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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

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.
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

[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

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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