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 & 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, (1992).   Google Scholar

[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.   Google Scholar

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V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics,", Amer. Math. Soc. Colloq. Publ., 49 (2002).   Google Scholar

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J. Cholewa and T. Dlotko, "Global Attractors in Abstract Parabolic Problems,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

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C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

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V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Non-classical Type,", Postdoctoral Thesis, (1988).   Google Scholar

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

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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.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

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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.   Google Scholar

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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.   Google Scholar

[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.  doi: 10.1098/rspa.2007.1831.  Google Scholar

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O. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", New York: Gordon and Breach Science Publisher, (1963).   Google Scholar

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O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Nauka, (1967).   Google Scholar

[20]

J. Lions, "Quelques Methodes de Resolution des Problemes aux Limits Nlineaires,", Dunod, (1969).   Google Scholar

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J. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", Spring-Verlag, (1972).   Google Scholar

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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.  doi: 10.1512/iumj.2002.51.2255.  Google Scholar

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T. Ma and S. Wang, "Bifurcation Theory and Applications,", World Scientific Series on Nonlinear Science, (2005).  doi: 10.1142/9789812701152.  Google Scholar

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T. Ma and S. Wang, "Stability and Bifurcation of Nonlinear Evolutions Equations,", Science Press, (2007).   Google Scholar

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T. Ma and S. Wang, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,", AMS Monograph and Mathematical Survey Series, 119 (2005).   Google Scholar

[26]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

[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., 38 (1973), 98.   Google Scholar

[28]

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

[29]

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

[30]

J. Robinson, "Infinite-dimensional Dynamical Systems,", Cambridge University Press, (2001).   Google Scholar

[31]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations,, Asymptotic Analysis, 59 (2008), 51.   Google Scholar

[32]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 3rd revised edition, (2001).   Google Scholar

[33]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics,", Springer-Verlag, (1997).   Google Scholar

[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.  doi: 10.1016/j.cam.2009.09.024.  Google Scholar

[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.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

show all references

References:
[1]

A. Babin and M. Vishik, "Attractors of Evolution Equations,", North Holland, (1992).   Google Scholar

[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.   Google Scholar

[3]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics,", Amer. Math. Soc. Colloq. Publ., 49 (2002).   Google Scholar

[4]

J. Cholewa and T. Dlotko, "Global Attractors in Abstract Parabolic Problems,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511526404.  Google Scholar

[5]

P. Constantin and C. Foias, "Navier-Stokes Equations,", The University of Chicago Press, (1988).   Google Scholar

[6]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

[7]

J. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[8]

V. Kalantarov, Attractors for some nonlinear problems of mathematical physics,, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50.   Google Scholar

[9]

V. Kalantarov, "Global Behavior of Solutions of Nonlinear Equation of Mathematical Physics of Classical and Non-classical Type,", Postdoctoral Thesis, (1988).   Google Scholar

[10]

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

[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.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1098/rspa.2007.1831.  Google Scholar

[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.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[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.   Google Scholar

[17]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolutions,", LezioniLincee, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar

[18]

O. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", New York: Gordon and Breach Science Publisher, (1963).   Google Scholar

[19]

O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Nauka, (1967).   Google Scholar

[20]

J. Lions, "Quelques Methodes de Resolution des Problemes aux Limits Nlineaires,", Dunod, (1969).   Google Scholar

[21]

J. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", Spring-Verlag, (1972).   Google Scholar

[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.  doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[23]

T. Ma and S. Wang, "Bifurcation Theory and Applications,", World Scientific Series on Nonlinear Science, (2005).  doi: 10.1142/9789812701152.  Google Scholar

[24]

T. Ma and S. Wang, "Stability and Bifurcation of Nonlinear Evolutions Equations,", Science Press, (2007).   Google Scholar

[25]

T. Ma and S. Wang, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,", AMS Monograph and Mathematical Survey Series, 119 (2005).   Google Scholar

[26]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

[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., 38 (1973), 98.   Google Scholar

[28]

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

[29]

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

[30]

J. Robinson, "Infinite-dimensional Dynamical Systems,", Cambridge University Press, (2001).   Google Scholar

[31]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations,, Asymptotic Analysis, 59 (2008), 51.   Google Scholar

[32]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 3rd revised edition, (2001).   Google Scholar

[33]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics,", Springer-Verlag, (1997).   Google Scholar

[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.  doi: 10.1016/j.cam.2009.09.024.  Google Scholar

[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.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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