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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.3934/dcds.2011.30.851.  Google Scholar

[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.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[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.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[5]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete and Continuous Dynamical Systems, 23 (2009), 1241.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[6]

J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Disc. Cont. Dynamical Systems, 23 (2009), 1253.  doi: 10.3934/dcds.2009.23.1253.  Google Scholar

[7]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, Selecta Math. New Ser., 3 (1997), 115.  doi: 10.1007/s000290050008.  Google Scholar

[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.  doi: 10.1007/BF01895688.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.030.  Google Scholar

[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.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[11]

L. Molinet, A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space,, , 141 (2013), 2793.  doi: 10.1090/S0002-9939-2013-11693-X.  Google Scholar

[12]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers Equation,, Int. Math. Research Notices, (2002), 1979.  doi: 10.1155/S1073792802112104.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141001385307.  Google Scholar

[15]

M. Panthee, On the ill-posedness result for the BBM equation,, Discrete Contin. Dyn. Syst., 30 (2011), 253.  doi: 10.3934/dcds.2011.30.253.  Google Scholar

[16]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043.  doi: 10.1016/S0764-4442(00)88471-2.  Google Scholar

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.   Google Scholar

[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.  doi: 10.3934/dcds.2011.30.851.  Google Scholar

[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.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[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.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[5]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete and Continuous Dynamical Systems, 23 (2009), 1241.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[6]

J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Disc. Cont. Dynamical Systems, 23 (2009), 1253.  doi: 10.3934/dcds.2009.23.1253.  Google Scholar

[7]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, Selecta Math. New Ser., 3 (1997), 115.  doi: 10.1007/s000290050008.  Google Scholar

[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.  doi: 10.1007/BF01895688.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.030.  Google Scholar

[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.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[11]

L. Molinet, A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space,, , 141 (2013), 2793.  doi: 10.1090/S0002-9939-2013-11693-X.  Google Scholar

[12]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers Equation,, Int. Math. Research Notices, (2002), 1979.  doi: 10.1155/S1073792802112104.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141001385307.  Google Scholar

[15]

M. Panthee, On the ill-posedness result for the BBM equation,, Discrete Contin. Dyn. Syst., 30 (2011), 253.  doi: 10.3934/dcds.2011.30.253.  Google Scholar

[16]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043.  doi: 10.1016/S0764-4442(00)88471-2.  Google Scholar

[1]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[2]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[3]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[4]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[5]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[6]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[7]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

[8]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[9]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[10]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[11]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[12]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[13]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[14]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[15]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[16]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[17]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[18]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[19]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[20]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]