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Attractors for autonomous and nonautonomous 3D NavierStokesVoight equations
1.  Department of Mathematics, Nanjing University, Nanjing 210093, China, China 
References:
[1] 
A. Babin and M. Vishik, "Attractors of Evolution Equations," North Holland, Amsterdam, 1992. 
[2] 
Y. Cao, E. Lunasin and E. Titi, Global wellposedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823848. 
[3] 
V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. 
[4] 
J. Cholewa and T. Dlotko, "Global Attractors in Abstract Parabolic Problems," Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. 
[5] 
P. Constantin and C. Foias, "NavierStokes Equations," The University of Chicago Press, 1988. 
[6] 
C. Foias, O. Manley, R. Rosa and R. Temam, "NavierStokes Equations and Turbulence," Cambridge University Press, New York, 2001. doi: 10.1017/CBO9780511546754. 
[7] 
J. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988. 
[8] 
V. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 5054. 
[9] 
V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Nonclassical Type," Postdoctoral Thesis, St. Petersburg, 1988. 
[10] 
V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D NavierStokesVoight equations, Chinese Annals of Mathematics, Series B, 30 (2009), 697714. doi: 10.1007/s1140100902053. 
[11] 
V. Kalantarov, B. Levant and E. Titi, Gevrey regularity of the global attractor of the 3D NavierStokesVoight equations, Journal of Nonlinear Science, 19 (2009), 133152. doi: 10.1007/s0033200890297. 
[12] 
B. Levant, F. Ramos and E. Titi, On the statistical properties of the 3D incompressible NavierStokesVoigt model, Communications in Mathematical Sciences, 8 (2010), 277293. 
[13] 
P. Kloeden, J. Langa and J. Real, Pullback $V$attractors of the 3dimensional globally modified NavierStokes equations, Commun. Pure Appl. Anal., 6 (2007), 937955. 
[14] 
P. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the threedimensional NavierStokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 14911508. doi: 10.1098/rspa.2007.1831. 
[15] 
P. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163181. doi: 10.1098/rspa.2006.1753. 
[16] 
T. Caraballo, J. Real and P. Kloeden, Unique strong solutions and $V$attractors of a three dimensional system of globally modified NavierStokes equations, Adv. Nonlinear Stud., 6 (2006), 411436. 
[17] 
O. Ladyzhenskaya, "Attractors for Semigroups and Evolutions," LezioniLincee, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. 
[18] 
O. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," New York: Gordon and Breach Science Publisher, 1963. 
[19] 
O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Nauka, 1967. 
[20] 
J. Lions, "Quelques Methodes de Resolution des Problemes aux Limits Nlineaires," Dunod, Paris, 1969. 
[21] 
J. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," SpringVerlag, Berlin, 1972. 
[22] 
Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math. J., 51 (2002), 15411559. doi: 10.1512/iumj.2002.51.2255. 
[23] 
T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science, Series A, Vol. 53, 2005. doi: 10.1142/9789812701152. 
[24] 
T. Ma and S. Wang, "Stability and Bifurcation of Nonlinear Evolutions Equations," Science Press, April, 2007. 
[25] 
T. Ma and S. Wang, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics," AMS Monograph and Mathematical Survey Series, vol. 119, 2005 
[26] 
I. Moise, R. Rosa and X. Wang, Attractors for noncompact semigroups via energy equations, Nonlinearity, 11 (1998), 13691393. doi: 10.1088/09517715/11/5/012. 
[27] 
A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI), 38 (1973), 98116. 
[28] 
V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511533. doi: 10.1007/s0022000412331. 
[29] 
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 14951506. doi: 10.1088/09517715/19/7/001. 
[30] 
J. Robinson, "Infinitedimensional Dynamical Systems," Cambridge University Press, "Texes in Applied Mathematics," Series, 2001. 
[31] 
C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 5181. 
[32] 
R. Temam, "NavierStokes Equations, Theory and Numerical Analysis," 3rd revised edition, North Holland, 2001. 
[33] 
R. Temam, "InfiniteDimensional Systems in Mechanics and Physics," SpringerVerlag, New York, 1997. 
[34] 
G. Yue and C. Zhong, On the convergence of the uniform attractor of 2D NS$\alpha$ model to the uniform attractor of 2D NS system, J. Comput. Appl. Math., 233 (2010), 18791887. doi: 10.1016/j.cam.2009.09.024. 
[35] 
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921934. doi: 10.3934/cpaa.2004.3.921. 
[36] 
C. Zhong, M. Yang and C. Sun, The existence of global attractors for the normtoweak continuous semigroup and application to the nonlinear reactiondiffusion equations, J. Diff. Equations, 223 (2006), 367399. doi: 10.1016/j.jde.2005.06.008. 
show all references
References:
[1] 
A. Babin and M. Vishik, "Attractors of Evolution Equations," North Holland, Amsterdam, 1992. 
[2] 
Y. Cao, E. Lunasin and E. Titi, Global wellposedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823848. 
[3] 
V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. 
[4] 
J. Cholewa and T. Dlotko, "Global Attractors in Abstract Parabolic Problems," Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. 
[5] 
P. Constantin and C. Foias, "NavierStokes Equations," The University of Chicago Press, 1988. 
[6] 
C. Foias, O. Manley, R. Rosa and R. Temam, "NavierStokes Equations and Turbulence," Cambridge University Press, New York, 2001. doi: 10.1017/CBO9780511546754. 
[7] 
J. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988. 
[8] 
V. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 5054. 
[9] 
V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Nonclassical Type," Postdoctoral Thesis, St. Petersburg, 1988. 
[10] 
V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D NavierStokesVoight equations, Chinese Annals of Mathematics, Series B, 30 (2009), 697714. doi: 10.1007/s1140100902053. 
[11] 
V. Kalantarov, B. Levant and E. Titi, Gevrey regularity of the global attractor of the 3D NavierStokesVoight equations, Journal of Nonlinear Science, 19 (2009), 133152. doi: 10.1007/s0033200890297. 
[12] 
B. Levant, F. Ramos and E. Titi, On the statistical properties of the 3D incompressible NavierStokesVoigt model, Communications in Mathematical Sciences, 8 (2010), 277293. 
[13] 
P. Kloeden, J. Langa and J. Real, Pullback $V$attractors of the 3dimensional globally modified NavierStokes equations, Commun. Pure Appl. Anal., 6 (2007), 937955. 
[14] 
P. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the threedimensional NavierStokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 14911508. doi: 10.1098/rspa.2007.1831. 
[15] 
P. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163181. doi: 10.1098/rspa.2006.1753. 
[16] 
T. Caraballo, J. Real and P. Kloeden, Unique strong solutions and $V$attractors of a three dimensional system of globally modified NavierStokes equations, Adv. Nonlinear Stud., 6 (2006), 411436. 
[17] 
O. Ladyzhenskaya, "Attractors for Semigroups and Evolutions," LezioniLincee, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. 
[18] 
O. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," New York: Gordon and Breach Science Publisher, 1963. 
[19] 
O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Nauka, 1967. 
[20] 
J. Lions, "Quelques Methodes de Resolution des Problemes aux Limits Nlineaires," Dunod, Paris, 1969. 
[21] 
J. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," SpringVerlag, Berlin, 1972. 
[22] 
Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math. J., 51 (2002), 15411559. doi: 10.1512/iumj.2002.51.2255. 
[23] 
T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science, Series A, Vol. 53, 2005. doi: 10.1142/9789812701152. 
[24] 
T. Ma and S. Wang, "Stability and Bifurcation of Nonlinear Evolutions Equations," Science Press, April, 2007. 
[25] 
T. Ma and S. Wang, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics," AMS Monograph and Mathematical Survey Series, vol. 119, 2005 
[26] 
I. Moise, R. Rosa and X. Wang, Attractors for noncompact semigroups via energy equations, Nonlinearity, 11 (1998), 13691393. doi: 10.1088/09517715/11/5/012. 
[27] 
A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI), 38 (1973), 98116. 
[28] 
V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511533. doi: 10.1007/s0022000412331. 
[29] 
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 14951506. doi: 10.1088/09517715/19/7/001. 
[30] 
J. Robinson, "Infinitedimensional Dynamical Systems," Cambridge University Press, "Texes in Applied Mathematics," Series, 2001. 
[31] 
C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 5181. 
[32] 
R. Temam, "NavierStokes Equations, Theory and Numerical Analysis," 3rd revised edition, North Holland, 2001. 
[33] 
R. Temam, "InfiniteDimensional Systems in Mechanics and Physics," SpringerVerlag, New York, 1997. 
[34] 
G. Yue and C. Zhong, On the convergence of the uniform attractor of 2D NS$\alpha$ model to the uniform attractor of 2D NS system, J. Comput. Appl. Math., 233 (2010), 18791887. doi: 10.1016/j.cam.2009.09.024. 
[35] 
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921934. doi: 10.3934/cpaa.2004.3.921. 
[36] 
C. Zhong, M. Yang and C. Sun, The existence of global attractors for the normtoweak continuous semigroup and application to the nonlinear reactiondiffusion equations, J. Diff. Equations, 223 (2006), 367399. doi: 10.1016/j.jde.2005.06.008. 
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