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doi: 10.3934/eect.2020099

Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay

Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam

Received  March 2019 Revised  September 2020 Published  October 2020

We consider the 3D Navier-Stokes-Voigt equations in a bounded domain with unbounded variable delay. We study the stability of stationary solutions by the classical direct method, and by an appropriate Lyapunov functional. We also give a sufficient condition of parameters for the polynomial stability of the stationary solution in a special case of unbounded variable delay. Finally, when the condition for polynomial stability is not satisfied, we stabilize the stationary by using the finite Fourier modes and by internal feedback control with a support large enough.

Citation: Vu Manh Toi. Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay. Evolution Equations & Control Theory, doi: 10.3934/eect.2020099
References:
[1]

C. T. Anh and D. T. P. Thanh, Existence and long-time behavior of solutions to Navier-Stokes-Voigt equations with infinite delay, Bull. Korean Math. Soc., 55 (2018), 379-403.  doi: 10.4134/BKMS.b170044.  Google Scholar

[2]

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

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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

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J. A. D. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Proc. 10'th Coll. Qualitative Theory of Diff. Equ., Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-32.  doi: 10.14232/ejqtde.2016.8.2.  Google Scholar

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L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130.  doi: 10.1016/j.na.2011.08.011.  Google Scholar

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C. Cao, D. D. Holm and E. S. Titi, On the Clark-$\alpha$ model of turbulence: Global regularity and long-time dynamics, J. Turbul., 6 (2005), 11 pp. doi: 10.1080/14685240500183756.  Google Scholar

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T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359.  doi: 10.4310/DPDE.2014.v11.n4.a3.  Google Scholar

[9]

T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101.  doi: 10.3934/dcdss.2015.8.1079.  Google Scholar

[10]

T. CaraballoA. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin. Dyn. Syst., 15 (2006), 559-578.  doi: 10.3934/dcds.2006.15.559.  Google Scholar

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T. CaraballoA. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423.  doi: 10.1016/j.jmaa.2007.08.011.  Google Scholar

[12]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

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T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar

[14]

T. CaraballoJ. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar

[15]

M. Coti 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

[16]

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

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J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[18]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[19]

M. J. HolstM. A. Ebrahimi and E. Lunasin, The Navier-Stokes-Voight model for image inpainting, IMA J. Appl. Math., 78 (2013), 869-894.  doi: 10.1093/imamat/hxr069.  Google Scholar

[20]

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

[21]

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

[22]

T. Kato, Asymptotic behavior of solutions of the functional differential equation $y'(x) = ay(\lambda x)+by(x)$, in Delay and Functional Differential Equations and Their Applications, Proc. Conf., Park City, Utah, Academic Press, New York, (1972), 197–217.  Google Scholar

[23]

T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.  doi: 10.1090/S0002-9904-1971-12805-7.  Google Scholar

[24]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 065504, 28 pp. doi: 10.1063/1.2360145.  Google Scholar

[25]

L. LiuT. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008.  Google Scholar

[26]

E. LunasinM. Holst and G. Tsogtgerel, Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567.  doi: 10.1007/s00332-010-9066-x.  Google Scholar

[27]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.  Google Scholar

[28]

P. Marín-RubioJ. Real and A. M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.  doi: 10.1515/ans-2011-0409.  Google Scholar

[29]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.  Google Scholar

[30]

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. Naučn. Sem. Leningrad. Otdel. Math. Inst. Steklov., (LOMI), 38 (173), 98–136. doi: 10.1007/BF01084613.  Google Scholar

[31]

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

[32]

G. Yue and C. 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

show all references

References:
[1]

C. T. Anh and D. T. P. Thanh, Existence and long-time behavior of solutions to Navier-Stokes-Voigt equations with infinite delay, Bull. Korean Math. Soc., 55 (2018), 379-403.  doi: 10.4134/BKMS.b170044.  Google Scholar

[2]

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

[3]

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

[4]

J. A. D. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Proc. 10'th Coll. Qualitative Theory of Diff. Equ., Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-32.  doi: 10.14232/ejqtde.2016.8.2.  Google Scholar

[5]

V. Barbu and C. Lefter, Internal stabilizability of the Navier-Stokes equations. Optimization and control of distributed systems, Systems Control Lett., 48 (2003), 161-167.  doi: 10.1016/S0167-6911(02)00261-X.  Google Scholar

[6]

L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130.  doi: 10.1016/j.na.2011.08.011.  Google Scholar

[7]

C. Cao, D. D. Holm and E. S. Titi, On the Clark-$\alpha$ model of turbulence: Global regularity and long-time dynamics, J. Turbul., 6 (2005), 11 pp. doi: 10.1080/14685240500183756.  Google Scholar

[8]

T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359.  doi: 10.4310/DPDE.2014.v11.n4.a3.  Google Scholar

[9]

T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101.  doi: 10.3934/dcdss.2015.8.1079.  Google Scholar

[10]

T. CaraballoA. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin. Dyn. Syst., 15 (2006), 559-578.  doi: 10.3934/dcds.2006.15.559.  Google Scholar

[11]

T. CaraballoA. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423.  doi: 10.1016/j.jmaa.2007.08.011.  Google Scholar

[12]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[13]

T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar

[14]

T. CaraballoJ. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar

[15]

M. Coti 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

[16]

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

[17]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[18]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[19]

M. J. HolstM. A. Ebrahimi and E. Lunasin, The Navier-Stokes-Voight model for image inpainting, IMA J. Appl. Math., 78 (2013), 869-894.  doi: 10.1093/imamat/hxr069.  Google Scholar

[20]

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

[21]

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

[22]

T. Kato, Asymptotic behavior of solutions of the functional differential equation $y'(x) = ay(\lambda x)+by(x)$, in Delay and Functional Differential Equations and Their Applications, Proc. Conf., Park City, Utah, Academic Press, New York, (1972), 197–217.  Google Scholar

[23]

T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.  doi: 10.1090/S0002-9904-1971-12805-7.  Google Scholar

[24]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 065504, 28 pp. doi: 10.1063/1.2360145.  Google Scholar

[25]

L. LiuT. Caraballo and P. Marín-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008.  Google Scholar

[26]

E. LunasinM. Holst and G. Tsogtgerel, Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567.  doi: 10.1007/s00332-010-9066-x.  Google Scholar

[27]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.  Google Scholar

[28]

P. Marín-RubioJ. Real and A. M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.  doi: 10.1515/ans-2011-0409.  Google Scholar

[29]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.  Google Scholar

[30]

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. Naučn. Sem. Leningrad. Otdel. Math. Inst. Steklov., (LOMI), 38 (173), 98–136. doi: 10.1007/BF01084613.  Google Scholar

[31]

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

[32]

G. Yue and C. 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

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