April  2011, 30(1): 253-259. doi: 10.3934/dcds.2011.30.253

On the ill-posedness result for the BBM equation

1. 

Centro de Matemática, Universidade do Minho, 4710-057, Braga, Portugal

Received  February 2010 Revised  September 2010 Published  February 2011

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.
Citation: Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
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,, preprint, (2010). Google Scholar

[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. doi: 10.1016/j.jfa.2005.08.004. Google Scholar

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

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

[6]

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

[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. doi: 10.1007/BF01896020. 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]

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

[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. doi: 10.1016/j.anihpc.2009.04.002. Google Scholar

[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. doi: 10.1002/cpa.3160460405. Google Scholar

[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. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[13]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417. Google Scholar

[14]

L. Molinet, Sharp ill-posedness result for the periodic Benjamin-Ono equation,, J. Funct. Anal., 257 (2009), 3488. doi: 10.1016/j.jfa.2009.08.018. Google Scholar

[15]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation,, Int. Math. Res. Not., 2002 (2002), 1979. doi: 10.1155/S1073792802112104. Google Scholar

[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. doi: 10.1137/S0036141001385307. Google Scholar

[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. doi: 10.1215/S0012-7094-02-11525-7. Google Scholar

[18]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The real line case,, preprint, (). Google Scholar

[19]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case,, preprint, (). Google Scholar

[20]

D. Roumegoux, A symplectic non-squeezing theorem for BBM equation,, preprint, (). Google Scholar

[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. Google Scholar

[22]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 1043. 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,, preprint, (2010). Google Scholar

[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. doi: 10.1016/j.jfa.2005.08.004. Google Scholar

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

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

[6]

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

[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. doi: 10.1007/BF01896020. 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]

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

[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. doi: 10.1016/j.anihpc.2009.04.002. Google Scholar

[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. doi: 10.1002/cpa.3160460405. Google Scholar

[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. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[13]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417. Google Scholar

[14]

L. Molinet, Sharp ill-posedness result for the periodic Benjamin-Ono equation,, J. Funct. Anal., 257 (2009), 3488. doi: 10.1016/j.jfa.2009.08.018. Google Scholar

[15]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation,, Int. Math. Res. Not., 2002 (2002), 1979. doi: 10.1155/S1073792802112104. Google Scholar

[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. doi: 10.1137/S0036141001385307. Google Scholar

[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. doi: 10.1215/S0012-7094-02-11525-7. Google Scholar

[18]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The real line case,, preprint, (). Google Scholar

[19]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case,, preprint, (). Google Scholar

[20]

D. Roumegoux, A symplectic non-squeezing theorem for BBM equation,, preprint, (). Google Scholar

[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. Google Scholar

[22]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 1043. Google Scholar

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