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The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension
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 |
References:
[1] |
J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Analysis, 9 (1985), 861-865.
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), 139-147.
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), 377-408.
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), 47-78.
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), 1241-1252.
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), 277-304.
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), 4565-4576.
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), 6209-6230.
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 $\mathbb{R}^{N}$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749.
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), 5377-5397.
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), 665-675.
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), 1128-1157.
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), 1766-1783.
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), 527-620.
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), 1610-1625.
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, June 2010. |
[18] |
P. J. Olver, Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85 (1979), 143-160.
doi: 10.1017/S0305004100055572. |
[19] |
M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.
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), 289-305.
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, Massachusetts Institute of Technology, April 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), 600-613.
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), 134-144.
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), 4852-4866.
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), 1458-1472.
doi: 10.1016/j.jde.2010.01.004. |
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), 861-865.
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), 139-147.
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), 377-408.
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), 47-78.
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), 1241-1252.
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), 277-304.
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), 4565-4576.
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), 6209-6230.
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 $\mathbb{R}^{N}$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749.
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), 5377-5397.
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), 665-675.
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), 1128-1157.
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), 1766-1783.
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), 527-620.
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), 1610-1625.
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, June 2010. |
[18] |
P. J. Olver, Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85 (1979), 143-160.
doi: 10.1017/S0305004100055572. |
[19] |
M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.
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), 289-305.
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, Massachusetts Institute of Technology, April 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), 600-613.
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), 134-144.
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), 4852-4866.
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), 1458-1472.
doi: 10.1016/j.jde.2010.01.004. |
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