On the ill-posedness result for the BBM equation

Mahendra Panthee

doi:10.3934/dcds.2011.30.253

Article Contents

2011, Volume 30, Issue 1: 253-259. Doi: 10.3934/dcds.2011.30.253

This issue Previous Article Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain Next Article Reducibility of skew-product systems with multidimensional Brjuno base flows

On the ill-posedness result for the BBM equation

Mahendra Panthee^1,

1.
Centro de Matemática, Universidade do Minho, 4710-057, Braga

Received: February 2010

Revised: September 2010

Published: January 2011

Abstract Related Papers Cited by

Abstract

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any fixed $t>0$ small enough. This result is sharp.

Keywords:

Mathematics Subject Classification: Primary: 35B45, 35Q53; Secondary: 76B15.

Citation:

Related Papers

Cited by

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, preprint, 2010.
[3]	I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, Jr. Functional Analysis, 233 (2006), 228-259.doi: 10.1016/j.jfa.2005.08.004.
[4]	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.
[5]	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.
[6]	J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete and Continuous Dynamical Systems, 23 (2009), 1241-1252.
[7]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.doi: 10.1007/BF01896020.
[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]	J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Sel. Math., New Ser., 3 (1997), 115-159.doi: 10.1007/s000290050008.
[10]	A. Grünrock, M. Panthee and J. D. Silva, On KP-II type equations on cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2335-2358.doi: 10.1016/j.anihpc.2009.04.002.
[11]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.doi: 10.1002/cpa.3160460405.
[12]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation, J. Amer. Math Soc., 9 (1996), 573-603.doi: 10.1090/S0894-0347-96-00200-7.
[13]	F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431.doi: 10.3934/cpaa.2004.3.417.
[14]	L. Molinet, Sharp ill-posedness result for the periodic Benjamin-Ono equation, J. Funct. Anal., 257 (2009), 3488-3516.doi: 10.1016/j.jfa.2009.08.018.
[15]	L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 2002 (2002), 1979-2005.doi: 10.1155/S1073792802112104.
[16]	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.
[17]	L. Molinet, J. C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.doi: 10.1215/S0012-7094-02-11525-7.
[18]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The real line case, preprint, arXiv:0911.5256.
[19]	L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case, preprint, arXiv:1005.4805.
[20]	D. Roumegoux, A symplectic non-squeezing theorem for BBM equation, preprint, arXiv:1007.1359.
[21]	H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electronic Jr. Diff. Eqn., 42 (2001), 1-23.
[22]	N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 1043-1047.