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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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