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A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations
Pullback attractors for a class of nonlinear lattices with delays
1. | School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China, China |
References:
[1] |
T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
[2] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[3] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., 2 (2009), 17-36.
doi: 10.3934/dcdss.2009.2.17. |
[4] |
T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[5] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[6] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[7] |
J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.
doi: 10.1016/j.jmaa.2010.04.074. |
[8] |
P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.
doi: 10.3934/dcds.2010.26.989. |
[9] |
B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[10] |
Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378. |
[11] |
Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.
doi: 10.3934/dcds.2014.34.4343. |
[12] |
C. D. Zhao, S. F. Zhou and W. M. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348.
doi: 10.1016/j.na.2008.02.015. |
[13] |
S. F. Zhou, Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43 (2002), 452-465.
doi: 10.1063/1.1418719. |
[14] |
S. F. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.
doi: 10.1016/j.jde.2004.02.005. |
show all references
References:
[1] |
T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
[2] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[3] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., 2 (2009), 17-36.
doi: 10.3934/dcdss.2009.2.17. |
[4] |
T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[5] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[6] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[7] |
J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.
doi: 10.1016/j.jmaa.2010.04.074. |
[8] |
P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.
doi: 10.3934/dcds.2010.26.989. |
[9] |
B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[10] |
Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378. |
[11] |
Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.
doi: 10.3934/dcds.2014.34.4343. |
[12] |
C. D. Zhao, S. F. Zhou and W. M. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348.
doi: 10.1016/j.na.2008.02.015. |
[13] |
S. F. Zhou, Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43 (2002), 452-465.
doi: 10.1063/1.1418719. |
[14] |
S. F. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.
doi: 10.1016/j.jde.2004.02.005. |
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