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On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China |
In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.
References:
[1] |
S. Bosia and S. Gatti,
Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dynamics of PDE, 11 (2014), 1-38.
doi: 10.4310/DPDE.2014.v11.n1.a1. |
[2] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez,
Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J.Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[3] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
[4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1114-1165.
|
[5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. |
[6] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. |
[7] |
I. Chueshov,
Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
|
[8] |
I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
[9] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. |
[10] |
R. Czaja and M. Efendiev,
Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
[11] |
R. Czaja,
Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Analysis, 104 (2014), 90-108.
doi: 10.1016/j.na.2014.03.020. |
[12] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential Attractors for a Nonlinear Reaction-Diffusion System in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[13] |
M. Efendiev, S. Zelik and A. Miranville,
Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[14] |
M. Efendiev, Y. Yamamoto and A. Yagi,
Exponential attractors for non-autonomous dissipative systems, Journal of the Mathematical Society of Japan, 63 (2011), 647-673.
doi: 10.2969/jmsj/06320647. |
[15] |
G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883. |
[16] |
P. Kloeden,
Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
doi: 10.1142/S0219493703000632. |
[17] |
K. Kuratowski,
Sur les espaces complets, Fund. Math., 15 (1930), 301-309.
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[18] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
|
[19] |
S. S. Lu, H. Q. Wu and C. K. Zhong,
Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[20] |
J. Simon,
Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65-96.
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[21] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. |
[22] |
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Sys., 33 (2013), 3189-3209.
doi: 10.3934/dcds.2013.33.3189. |
[23] |
Z. J. Yang and Y. Q. Wang,
Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.
doi: 10.1016/j.jde.2010.09.024. |
[24] |
Z. J. Yang, P. Y. Ding and L. Li,
Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.
doi: 10.1016/j.jmaa.2016.04.079. |
[25] |
S. F. Zhou and X. Y. Han,
Pullback exponential attractors for non-autonomous lattice systems, J Dyn. Diff. Equat., 24 (2012), 601-631.
doi: 10.1007/s10884-012-9260-7. |
show all references
References:
[1] |
S. Bosia and S. Gatti,
Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dynamics of PDE, 11 (2014), 1-38.
doi: 10.4310/DPDE.2014.v11.n1.a1. |
[2] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez,
Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J.Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[3] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
[4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1114-1165.
|
[5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. |
[6] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999. |
[7] |
I. Chueshov,
Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
|
[8] |
I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
[9] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. |
[10] |
R. Czaja and M. Efendiev,
Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
[11] |
R. Czaja,
Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Analysis, 104 (2014), 90-108.
doi: 10.1016/j.na.2014.03.020. |
[12] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential Attractors for a Nonlinear Reaction-Diffusion System in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[13] |
M. Efendiev, S. Zelik and A. Miranville,
Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[14] |
M. Efendiev, Y. Yamamoto and A. Yagi,
Exponential attractors for non-autonomous dissipative systems, Journal of the Mathematical Society of Japan, 63 (2011), 647-673.
doi: 10.2969/jmsj/06320647. |
[15] |
G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883. |
[16] |
P. Kloeden,
Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
doi: 10.1142/S0219493703000632. |
[17] |
K. Kuratowski,
Sur les espaces complets, Fund. Math., 15 (1930), 301-309.
|
[18] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
|
[19] |
S. S. Lu, H. Q. Wu and C. K. Zhong,
Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[20] |
J. Simon,
Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65-96.
|
[21] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. |
[22] |
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Sys., 33 (2013), 3189-3209.
doi: 10.3934/dcds.2013.33.3189. |
[23] |
Z. J. Yang and Y. Q. Wang,
Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.
doi: 10.1016/j.jde.2010.09.024. |
[24] |
Z. J. Yang, P. Y. Ding and L. Li,
Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.
doi: 10.1016/j.jmaa.2016.04.079. |
[25] |
S. F. Zhou and X. Y. Han,
Pullback exponential attractors for non-autonomous lattice systems, J Dyn. Diff. Equat., 24 (2012), 601-631.
doi: 10.1007/s10884-012-9260-7. |
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