American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations
  • DCDS Home
  • This Issue
  • Next Article
    The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension
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 and Continuous Dynamical Systems, 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), 861-865. 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.

[1]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
[2]

Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
[3]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[4]

Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011
[5]

Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149
[6]

Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365
[7]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[8]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure and Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
[9]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[10]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[11]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[12]

Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367
[13]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure and Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[14]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[15]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[16]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012
[17]

Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725
[18]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
[19]

Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations and Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002
[20]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (167)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy