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September  2020, 25(9): 3393-3436. doi: 10.3934/dcdsb.2020067

Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

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

*xCorresponding author: Manil T. Mohan

Received  June 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the $ m $-accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

Citation: Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3393-3436. doi: 10.3934/dcdsb.2020067
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.

[2]

C. T. Anh and P. T. Trang, On the 3D Kelvin-Voigt-Brinkman-Forchheimer equations in some unbounded domains, Nonlinear Anal., 89 (2013), 36-54.  doi: 10.1016/j.na.2013.04.014.

[3]

C. T. Anh and N. V. Thanh, Asymptotic behavior of the stochastic Kelvin-Voigt-Brinkman-Forchheimer equations, Stoch. Anal. Appl., 34 (2016), 441-455.  doi: 10.1080/07362994.2016.1149775.

[4]

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

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[7]

F. E. Browder, Remarks on nonlinear functional equations, Proc. Natl. Acad. Sci. U.S.A., 51 (1964), 985-989.  doi: 10.1073/pnas.51.6.985.

[8]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[9]

A. O. ÇelebiV. K. Kalantarovb and M. Polata, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[10]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4684-0364-0.

[11]

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

[12]

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.

[13] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematical Series, University of Chicago Press, Chicago, IL, 1989. 
[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Application Mathematics, 37, John Wiley & Sons, Ltd., Chichester, 1994.

[15]

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.

[16]

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.

[17]

M. Efendiev, Attractors for Degenerate Parabolic Type Equations, Mathematical Surveys and Monographs, 192, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/192.

[18] C. FoiasO. ManleyR. Rosa and and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2008.  doi: 10.1017/CBO9780511546754.
[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[20]

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

[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.

[22]

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.

[23]

V. K. KalantarovB. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stoke-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[24]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[25]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[27]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.  doi: 10.1215/s0012-7094-62-02933-2.

[28]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Commun. Stoch. Anal., 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.

[29]

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, 2019. doi: 10.3934/eect.2020007.

[30]

M. T. Mohan, Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids, submitted.

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[32]

S. S. Sritharan, Invariant Manifold Theory of Hydrodynamic Transition, Pitman Research Notes in Mathematics Series, 241, Longman Scientific & Technical, Harlow, 1990.

[33]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

[34]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[36]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Methods Appl. Sci., 31 (2008), 1479-1495.  doi: 10.1002/mma.985.

[37]

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.

[38]

V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci. (N.Y.), 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.

[2]

C. T. Anh and P. T. Trang, On the 3D Kelvin-Voigt-Brinkman-Forchheimer equations in some unbounded domains, Nonlinear Anal., 89 (2013), 36-54.  doi: 10.1016/j.na.2013.04.014.

[3]

C. T. Anh and N. V. Thanh, Asymptotic behavior of the stochastic Kelvin-Voigt-Brinkman-Forchheimer equations, Stoch. Anal. Appl., 34 (2016), 441-455.  doi: 10.1080/07362994.2016.1149775.

[4]

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

[5]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[7]

F. E. Browder, Remarks on nonlinear functional equations, Proc. Natl. Acad. Sci. U.S.A., 51 (1964), 985-989.  doi: 10.1073/pnas.51.6.985.

[8]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[9]

A. O. ÇelebiV. K. Kalantarovb and M. Polata, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[10]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4684-0364-0.

[11]

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

[12]

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.

[13] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematical Series, University of Chicago Press, Chicago, IL, 1989. 
[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Application Mathematics, 37, John Wiley & Sons, Ltd., Chichester, 1994.

[15]

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.

[16]

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.

[17]

M. Efendiev, Attractors for Degenerate Parabolic Type Equations, Mathematical Surveys and Monographs, 192, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/192.

[18] C. FoiasO. ManleyR. Rosa and and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2008.  doi: 10.1017/CBO9780511546754.
[19]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[20]

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

[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.

[22]

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.

[23]

V. K. KalantarovB. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stoke-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[24]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[25]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[27]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.  doi: 10.1215/s0012-7094-62-02933-2.

[28]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Commun. Stoch. Anal., 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.

[29]

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, 2019. doi: 10.3934/eect.2020007.

[30]

M. T. Mohan, Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids, submitted.

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[32]

S. S. Sritharan, Invariant Manifold Theory of Hydrodynamic Transition, Pitman Research Notes in Mathematics Series, 241, Longman Scientific & Technical, Harlow, 1990.

[33]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

[34]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[36]

B. Wang and S. Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Math. Methods Appl. Sci., 31 (2008), 1479-1495.  doi: 10.1002/mma.985.

[37]

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.

[38]

V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci. (N.Y.), 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.

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