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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:
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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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##### 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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