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

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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

Abstract
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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:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
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[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.
[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.
[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.
[17]	J. Nahas, A Decay Property of Solutions to the mKdV Equation,, PhD. Thesis University of California-Santa Barbara, (2010).
[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.
[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.
[20]	M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness,, Elsevier, (1975).
[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.
[22]	W. Rudin, Functional Analysis,, International series in pure and applied mathematics, (1991).
[23]	V. Sohinger, Bounds on the Growth of High Sobolev Norms of Solutions to Nonlinear Schrödinger Equations,, PhD. Thesis, (2011).
[24]	E. M. Stein, Singular Integral and Differential Property of Functions,, New Jersey: Princeton Univ. Press, (1970).
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	L. Grafakos, Modern Fourier Analysis, Spinger 2nd ed.,, New York, (2009). doi: 10.1007/978-1-4939-1230-8.
[13]	L. Grafakos and S. Oh, The Kato-Ponce Inequality,, Communications in Partial Differential Equations, 39 (2014), 1128. doi: 10.1080/03605302.2013.822885.
[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.
[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.
[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.
[17]	J. Nahas, A Decay Property of Solutions to the mKdV Equation,, PhD. Thesis University of California-Santa Barbara, (2010).
[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.
[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.
[20]	M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness,, Elsevier, (1975).
[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.
[22]	W. Rudin, Functional Analysis,, International series in pure and applied mathematics, (1991).
[23]	V. Sohinger, Bounds on the Growth of High Sobolev Norms of Solutions to Nonlinear Schrödinger Equations,, PhD. Thesis, (2011).
[24]	E. M. Stein, Singular Integral and Differential Property of Functions,, New Jersey: Princeton Univ. Press, (1970).
[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.
[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.
[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.
[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.

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