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Pullback attractors for generalized evolutionary systems
1. | Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045, United States, United States |
References:
[1] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[2] |
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. |
[3] |
T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.
doi: 10.1023/A:1024422619616. |
[4] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[5] |
D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, Interdisciplinary Mathematical Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.
doi: 10.1142/9789812563088. |
[6] |
V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.
doi: 10.1512/iumj.1993.42.42049. |
[7] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[9] |
A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.
doi: 10.1016/j.jde.2006.08.021. |
[10] |
A. Cheskidov and S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.
doi: 10.1016/j.aim.2014.09.005. |
[11] |
A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.
doi: 10.1007/s10884-009-9133-x. |
[12] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55-66.
doi: 10.3934/dcdss.2009.2.55. |
[13] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. |
[14] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[15] |
F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398.
doi: 10.1023/A:1021937715194. |
[16] |
C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, 1987, 55-73. |
[17] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 17, Masson, Paris, 1991. |
[18] |
A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.
doi: 10.1016/j.jde.2007.06.008. |
[19] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011.
doi: 10.1090/surv/176. |
[20] |
P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Dynamical Numerical Analysis (Atlanta, GA, 1995), Numer. Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[21] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[22] |
S. Lu, H. Wu and C. 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. |
[23] |
V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.
doi: 10.1023/A:1008608431399. |
[24] |
J. C. Robinson, Infinite-dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[25] |
R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269.
doi: 10.1016/j.jde.2006.03.004. |
[26] |
G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.
doi: 10.1007/BF02218613. |
[27] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[28] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[29] |
M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system, Mat. Zametki, 71 (2002), 194-213.
doi: 10.1023/A:1014190629738. |
[30] |
D. Vorotnikov, Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differential Equations, 251 (2011), 2209-2225.
doi: 10.1016/j.jde.2011.07.008. |
show all references
References:
[1] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[2] |
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. |
[3] |
T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.
doi: 10.1023/A:1024422619616. |
[4] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[5] |
D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, Interdisciplinary Mathematical Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.
doi: 10.1142/9789812563088. |
[6] |
V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.
doi: 10.1512/iumj.1993.42.42049. |
[7] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[9] |
A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.
doi: 10.1016/j.jde.2006.08.021. |
[10] |
A. Cheskidov and S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.
doi: 10.1016/j.aim.2014.09.005. |
[11] |
A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.
doi: 10.1007/s10884-009-9133-x. |
[12] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55-66.
doi: 10.3934/dcdss.2009.2.55. |
[13] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. |
[14] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[15] |
F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398.
doi: 10.1023/A:1021937715194. |
[16] |
C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, 1987, 55-73. |
[17] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 17, Masson, Paris, 1991. |
[18] |
A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.
doi: 10.1016/j.jde.2007.06.008. |
[19] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011.
doi: 10.1090/surv/176. |
[20] |
P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Dynamical Numerical Analysis (Atlanta, GA, 1995), Numer. Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[21] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[22] |
S. Lu, H. Wu and C. 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. |
[23] |
V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.
doi: 10.1023/A:1008608431399. |
[24] |
J. C. Robinson, Infinite-dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[25] |
R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269.
doi: 10.1016/j.jde.2006.03.004. |
[26] |
G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.
doi: 10.1007/BF02218613. |
[27] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[28] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[29] |
M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system, Mat. Zametki, 71 (2002), 194-213.
doi: 10.1023/A:1014190629738. |
[30] |
D. Vorotnikov, Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differential Equations, 251 (2011), 2209-2225.
doi: 10.1016/j.jde.2011.07.008. |
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