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On ill-posedness for the generalized BBM equation
1. | Instituto de Matemática - UFRJ Av. Horácio Macedo, Centro de Tecnologia, Cidade Universitária, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, Brazil |
2. | Departamento de Matemática, Universidade Estadual de Campinas (UNICAMP), Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil |
References:
[1] |
A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166. |
[2] |
J. Angulo Pava, C. Banquet and M. Scialom, Stability for the modified and fourth Benjamin-Bona-Mahony equations, Discrete Contin. Dyn. Syst., 30 (2011), 851-871.
doi: 10.3934/dcds.2011.30.851. |
[3] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. London, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[4] |
J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. London Series A, 302 (1981), 457-510.
doi: 10.1098/rsta.1981.0178. |
[5] |
J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete and Continuous Dynamical Systems, 23 (2009), 1241-1252.
doi: 10.3934/dcds.2009.23.1241. |
[6] |
J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Disc. Cont. Dynamical Systems, 23 (2009), 1253-1275.
doi: 10.3934/dcds.2009.23.1253. |
[7] |
J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. New Ser., 3 (1997), 115-159.
doi: 10.1007/s000290050008. |
[8] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[9] |
W. Chen and J. Li, On the low regularity of the modified Korteweg-de Vries equation with a dissipative term, J. Diff. Equations, 240 (2007), 125-144.
doi: 10.1016/j.jde.2007.05.030. |
[10] |
M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity illposedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
[11] |
L. Molinet, A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space, arXiv:1110.2352v1, Proc. Amer. Math. Soc., 141 (2013), 2793-2798.
doi: 10.1090/S0002-9939-2013-11693-X. |
[12] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers Equation, Int. Math. Research Notices, (2002), 1979-2005.
doi: 10.1155/S1073792802112104. |
[13] |
L. Molinet, F. Ribaud and A Youssfi, Ill-posedness issue for a class of parabolic equations, Proceedings of the Royal Society of Edinburgh, 132 (2002), 1407-1416. |
[14] |
L. Molinet, J. -C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM. J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[15] |
M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.
doi: 10.3934/dcds.2011.30.253. |
[16] |
N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043-1047.
doi: 10.1016/S0764-4442(00)88471-2. |
show all references
References:
[1] |
A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166. |
[2] |
J. Angulo Pava, C. Banquet and M. Scialom, Stability for the modified and fourth Benjamin-Bona-Mahony equations, Discrete Contin. Dyn. Syst., 30 (2011), 851-871.
doi: 10.3934/dcds.2011.30.851. |
[3] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. London, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[4] |
J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. London Series A, 302 (1981), 457-510.
doi: 10.1098/rsta.1981.0178. |
[5] |
J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete and Continuous Dynamical Systems, 23 (2009), 1241-1252.
doi: 10.3934/dcds.2009.23.1241. |
[6] |
J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Disc. Cont. Dynamical Systems, 23 (2009), 1253-1275.
doi: 10.3934/dcds.2009.23.1253. |
[7] |
J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. New Ser., 3 (1997), 115-159.
doi: 10.1007/s000290050008. |
[8] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[9] |
W. Chen and J. Li, On the low regularity of the modified Korteweg-de Vries equation with a dissipative term, J. Diff. Equations, 240 (2007), 125-144.
doi: 10.1016/j.jde.2007.05.030. |
[10] |
M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity illposedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
[11] |
L. Molinet, A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space, arXiv:1110.2352v1, Proc. Amer. Math. Soc., 141 (2013), 2793-2798.
doi: 10.1090/S0002-9939-2013-11693-X. |
[12] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers Equation, Int. Math. Research Notices, (2002), 1979-2005.
doi: 10.1155/S1073792802112104. |
[13] |
L. Molinet, F. Ribaud and A Youssfi, Ill-posedness issue for a class of parabolic equations, Proceedings of the Royal Society of Edinburgh, 132 (2002), 1407-1416. |
[14] |
L. Molinet, J. -C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM. J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[15] |
M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.
doi: 10.3934/dcds.2011.30.253. |
[16] |
N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043-1047.
doi: 10.1016/S0764-4442(00)88471-2. |
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