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

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December 2021, 10(4): 1007-1023. 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 December 2021 Early access 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.

Keywords: Navier-Stokes-Voigt equations, unbounded variable delay, stability, stabilization, Fourier modes, internal feedback control, thin domain.

Mathematics Subject Classification: 35Q35, 35Q30, 35B35, 35B40, 35A01, 93D15.

Citation: Vu Manh Toi. Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay. Evolution Equations and Control Theory, 2021, 10 (4) : 1007-1023. doi: 10.3934/eect.2020099

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	T. Caraballo, A. 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.
[11]	T. Caraballo, A. 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.
[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.
[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.
[14]	T. Caraballo, J. 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.
[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.
[16]	J. García-Luengo, P. 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.
[17]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[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.
[19]	M. J. Holst, M. 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.
[20]	V. K. Kalantarov, B. 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.
[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.
[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.
[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.
[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.
[25]	L. Liu, T. 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.
[26]	E. Lunasin, M. 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.
[27]	P. Marín-Rubio, A. 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.
[28]	P. Marín-Rubio, J. 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.
[29]	P. Marín-Rubio, A. 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.
[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.
[31]	Y. Qin, X. 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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	T. Caraballo, A. 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.
[11]	T. Caraballo, A. 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.
[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.
[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.
[14]	T. Caraballo, J. 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.
[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.
[16]	J. García-Luengo, P. 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.
[17]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[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.
[19]	M. J. Holst, M. 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.
[20]	V. K. Kalantarov, B. 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.
[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.
[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.
[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.
[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.
[25]	L. Liu, T. 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.
[26]	E. Lunasin, M. 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.
[27]	P. Marín-Rubio, A. 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.
[28]	P. Marín-Rubio, J. 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.
[29]	P. Marín-Rubio, A. 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.
[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.
[31]	Y. Qin, X. 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.
[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.

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