0894-0347-06-00551-0 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. MR2291918 1123.35055 10.1090/S0894-0347-06-00551-0
A. D. Ionescu, C. E. Kenig, and D. Tataru, “Global well-posed-ness of the KP-I initial-value problem in the energy space,” Inventiones Mathematicae, vol. 173, no. 2, pp. 265–304, 2008. MR2415308 1188.35163 10.1007/s00222-008-0115-0 A. D. Ionescu, C. E. Kenig, and D. Tataru, “Global well-posed-ness of the KP-I initial-value problem in the energy space,” Inventiones Mathematicae, vol. 173, no. 2, pp. 265–304, 2008. MR2415308 1188.35163 10.1007/s00222-008-0115-0
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. MR1215780 0787.35098 10.1007/BF01895688 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. MR1215780 0787.35098 10.1007/BF01895688
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. MR2204680 10.1016/j.jfa.2005.08.004 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. MR2204680 10.1016/j.jfa.2005.08.004
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. MR1230283 0787.35090 10.1215/S0012-7094-93-07101-3 euclid.dmj/1077289834
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. MR1230283 0787.35090 10.1215/S0012-7094-93-07101-3 euclid.dmj/1077289834
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. MR1101221 0738.35022 10.1512/iumj.1991.40.40003 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. MR1101221 0738.35022 10.1512/iumj.1991.40.40003
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. MR1423623 0870.35096 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. MR1423623 0870.35096
N. Tzvetkov, “Remark on the local ill-posedness for KdV equa-tion,” Comptes Rendus de l'Académie des Sciences. Série I. Math-ématique, vol. 329, no. 12, pp. 1043–1047, 1999. \endinput MR1735881 0941.35097 10.1016/S0764-4442(00)88471-2 N. Tzvetkov, “Remark on the local ill-posedness for KdV equa-tion,” Comptes Rendus de l'Académie des Sciences. Série I. Math-ématique, vol. 329, no. 12, pp. 1043–1047, 1999. \endinput MR1735881 0941.35097 10.1016/S0764-4442(00)88471-2
