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October  2016, 36(10): 5763-5788. doi: 10.3934/dcds.2016053

Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces

Ming Wang 1,

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Received  August 2015 Revised  February 2016 Published  July 2016

The global well-posedness of the BBM equation is established in $H^{s,p}(\textbf{R})$ with $s\geq \max\{0,\frac{1}{p}-\frac{1}{2}\}$ and $1\leq p<\infty$. Moreover, the well-posedness results are shown to be sharp in the sense that the solution map is no longer $C^2$ from $H^{s,p}(\textbf{R})$ to $C([0,T];H^{s,p}(\textbf{R}))$ for smaller $s$ or $p$. Finally, some growth bounds of global solutions in terms of time $T$ are proved.
Keywords: $L^p$ space., well posedness, BBM equation, ill posedness.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A0.
Citation: Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
References:
[1]

J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions,, Nonlinear Analysis, 9 (1985), 861.  doi: 10.1016/0362-546X(85)90023-9.  Google Scholar

[2]

J. Avrin, The generalized Benjamin-Bona-Mahony equation in $R^n$ with singular initial data,, Nonlinear Analysis, 11 (1987), 139.  doi: 10.1016/0362-546X(87)90032-0.  Google Scholar

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T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil.Trans. R. Soc., 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[5]

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

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J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, International Mathematics Research Notices, 6 (1996), 277.  doi: 10.1155/S1073792896000207.  Google Scholar

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X. Carvajal and M. Panthee, On ill-posedness for the generalized BBM equation,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 4565.  doi: 10.3934/dcds.2014.34.4565.  Google Scholar

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W. Chen, Z. Guo and J. Xiao, Sharp well-posedness for the Benjamin equation,, Nonlinear Analysis, 74 (2011), 6209.  doi: 10.1016/j.na.2011.06.002.  Google Scholar

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, Journal of the American Mathematical Society, 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

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Q. Deng, Y. Ding and X. Yao, Gaussian bounds for higher-order elliptic differential operators with Kato type potentials,, J. Funct. Anal., 266 (2014), 5377.  doi: 10.1016/j.jfa.2014.02.014.  Google Scholar

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J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions,, Nonlinear Analysis, 4 (1980), 665.  doi: 10.1016/0362-546X(80)90067-X.  Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis, Spinger 2nd ed.,, New York, (2009).  doi: 10.1007/978-1-4939-1230-8.  Google Scholar

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L. Grafakos and S. Oh, The Kato-Ponce Inequality,, Communications in Partial Differential Equations, 39 (2014), 1128.  doi: 10.1080/03605302.2013.822885.  Google Scholar

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Y. Guo, M. Wang and Y. Tang, Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R,, Applicable Analysis: An International Journal, 94 (2015), 1766.  doi: 10.1080/00036811.2014.946561.  Google Scholar

[15]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle,, Communications on Pure and Applied Mathematics, 46 (1993), 527.  doi: 10.1002/cpa.3160460405.  Google Scholar

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Y. Li and Y. Wu, Global well-posedness for the Benjamin eqaution in low regularity,, Nonlinear Anal., 73 (2010), 1610.  doi: 10.1016/j.na.2010.04.068.  Google Scholar

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J. Nahas, A Decay Property of Solutions to the mKdV Equation,, PhD. Thesis University of California-Santa Barbara, (2010).   Google Scholar

[18]

P. J. Olver, Euler operators and conservation laws of the BBM equation,, Mathematical Proceedings of the Cambridge Philosophical Society, 85 (1979), 143.  doi: 10.1017/S0305004100055572.  Google Scholar

[19]

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

[20]

M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness,, Elsevier, (1975).   Google Scholar

[21]

D. Roumégoux, A symplectic non-squeezing theorem for BBM equation,, Dyn. Partial Differ. Equ., 7 (2010), 289.  doi: 10.4310/DPDE.2010.v7.n4.a1.  Google Scholar

[22]

W. Rudin, Functional Analysis,, International series in pure and applied mathematics, (1991).   Google Scholar

[23]

V. Sohinger, Bounds on the Growth of High Sobolev Norms of Solutions to Nonlinear Schrödinger Equations,, PhD. Thesis, (2011).   Google Scholar

[24]

E. M. Stein, Singular Integral and Differential Property of Functions,, New Jersey: Princeton Univ. Press, (1970).   Google Scholar

[25]

H. Wang and S. Cui, Global well-posedness of the Cauchy problem of the fifth-order shallow water equation,, Journal of Differential Equations, 230 (2006), 600.  doi: 10.1016/j.jde.2006.04.008.  Google Scholar

[26]

M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces,, Nonlinear Analysis, 105 (2014), 134.  doi: 10.1016/j.na.2014.04.013.  Google Scholar

[27]

M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces,, Mathematical Methods in the Applied Sciences, 38 (2015), 4852.  doi: 10.1002/mma.3400.  Google Scholar

[28]

X. Yang and Y. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces,, Journal of Differential Equations, 248 (2010), 1458.  doi: 10.1016/j.jde.2010.01.004.  Google Scholar

show all references

References:
[1]

J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions,, Nonlinear Analysis, 9 (1985), 861.  doi: 10.1016/0362-546X(85)90023-9.  Google Scholar

[2]

J. Avrin, The generalized Benjamin-Bona-Mahony equation in $R^n$ with singular initial data,, Nonlinear Analysis, 11 (1987), 139.  doi: 10.1016/0362-546X(87)90032-0.  Google Scholar

[3]

H. Bae and A. Biswas, Gevrey regularity for a class of dissipative equations with analytic nonlinearity,, Methods Appl. Anal., 22 (2015), 377.  doi: 10.4310/MAA.2015.v22.n4.a3.  Google Scholar

[4]

T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil.Trans. R. Soc., 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[5]

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

[6]

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, International Mathematics Research Notices, 6 (1996), 277.  doi: 10.1155/S1073792896000207.  Google Scholar

[7]

X. Carvajal and M. Panthee, On ill-posedness for the generalized BBM equation,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 4565.  doi: 10.3934/dcds.2014.34.4565.  Google Scholar

[8]

W. Chen, Z. Guo and J. Xiao, Sharp well-posedness for the Benjamin equation,, Nonlinear Analysis, 74 (2011), 6209.  doi: 10.1016/j.na.2011.06.002.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, Journal of the American Mathematical Society, 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[10]

Q. Deng, Y. Ding and X. Yao, Gaussian bounds for higher-order elliptic differential operators with Kato type potentials,, J. Funct. Anal., 266 (2014), 5377.  doi: 10.1016/j.jfa.2014.02.014.  Google Scholar

[11]

J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions,, Nonlinear Analysis, 4 (1980), 665.  doi: 10.1016/0362-546X(80)90067-X.  Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis, Spinger 2nd ed.,, New York, (2009).  doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[13]

L. Grafakos and S. Oh, The Kato-Ponce Inequality,, Communications in Partial Differential Equations, 39 (2014), 1128.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[14]

Y. Guo, M. Wang and Y. Tang, Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R,, Applicable Analysis: An International Journal, 94 (2015), 1766.  doi: 10.1080/00036811.2014.946561.  Google Scholar

[15]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle,, Communications on Pure and Applied Mathematics, 46 (1993), 527.  doi: 10.1002/cpa.3160460405.  Google Scholar

[16]

Y. Li and Y. Wu, Global well-posedness for the Benjamin eqaution in low regularity,, Nonlinear Anal., 73 (2010), 1610.  doi: 10.1016/j.na.2010.04.068.  Google Scholar

[17]

J. Nahas, A Decay Property of Solutions to the mKdV Equation,, PhD. Thesis University of California-Santa Barbara, (2010).   Google Scholar

[18]

P. J. Olver, Euler operators and conservation laws of the BBM equation,, Mathematical Proceedings of the Cambridge Philosophical Society, 85 (1979), 143.  doi: 10.1017/S0305004100055572.  Google Scholar

[19]

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

[20]

M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness,, Elsevier, (1975).   Google Scholar

[21]

D. Roumégoux, A symplectic non-squeezing theorem for BBM equation,, Dyn. Partial Differ. Equ., 7 (2010), 289.  doi: 10.4310/DPDE.2010.v7.n4.a1.  Google Scholar

[22]

W. Rudin, Functional Analysis,, International series in pure and applied mathematics, (1991).   Google Scholar

[23]

V. Sohinger, Bounds on the Growth of High Sobolev Norms of Solutions to Nonlinear Schrödinger Equations,, PhD. Thesis, (2011).   Google Scholar

[24]

E. M. Stein, Singular Integral and Differential Property of Functions,, New Jersey: Princeton Univ. Press, (1970).   Google Scholar

[25]

H. Wang and S. Cui, Global well-posedness of the Cauchy problem of the fifth-order shallow water equation,, Journal of Differential Equations, 230 (2006), 600.  doi: 10.1016/j.jde.2006.04.008.  Google Scholar

[26]

M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces,, Nonlinear Analysis, 105 (2014), 134.  doi: 10.1016/j.na.2014.04.013.  Google Scholar

[27]

M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces,, Mathematical Methods in the Applied Sciences, 38 (2015), 4852.  doi: 10.1002/mma.3400.  Google Scholar

[28]

X. Yang and Y. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces,, Journal of Differential Equations, 248 (2010), 1458.  doi: 10.1016/j.jde.2010.01.004.  Google Scholar

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