doi: 10.3934/dcdsb.2021284
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Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping

College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

*Corresponding author: Xiaona Cui

Received  April 2021 Revised  September 2021 Early access December 2021

Fund Project: Research supported by the Young Backbone Teacher in Henan Province (No. 2018GGJS039), Cultivation Fund of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).

This paper is concerned with the tempered pullback attractors for 3D incompressible Navier-Stokes model with a double time-delays and a damping term. The delays are in the convective term and external force, which originate from the control in engineer and application. Based on the existence of weak and strong solutions for three dimensional hydrodynamical model with subcritical nonlinearity, we proved the existence of minimal family for pullback attractors with respect to tempered universes for the non-autonomous dynamical systems.

Citation: Wei Shi, Xiaona Cui, Xuezhi Li, Xin-Guang Yang. Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021284
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P. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

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H. Liu and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Appl. Math. Lett., 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.  Google Scholar

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[30]

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G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications. Collge de France Seminar, Vol. XI (Paris, 1989-1991), Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 299 (1994), 205-247.  Google Scholar

[32]

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X. YangY. QinY. Lu and T. Ma, Dynamics of 2D incompressible non-autonomous Navier-Stokes equations on Lipschitz-like domains, Appl. Math. Optim., 83 (2021), 2129-2183.  doi: 10.1007/s00245-019-09622-w.  Google Scholar

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X. Yang, W. Shi, A. Miranville and X. Yan, Dynamics and singular limit of the 3D incompressible Navier-Stokes equations with nonlinear damping and oscillating forces, preprint, 2021. Google Scholar

[44]

X. YangR. WangX. Yan and A. Miranville, Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains, Discrete Contin. Dyn. Syst., 41 (2021), 3343-3366.  doi: 10.3934/dcds.2020408.  Google Scholar

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R. Yang and X. Yang, Asymptotic stability of 3D Navier-Stokes equations with damping, Appl. Math. Lett., 116 (2021), 107012.  doi: 10.1016/j.aml.2020.107012.  Google Scholar

[46]

Z. ZhangX. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.  Google Scholar

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Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.  Google Scholar

[3]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[4]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.  Google Scholar

[5]

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

[6]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.   Google Scholar

[7]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[8] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[9]

X. CuiW. ShiX. Li and X. Yang, Pullback dynamics for the 3D incompressible Navier-Stokes equations with damping and delay, Math. Methods Appl. Sci., 44 (2021), 7031-7047.  doi: 10.1002/mma.7239.  Google Scholar

[10] C. FoiasO. ManleyR. Rosa and R. Temam, NavierStokes Equations and Turbulence, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[11]

J. García-LuengoP. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.  Google Scholar

[12]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors in V for non-autonomous 2D-NavierStokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010.  Google Scholar

[13]

M. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.  doi: 10.1016/j.na.2005.05.057.  Google Scholar

[14]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.  Google Scholar

[15]

L. Hoang and G. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations, 22 (2010), 563-616.  doi: 10.1007/s10884-010-9189-7.  Google Scholar

[16]

E. Hopf, Üeber die Anfangswertaufgable für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[17]

Y. JiaX. Zhang and B. Dong, The asymptotic behavior of solutions to three-dimensional NavierStokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747.  doi: 10.1016/j.nonrwa.2010.11.006.  Google Scholar

[18]

Z. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.  doi: 10.1016/j.na.2012.04.014.  Google Scholar

[19]

Z. Jiang and M. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102.  doi: 10.1002/mma.1540.  Google Scholar

[20]

N. Kim and M. Kwak, Global existence for 3D Navier-Stokes equations in a long periodic domain, J. Korean Math. Soc., 49 (2012), 315-324.  doi: 10.4134/JKMS.2012.49.2.315.  Google Scholar

[21]

P. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[22]

J. Leray, Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pures Appl., 13 (1934), 331-418.   Google Scholar

[23]

J. Leray, Essai sur le mouvement dun liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.   Google Scholar

[24]

J. Leray, Etude de diverses equations integrales nonlineaires et de quelques problemes que pose lhydrodynamique, J. Math. Pures Appl., 12 (1933), 1-82.   Google Scholar

[25]

F. LiB. You and Y. Xu, Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4267-4284.  doi: 10.3934/dcdsb.2018137.  Google Scholar

[26]

J. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[27]

H. Liu and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Appl. Math. Lett., 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.  Google Scholar

[28]

L. Paumond, A rigorous link between KP and a Benney-Luke equation (English summary), Differential Integral Equations, 16 (2003), 1039-1064.   Google Scholar

[29]

G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776.  Google Scholar

[30]

G. Raugel and G. Sell, Navier-Stokes equations in thin 3D domains. III. Existence of a global attractor, Turbulence in Fluid Flows, 55 (1993), 137-163.  doi: 10.1007/978-1-4612-4346-5_9.  Google Scholar

[31]

G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications. Collge de France Seminar, Vol. XI (Paris, 1989-1991), Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 299 (1994), 205-247.  Google Scholar

[32]

R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[33]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.  Google Scholar

[34]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[35]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkhäuser/Springer Basel AG, Basel, 2001.  Google Scholar

[36]

X. Song and Y. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252.  doi: 10.3934/dcds.2011.31.239.  Google Scholar

[37]

X. Song and Y. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351.  doi: 10.1016/j.jmaa.2014.08.044.  Google Scholar

[38]

T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018.  doi: 10.3934/dcds.2005.12.997.  Google Scholar

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[40]

S. WangG. Xu and G. Chen, Cauchy problem for the generalized Benney-Luke equation, J. Math. Phys., 48 (2007), 073521, 16pp.  doi: 10.1063/1.2751280.  Google Scholar

[41]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations, Dyn. Syst., 23 (2008), 1-16.  doi: 10.1080/14689360701611821.  Google Scholar

[42]

X. YangY. QinY. Lu and T. Ma, Dynamics of 2D incompressible non-autonomous Navier-Stokes equations on Lipschitz-like domains, Appl. Math. Optim., 83 (2021), 2129-2183.  doi: 10.1007/s00245-019-09622-w.  Google Scholar

[43]

X. Yang, W. Shi, A. Miranville and X. Yan, Dynamics and singular limit of the 3D incompressible Navier-Stokes equations with nonlinear damping and oscillating forces, preprint, 2021. Google Scholar

[44]

X. YangR. WangX. Yan and A. Miranville, Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains, Discrete Contin. Dyn. Syst., 41 (2021), 3343-3366.  doi: 10.3934/dcds.2020408.  Google Scholar

[45]

R. Yang and X. Yang, Asymptotic stability of 3D Navier-Stokes equations with damping, Appl. Math. Lett., 116 (2021), 107012.  doi: 10.1016/j.aml.2020.107012.  Google Scholar

[46]

Z. ZhangX. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.  Google Scholar

[47]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.  Google Scholar

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