October  2011, 16(3): 985-1002. doi: 10.3934/dcdsb.2011.16.985

Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China, China

Received  October 2010 Revised  December 2010 Published  June 2011

In this paper we study the long time behavior of the three dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Navier-Stokes-Voight equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to $V'.$ For the nonautonomous case, we only assume that $f(x,t)$ is translation bounded instead of translation compact, where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal projection. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Navier-Stokes-Voight equations and also obtain the existence of the uniform attractor. Finally, we describe the structure of the uniform attractor and its regularity.
Citation: Gaocheng Yue, Chengkui Zhong. Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 985-1002. doi: 10.3934/dcdsb.2011.16.985
References:
[1]

A. Babin and M. Vishik, "Attractors of Evolution Equations," North Holland, Amsterdam, 1992.

[2]

Y. Cao, E. Lunasin and E. Titi, Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823-848.

[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, "Navier-Stokes Equations," The University of Chicago Press, 1988.

[6]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes 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), 50-54.

[9]

V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Non-classical Type," Postdoctoral Thesis, St. Petersburg, 1988.

[10]

V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chinese Annals of Mathematics, Series B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3.

[11]

V. Kalantarov, B. Levant and E. Titi, Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations, Journal of Nonlinear Science, 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7.

[12]

B. Levant, F. Ramos and E. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Communications in Mathematical Sciences, 8 (2010), 277-293.

[13]

P. Kloeden, J. Langa and J. Real, Pullback $V$-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955.

[14]

P. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. 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), 163-181. 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 Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.

[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, "Non-homogeneous Boundary Value Problems and Applications," Spring-Verlag, 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), 1541-1559. 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 non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012.

[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), 98-116.

[28]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.

[29]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.

[30]

J. Robinson, "Infinite-dimensional 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), 51-81.

[32]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," 3rd revised edition, North Holland, 2001.

[33]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Springer-Verlag, 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), 1879-1887. 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), 921-934. doi: 10.3934/cpaa.2004.3.921.

[36]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Diff. Equations, 223 (2006), 367-399. 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 well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823-848.

[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, "Navier-Stokes Equations," The University of Chicago Press, 1988.

[6]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes 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), 50-54.

[9]

V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Non-classical Type," Postdoctoral Thesis, St. Petersburg, 1988.

[10]

V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chinese Annals of Mathematics, Series B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3.

[11]

V. Kalantarov, B. Levant and E. Titi, Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations, Journal of Nonlinear Science, 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7.

[12]

B. Levant, F. Ramos and E. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Communications in Mathematical Sciences, 8 (2010), 277-293.

[13]

P. Kloeden, J. Langa and J. Real, Pullback $V$-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955.

[14]

P. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. 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), 163-181. 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 Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.

[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, "Non-homogeneous Boundary Value Problems and Applications," Spring-Verlag, 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), 1541-1559. 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 non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012.

[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), 98-116.

[28]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.

[29]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.

[30]

J. Robinson, "Infinite-dimensional 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), 51-81.

[32]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," 3rd revised edition, North Holland, 2001.

[33]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Springer-Verlag, 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), 1879-1887. 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), 921-934. doi: 10.3934/cpaa.2004.3.921.

[36]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Diff. Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.

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