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The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain
The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
2. | Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main |
References:
[1] |
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.
doi: 10.1023/A:1022902802385. |
[2] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[3] |
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. |
[4] |
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. |
[5] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Global attractors for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), 183-201.
doi: 10.1016/j.na.2010.03.012. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. |
[8] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London, 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[9] |
P. E. Kloeden and B. Schmalfuss, Asymptotical behaviour of nonautonomous difference inclusions, Syst. Cont. Lett., 33 (1998), 275-280.
doi: 10.1016/S0167-6911(97)00107-2. |
[10] |
D. S. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions, J. Differential Equations, 224 (2006), 1-38.
doi: 10.1016/j.jde.2005.07.012. |
[11] |
D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of non-autonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671.
doi: 10.1007/s11228-007-0054-8. |
[12] |
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. |
[13] |
Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.
doi: 10.1512/iumj.2002.51.2255. |
[14] |
J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440.
doi: 10.1006/jdeq.1996.0036. |
[15] |
J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[16] |
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. |
[17] |
F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^N2$ with continuous nonlinearity, Asymptotic Anal., 44 (2005), 111-130. |
[18] |
R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. |
[19] |
M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193; translated from Mat. Zametki, 71 (2002), 194-213.
doi: 10.1023/A:1014190629738. |
[20] |
Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
[21] |
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. |
[22] |
Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quart. Applied Math., 71 (2013), 369-399.
doi: 10.1090/S0033-569X-2013-01306-1. |
show all references
References:
[1] |
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.
doi: 10.1023/A:1022902802385. |
[2] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[3] |
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. |
[4] |
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. |
[5] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Global attractors for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), 183-201.
doi: 10.1016/j.na.2010.03.012. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. |
[8] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London, 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[9] |
P. E. Kloeden and B. Schmalfuss, Asymptotical behaviour of nonautonomous difference inclusions, Syst. Cont. Lett., 33 (1998), 275-280.
doi: 10.1016/S0167-6911(97)00107-2. |
[10] |
D. S. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions, J. Differential Equations, 224 (2006), 1-38.
doi: 10.1016/j.jde.2005.07.012. |
[11] |
D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of non-autonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671.
doi: 10.1007/s11228-007-0054-8. |
[12] |
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. |
[13] |
Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.
doi: 10.1512/iumj.2002.51.2255. |
[14] |
J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440.
doi: 10.1006/jdeq.1996.0036. |
[15] |
J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[16] |
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. |
[17] |
F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^N2$ with continuous nonlinearity, Asymptotic Anal., 44 (2005), 111-130. |
[18] |
R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. |
[19] |
M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193; translated from Mat. Zametki, 71 (2002), 194-213.
doi: 10.1023/A:1014190629738. |
[20] |
Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
[21] |
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. |
[22] |
Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quart. Applied Math., 71 (2013), 369-399.
doi: 10.1090/S0033-569X-2013-01306-1. |
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