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 and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021284
References:
[1]

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

[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.

[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.

[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.

[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.

[6]

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

[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.

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

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

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

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

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