March  2021, 10(1): 1-23. doi: 10.3934/eect.2020039

Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces

1. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

2. 

Department of Mathematics, Hung Vuong University, Nong Trang, Viet Tri, Phu Tho, Vietnam

3. 

Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

* Corresponding author: anhctmath@hnue.edu.vn

Dedicated to the memory of Professor Geneviève Raugel

Received  June 2019 Revised  December 2019 Published  March 2020

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.303

We consider a three-dimensional Navier-Stokes-Voigt equations with memory in lacking instantaneous kinematic viscosity, in presence of Ekman type damping and singularly oscillating external forces depending on a positive parameter $ \varepsilon $. Under suitable assumptions on the memory term and on the external forces, we prove the existence and the uniform (w.r.t. $ \varepsilon $) boundedness as well as the convergence as $ \varepsilon $ tends to $ 0 $ of uniform attractors $ \mathcal A ^\varepsilon $ of a family of processes associated to the model.

Citation: Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
References:
[1]

C. T. AnhD. T. P. Thanh and N. D. Toan, Averaging of nonclassical diffusion equations with memory and singularly oscillating forces, Z. Anal. Anwend., 37 (2018), 299-314.  doi: 10.4171/ZAA/1615.  Google Scholar

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three dimensional Navier-Stokes-Voigt equations in some unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.  Google Scholar

[3]

C. T. Anh and P. T. Trang, Decay rate of solutions to the 3D Navier-Stokes-Voigt equations in $H^m$ spaces, Appl. Math. Lett., 61 (2016), 1-7.  doi: 10.1016/j.aml.2016.04.015.  Google Scholar

[4]

C. T. Anh and P. T. Trang, On the regularity and convergence of solutions to the 3D Navier-Stokes-Voigt equations, Comput. Math. Appl., 73 (2017), 601-615.  doi: 10.1016/j.camwa.2016.12.023.  Google Scholar

[5]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

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Y. CaoE. LunasinE. S. Titi and S. Edriss, 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

[8]

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

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[10]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Methods Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[11]

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

[12]

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

[13]

S. GattiA. MiranvilleV. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain Journal of Mathematics, 38 (2008), 1117-1138.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[14]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.  Google Scholar

[15]

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

[16]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Lenigrad. Otdel. Math. Inst. Steklov. (LOMI), 152 (1986), 50-54.  doi: 10.1007/BF01094186.  Google Scholar

[17]

V. K. Kalantarov and E. S. Titi, Global attractor 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

[18]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. Ⅲ, Springer-Verlag, Berlin-Heidelberg, 1973.  Google Scholar

[19]

C. J. Niche, Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum, J. Differential Equations, 260 (2016), 4440-4453.  doi: 10.1016/j.jde.2015.11.014.  Google Scholar

[20]

A. P. 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. Math. Inst. Steklov. (LOMI), 38 (1973), 98-136.   Google Scholar

[21]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[22]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270.  doi: 10.1016/j.jmaa.2010.07.006.  Google Scholar

[23]

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

[24]

Y. QinX. Yang and X. Liu, Averaging of a 3D Navier-Stokes-Voight equation with singularly oscillating forces, Nonlinear Anal. Real World Appl., 13 (2012), 893-904.  doi: 10.1016/j.nonrwa.2011.08.025.  Google Scholar

[25]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, Chelsea Publishing, Providence, RI, 2001. Google Scholar

[26]

X. G. YangL. Li and Y. Lu, Regularity of uniform attractor for 3D non-autonomous Navier-Stokes-Voigt equation, Appl. Math. Comput., 334 (2018), 11-29.  doi: 10.1016/j.amc.2018.03.096.  Google Scholar

[27]

G. Yue and C. K. Zhong, Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 985-1002.  doi: 10.3934/dcdsb.2011.16.985.  Google Scholar

[28]

C. Zhao and H. Zhu, Upper bound of decay rate for solutions to the Navier-Stokes-Voigt equations in $ \mathbb{R}^3$, Appl. Math. Comp., 256 (2015), 183-191.  doi: 10.1016/j.amc.2014.12.131.  Google Scholar

show all references

References:
[1]

C. T. AnhD. T. P. Thanh and N. D. Toan, Averaging of nonclassical diffusion equations with memory and singularly oscillating forces, Z. Anal. Anwend., 37 (2018), 299-314.  doi: 10.4171/ZAA/1615.  Google Scholar

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three dimensional Navier-Stokes-Voigt equations in some unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.  Google Scholar

[3]

C. T. Anh and P. T. Trang, Decay rate of solutions to the 3D Navier-Stokes-Voigt equations in $H^m$ spaces, Appl. Math. Lett., 61 (2016), 1-7.  doi: 10.1016/j.aml.2016.04.015.  Google Scholar

[4]

C. T. Anh and P. T. Trang, On the regularity and convergence of solutions to the 3D Navier-Stokes-Voigt equations, Comput. Math. Appl., 73 (2017), 601-615.  doi: 10.1016/j.camwa.2016.12.023.  Google Scholar

[5]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[7]

Y. CaoE. LunasinE. S. Titi and S. Edriss, 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

[8]

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

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[10]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Methods Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[11]

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

[12]

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

[13]

S. GattiA. MiranvilleV. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain Journal of Mathematics, 38 (2008), 1117-1138.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[14]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.  Google Scholar

[15]

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

[16]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Lenigrad. Otdel. Math. Inst. Steklov. (LOMI), 152 (1986), 50-54.  doi: 10.1007/BF01094186.  Google Scholar

[17]

V. K. Kalantarov and E. S. Titi, Global attractor 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

[18]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. Ⅲ, Springer-Verlag, Berlin-Heidelberg, 1973.  Google Scholar

[19]

C. J. Niche, Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum, J. Differential Equations, 260 (2016), 4440-4453.  doi: 10.1016/j.jde.2015.11.014.  Google Scholar

[20]

A. P. 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. Math. Inst. Steklov. (LOMI), 38 (1973), 98-136.   Google Scholar

[21]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[22]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270.  doi: 10.1016/j.jmaa.2010.07.006.  Google Scholar

[23]

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

[24]

Y. QinX. Yang and X. Liu, Averaging of a 3D Navier-Stokes-Voight equation with singularly oscillating forces, Nonlinear Anal. Real World Appl., 13 (2012), 893-904.  doi: 10.1016/j.nonrwa.2011.08.025.  Google Scholar

[25]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, Chelsea Publishing, Providence, RI, 2001. Google Scholar

[26]

X. G. YangL. Li and Y. Lu, Regularity of uniform attractor for 3D non-autonomous Navier-Stokes-Voigt equation, Appl. Math. Comput., 334 (2018), 11-29.  doi: 10.1016/j.amc.2018.03.096.  Google Scholar

[27]

G. Yue and C. K. Zhong, Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 985-1002.  doi: 10.3934/dcdsb.2011.16.985.  Google Scholar

[28]

C. Zhao and H. Zhu, Upper bound of decay rate for solutions to the Navier-Stokes-Voigt equations in $ \mathbb{R}^3$, Appl. Math. Comp., 256 (2015), 183-191.  doi: 10.1016/j.amc.2014.12.131.  Google Scholar

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