American Institute of Mathematical Sciences

November  2014, 34(11): 4565-4576. doi: 10.3934/dcds.2014.34.4565

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

Received  October 2013 Revised  December 2013 Published  May 2014

We consider the Cauchy problem associated to the generalized Benjamin-Bona-Mahony (BBM) equation for given data in the $L^2$-based Sobolev spaces. Depending on the order of nonlinearity and dispersion, we prove that the Cauchy problem is ill-posed for data with lower order Sobolev regularity. We also prove that, in certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.
Citation: Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
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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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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