doi: 10.3934/eect.2020105

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

*Corresponding author: Manil T. Mohan

Received  October 2019 Revised  September 2020 Published  December 2020

The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension.

Citation: Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, doi: 10.3934/eect.2020105
References:
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V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[2]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.  Google Scholar

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A. O. CelebiV. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Applicable Analysis, 88 (2009), 381-392.  doi: 10.1080/00036810902766682.  Google Scholar

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V. V. ChepyzhovM. Conti and V. Pata, Averaging of equations of viscoelasticity with singularly oscillating external forces, Journal de Mathématiques Pures et Appliquées, 108 (2017), 841-868.  doi: 10.1016/j.matpur.2017.05.007.  Google Scholar

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B. CockburnD. A. Jones and E. S. Titi, Determining degrees of freedom for nonlinear dissipative equations, C. R. Acad. Sci. Paris Ser. I Math., 321 (1995), 563-568.   Google Scholar

[7]

P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Commun. Pure Appl. Math., 38 (1985), 1-27.  doi: 10.1002/cpa.3160380102.  Google Scholar

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P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

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C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

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V. DaneseP. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discret. Contin. Dyn. Syst., 35 (2015), 2881-2904.  doi: 10.3934/dcds.2015.35.2881.  Google Scholar

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F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.  Google Scholar

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A. EdenC. Foias and V. Kalantarov, A remark for two constructions of exponential attractors for $\alpha$-contractions, J. Dynam. Differential Equations, 10 (1998), 37-45.  doi: 10.1023/A:1022636328133.  Google Scholar

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M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

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M. Efendiev, Attractors for Degenerate Parabolic Type Equations, American Mathematical Society, Providence, Rhode Island, 2013. doi: 10.1090/surv/192.  Google Scholar

[15]

C. FoiasO. P. ManleyR. Temam and et al., Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.  Google Scholar

[16]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[17]

C. FoiasO. P. ManleyR. Temam and Y. M. Tréve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.  Google Scholar

[18]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.  Google Scholar

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[20]

A. A. Ilyin, On the spectrum of the Stokes operator, Funct. Anal. Appl., 43 (2009), 254-263.  doi: 10.1007/s10688-009-0034-x.  Google Scholar

[21]

A. A. Ilyin, Lower bounds for the spectrum of the Laplace and Stokes operators, Discrete Contin. Dyn. Syst., 28 (2010), 131-146.  doi: 10.3934/dcds.2010.28.131.  Google Scholar

[22]

D. A. Jones and and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 875-887.  doi: 10.1512/iumj.1993.42.42039.  Google Scholar

[23]

D. A. Jones and E. S. Titi, On the number of determining nodes for the 2D Navier-Stokes equations, J. Math. Anal. Appl., 168 (1992), 72-88.  doi: 10.1016/0022-247X(92)90190-O.  Google Scholar

[24]

D. A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D, 60 (1992), 165-174.  doi: 10.1016/0167-2789(92)90233-D.  Google Scholar

[25]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.  Google Scholar

[26]

N. A. KarazeevaA. A. Kotsiolis and A. P. Oskolkov, Dynamical systems generated by initial-boundary value problems for equations of motion of linear viscoelastic fluids, Proc. Steklov Inst. Math., 3 (1991), 73-108.   Google Scholar

[27]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[28]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[29]

B. LevantF. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293.  doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar

[30]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: Global solvability, exponential stability and exact controllability of Galerkin approximations, Evolution Equations and Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[31]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equations perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.  Google Scholar

[32]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202.   Google Scholar

[33]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.  Google Scholar

[34]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1990), 2699-2723.  doi: 10.1007/BF01102639.  Google Scholar

[35]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[36]

R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[37]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[38]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^nd$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[40]

M. C. Zelati and C. G. Gal, Singular limits of Voigt models in fluid dynamics, J. Math. Fluid Mech., 17 (2015), 233-259.  doi: 10.1007/s00021-015-0201-1.  Google Scholar

[41]

V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

show all references

References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[2]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.  Google Scholar

[3]

A. O. CelebiV. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Applicable Analysis, 88 (2009), 381-392.  doi: 10.1080/00036810902766682.  Google Scholar

[4]

V. V. ChepyzhovM. Conti and V. Pata, Averaging of equations of viscoelasticity with singularly oscillating external forces, Journal de Mathématiques Pures et Appliquées, 108 (2017), 841-868.  doi: 10.1016/j.matpur.2017.05.007.  Google Scholar

[5]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar

[6]

B. CockburnD. A. Jones and E. S. Titi, Determining degrees of freedom for nonlinear dissipative equations, C. R. Acad. Sci. Paris Ser. I Math., 321 (1995), 563-568.   Google Scholar

[7]

P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Commun. Pure Appl. Math., 38 (1985), 1-27.  doi: 10.1002/cpa.3160380102.  Google Scholar

[8]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[10]

V. DaneseP. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discret. Contin. Dyn. Syst., 35 (2015), 2881-2904.  doi: 10.3934/dcds.2015.35.2881.  Google Scholar

[11]

F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.  Google Scholar

[12]

A. EdenC. Foias and V. Kalantarov, A remark for two constructions of exponential attractors for $\alpha$-contractions, J. Dynam. Differential Equations, 10 (1998), 37-45.  doi: 10.1023/A:1022636328133.  Google Scholar

[13]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $ \mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[14]

M. Efendiev, Attractors for Degenerate Parabolic Type Equations, American Mathematical Society, Providence, Rhode Island, 2013. doi: 10.1090/surv/192.  Google Scholar

[15]

C. FoiasO. P. ManleyR. Temam and et al., Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.  Google Scholar

[16]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[17]

C. FoiasO. P. ManleyR. Temam and Y. M. Tréve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.  Google Scholar

[18]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.  Google Scholar

[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[20]

A. A. Ilyin, On the spectrum of the Stokes operator, Funct. Anal. Appl., 43 (2009), 254-263.  doi: 10.1007/s10688-009-0034-x.  Google Scholar

[21]

A. A. Ilyin, Lower bounds for the spectrum of the Laplace and Stokes operators, Discrete Contin. Dyn. Syst., 28 (2010), 131-146.  doi: 10.3934/dcds.2010.28.131.  Google Scholar

[22]

D. A. Jones and and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 875-887.  doi: 10.1512/iumj.1993.42.42039.  Google Scholar

[23]

D. A. Jones and E. S. Titi, On the number of determining nodes for the 2D Navier-Stokes equations, J. Math. Anal. Appl., 168 (1992), 72-88.  doi: 10.1016/0022-247X(92)90190-O.  Google Scholar

[24]

D. A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D, 60 (1992), 165-174.  doi: 10.1016/0167-2789(92)90233-D.  Google Scholar

[25]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.  Google Scholar

[26]

N. A. KarazeevaA. A. Kotsiolis and A. P. Oskolkov, Dynamical systems generated by initial-boundary value problems for equations of motion of linear viscoelastic fluids, Proc. Steklov Inst. Math., 3 (1991), 73-108.   Google Scholar

[27]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[28]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[29]

B. LevantF. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293.  doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar

[30]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: Global solvability, exponential stability and exact controllability of Galerkin approximations, Evolution Equations and Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[31]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equations perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.  Google Scholar

[32]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202.   Google Scholar

[33]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.  Google Scholar

[34]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1990), 2699-2723.  doi: 10.1007/BF01102639.  Google Scholar

[35]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[36]

R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[37]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[38]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^nd$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[40]

M. C. Zelati and C. G. Gal, Singular limits of Voigt models in fluid dynamics, J. Math. Fluid Mech., 17 (2015), 233-259.  doi: 10.1007/s00021-015-0201-1.  Google Scholar

[41]

V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

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