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Singular support of the global attractor for a damped BBM equation
a. | College of Science, National University of Defense Technology, Changsha, 410073, China |
b. | TAG_SUPSchool of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China |
c. | School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China |
The singular support of the global attractor is introduced. It is shown that the singular support of the global attractor for a damped BBM equation equals to the singular support of the force term. This gives a delicate description of the local regularity, which roughly says that the attractor is smooth exactly where the force is smooth.
References:
[1] |
K. Ammari and E. Crépeau,
Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.
doi: 10.1016/j.sysconle.2019.03.005. |
[2] |
T. B. Benjamin, J. L. Bona and J. 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. |
[3] |
J. L. Bona and V. A. Dougalis,
An initial-and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.
doi: 10.1016/0022-247X(80)90098-0. |
[4] |
J. L. Bona and R. Smith,
The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[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] |
F. Dell'Oro, O. Goubet, Y. Mammeri and V. Pata,
Global attractors for the Benjamin-Bona-Mahony equation with memory, Indiana Univ. Math. J., 69 (2020), 749-783.
doi: 10.1512/iumj.2020.69.7906. |
[7] |
F. Dell'Oro and Y. Mammeri, Benjamin-Bona-Mahony equations with memory and rayleigh friction, Applied Mathematics & Optimization, (2019), in press.
doi: 10.1007/s00245-019-09568-z. |
[8] |
F. Dell'Oro, Y. Mammeri and V. Pata,
The Benjamin-Bona-Mahony equation with dissipative memory, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 899-910.
doi: 10.1007/s00030-014-0308-8. |
[9] |
L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010.
doi: 10.1090/GSM/019. |
[10] |
O. Goubet,
Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644.
doi: 10.3934/dcds.2000.6.625. |
[11] |
O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25–53.
doi: 10.1006/jdeq.2001.4163. |
[12] |
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. |
[13] |
J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
doi: 10.1090/surv/025. |
[14] |
L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 1990.
doi: 10.1007/978-3-642-61497-2. |
[15] |
L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005.
doi: 10.1007/b138375. |
[16] |
J.-R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Applied Mathematics and Computation, 274 (2016), 343–352.
doi: 10.1016/j.amc.2015.10.086. |
[17] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991) 323–347.
doi: 10.1090/S0894-0347-1991-1086966-0. |
[18] |
D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100.
doi: 10.4171/rmi/1049. |
[19] |
Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stochastics and Dynamics, 18 (2018), 1850004.
doi: 10.1142/S0219493718500041. |
[20] |
Y. Qin, X. Yang and X. Liu,
Pullback attractor of Benjamin-Bona-Mahony equations in $H^2$, Acta. Math. Sci., 32 (2012), 1338-1348.
doi: 10.1016/S0252-9602(12)60103-9. |
[21] |
M. Stanislavova,
On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832.
|
[22] |
M. Stanislavova, A. Stefanov and B. Wang,
Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on ${\mathbb{R}}^3$, J. Differ. Equations, 219 (2005), 451-483.
doi: 10.1016/j.jde.2005.08.004. |
[23] |
C. Sun, M. Yang and C. Zhong,
Global attractors for the wave equation with nonlinear damping, J. Differ. Equations, 227 (2006), 427-443.
doi: 10.1016/j.jde.2005.09.010. |
[24] |
B. Wang,
Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10 (1997), 23-28.
doi: 10.1016/S0893-9659(97)00005-0. |
[25] |
B. Wang,
Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 31 (1998), 7635-7645.
doi: 10.1088/0305-4470/31/37/021. |
[26] |
B. Wang,
Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, Journal of Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[27] |
B. Wang, D. W. Fussner and C. Bi,
Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A Math. Theor., 40 (2007), 10491-10504.
doi: 10.1088/1751-8113/40/34/007. |
[28] |
B. Wang and W. Yang,
Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 30 (1997), 4877-4885.
doi: 10.1088/0305-4470/30/13/035. |
[29] |
M. Wang,
Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Analysis: Theory, Methods & Applications, 105 (2014), 134-144.
doi: 10.1016/j.na.2014.04.013. |
[30] |
M. Wang,
Long time behavior of a damped generalized BBM equation in low regularity spaces, Math. Method App. Sci., 38 (2015), 4852-4866.
doi: 10.1002/mma.3400. |
[31] |
M. Wang,
Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst.-A., 35 (2015), 3799-3825.
doi: 10.3934/dcds.2015.35.3799. |
[32] |
M. Wang,
Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces, Discrete Cont. Dyn-A., 36 (2016), 5763-5788.
doi: 10.3934/dcds.2016053. |
[33] |
M. Wang and A. Liu,
Dynamics of the BBM equation with a distribution force in low regularity spaces, Topological Methods in Nonlinear Analysis, 51 (2018), 91-109.
doi: 10.12775/TMNA.2017.058. |
[34] |
M. Wang and Z. Zhang,
Sharp global well-posedness for the fractional BBM equation, Mathematical Methods in the Applied Sciences, 41 (2018), 5906-5918.
doi: 10.1002/mma.5109. |
[35] |
Q. Zhang and Y. Li,
Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, Journal of Dynamical and Control Systems, 26 (2020), 423-441.
doi: 10.1007/s10883-019-09450-9. |
[36] |
M. Zhao, X.-G. Yang, X. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptotic Analysis, (2020), in press.
doi: 10.3233/ASY-201601. |
show all references
References:
[1] |
K. Ammari and E. Crépeau,
Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.
doi: 10.1016/j.sysconle.2019.03.005. |
[2] |
T. B. Benjamin, J. L. Bona and J. 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. |
[3] |
J. L. Bona and V. A. Dougalis,
An initial-and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.
doi: 10.1016/0022-247X(80)90098-0. |
[4] |
J. L. Bona and R. Smith,
The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[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] |
F. Dell'Oro, O. Goubet, Y. Mammeri and V. Pata,
Global attractors for the Benjamin-Bona-Mahony equation with memory, Indiana Univ. Math. J., 69 (2020), 749-783.
doi: 10.1512/iumj.2020.69.7906. |
[7] |
F. Dell'Oro and Y. Mammeri, Benjamin-Bona-Mahony equations with memory and rayleigh friction, Applied Mathematics & Optimization, (2019), in press.
doi: 10.1007/s00245-019-09568-z. |
[8] |
F. Dell'Oro, Y. Mammeri and V. Pata,
The Benjamin-Bona-Mahony equation with dissipative memory, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 899-910.
doi: 10.1007/s00030-014-0308-8. |
[9] |
L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010.
doi: 10.1090/GSM/019. |
[10] |
O. Goubet,
Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644.
doi: 10.3934/dcds.2000.6.625. |
[11] |
O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25–53.
doi: 10.1006/jdeq.2001.4163. |
[12] |
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. |
[13] |
J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
doi: 10.1090/surv/025. |
[14] |
L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 1990.
doi: 10.1007/978-3-642-61497-2. |
[15] |
L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005.
doi: 10.1007/b138375. |
[16] |
J.-R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Applied Mathematics and Computation, 274 (2016), 343–352.
doi: 10.1016/j.amc.2015.10.086. |
[17] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991) 323–347.
doi: 10.1090/S0894-0347-1991-1086966-0. |
[18] |
D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100.
doi: 10.4171/rmi/1049. |
[19] |
Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stochastics and Dynamics, 18 (2018), 1850004.
doi: 10.1142/S0219493718500041. |
[20] |
Y. Qin, X. Yang and X. Liu,
Pullback attractor of Benjamin-Bona-Mahony equations in $H^2$, Acta. Math. Sci., 32 (2012), 1338-1348.
doi: 10.1016/S0252-9602(12)60103-9. |
[21] |
M. Stanislavova,
On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832.
|
[22] |
M. Stanislavova, A. Stefanov and B. Wang,
Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on ${\mathbb{R}}^3$, J. Differ. Equations, 219 (2005), 451-483.
doi: 10.1016/j.jde.2005.08.004. |
[23] |
C. Sun, M. Yang and C. Zhong,
Global attractors for the wave equation with nonlinear damping, J. Differ. Equations, 227 (2006), 427-443.
doi: 10.1016/j.jde.2005.09.010. |
[24] |
B. Wang,
Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10 (1997), 23-28.
doi: 10.1016/S0893-9659(97)00005-0. |
[25] |
B. Wang,
Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 31 (1998), 7635-7645.
doi: 10.1088/0305-4470/31/37/021. |
[26] |
B. Wang,
Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, Journal of Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[27] |
B. Wang, D. W. Fussner and C. Bi,
Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A Math. Theor., 40 (2007), 10491-10504.
doi: 10.1088/1751-8113/40/34/007. |
[28] |
B. Wang and W. Yang,
Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 30 (1997), 4877-4885.
doi: 10.1088/0305-4470/30/13/035. |
[29] |
M. Wang,
Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Analysis: Theory, Methods & Applications, 105 (2014), 134-144.
doi: 10.1016/j.na.2014.04.013. |
[30] |
M. Wang,
Long time behavior of a damped generalized BBM equation in low regularity spaces, Math. Method App. Sci., 38 (2015), 4852-4866.
doi: 10.1002/mma.3400. |
[31] |
M. Wang,
Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst.-A., 35 (2015), 3799-3825.
doi: 10.3934/dcds.2015.35.3799. |
[32] |
M. Wang,
Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces, Discrete Cont. Dyn-A., 36 (2016), 5763-5788.
doi: 10.3934/dcds.2016053. |
[33] |
M. Wang and A. Liu,
Dynamics of the BBM equation with a distribution force in low regularity spaces, Topological Methods in Nonlinear Analysis, 51 (2018), 91-109.
doi: 10.12775/TMNA.2017.058. |
[34] |
M. Wang and Z. Zhang,
Sharp global well-posedness for the fractional BBM equation, Mathematical Methods in the Applied Sciences, 41 (2018), 5906-5918.
doi: 10.1002/mma.5109. |
[35] |
Q. Zhang and Y. Li,
Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, Journal of Dynamical and Control Systems, 26 (2020), 423-441.
doi: 10.1007/s10883-019-09450-9. |
[36] |
M. Zhao, X.-G. Yang, X. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptotic Analysis, (2020), in press.
doi: 10.3233/ASY-201601. |
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