• 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

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, 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.  Google Scholar

[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.  Google Scholar

[3]

H. Bae and A. Biswas, Gevrey regularity for a class of dissipative equations with analytic nonlinearity, Methods Appl. Anal., 22 (2015), 377-408. 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), 47-78. 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), 1241-1252. 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), 277-304. 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), 4565-4576. 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), 6209-6230. 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), 705-749. 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), 5377-5397. 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), 665-675. 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), 1128-1157. 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), 1766-1783. 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), 527-620. 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), 1610-1625. 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, June 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), 143-160. doi: 10.1017/S0305004100055572.  Google Scholar

[19]

M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259. 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), 289-305. 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, Massachusetts Institute of Technology, April 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), 600-613. 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), 134-144. 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), 4852-4866. 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), 1458-1472. 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), 861-865. 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), 139-147. 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), 377-408. 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), 47-78. 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), 1241-1252. 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), 277-304. 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), 4565-4576. 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), 6209-6230. 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), 705-749. 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), 5377-5397. 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), 665-675. 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), 1128-1157. 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), 1766-1783. 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), 527-620. 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), 1610-1625. 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, June 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), 143-160. doi: 10.1017/S0305004100055572.  Google Scholar

[19]

M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259. 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), 289-305. 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, Massachusetts Institute of Technology, April 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), 600-613. 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), 134-144. 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), 4852-4866. 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), 1458-1472. doi: 10.1016/j.jde.2010.01.004.  Google Scholar

[1]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]