# American Institute of Mathematical Sciences

December  2021, 29(6): 4137-4157. doi: 10.3934/era.2021076

## Pullback dynamics of a 3D modified Navier-Stokes equations with double delays

 1 College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China 2 Department of Mathematics, China University of Mining and Technology, Beijing 100083, China 3 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

* Corresponding author: Lan Huang

Received  May 2021 Revised  July 2021 Published  December 2021 Early access  October 2021

Fund Project: Research partly supported by the NSFC (No. 11501199 and No. 11871212) and the Young Key Teachers Project in Higher Vocational Colleges of Henan Province (No. 2020GZGG109)

This paper is concerned with the tempered pullback dynamics for a 3D modified Navier-Stokes equations with double time-delays, which includes delays on external force and convective terms respectively. Based on the property of monotone operator and some suitable hypotheses on the external forces, the existence and uniqueness of weak solutions can be shown in an appropriate functional Banach space. By using the energy equation technique and weak convergence method to achieve asymptotic compactness for the process, the existence of minimal family of pullback attractors has also been derived.

Citation: Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang. Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29 (6) : 4137-4157. doi: 10.3934/era.2021076
##### References:
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##### References:
 [1] H. Bae, Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.  Google Scholar [2] J. M. Ball, Global attractors for damped semi-linear wave equations, Disc. Cont. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar [3] T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Ann. Math., 189 (2019), 101-144.  doi: 10.4007/annals.2019.189.1.3.  Google Scholar [4] 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 [5] 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 [6] T. Caraballo and J. Real, Asymptotic behaviour of 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 [7] 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 [8] T. Caraballo, J. Real and A. M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.  Google Scholar [9] Y. Chen, X. Yang and M. Si, The long-time dynamics of 3D non-autonomous Navier-Stokes equations with variable viscosity, ScienceAsia, 44 (2018), 18-26.  doi: 10.2306/scienceasia1513-1874.2018.44.018.  Google Scholar [10] B. Dong and W. Jiang, On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows, Sci. China Ser. A, 51 (2008), 925-934.  doi: 10.1007/s11425-007-0196-z.  Google Scholar [11] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar [12] J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Disc. Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.  Google Scholar [13] 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.  Google Scholar [14] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for nonautonomous 2D-Navier-Stokes equations and their tempered behavior, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010.  Google Scholar [15] J. García-Luengo, P. 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 [16] C. Guo, R. Lu, X. Yang and P. Zhang, Dynamics for three dimensional generalized Navier-Stokes equations with delay, Preprint, (2021). Google Scholar [17] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238.  doi: 10.3934/dcdsb.2011.16.225.  Google Scholar [18] X. Han, P. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.  Google Scholar [19] E. Hopf, Üeber die Anfangswertaufgable für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar [20] A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar [21] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar [22] O. A. Ladyzhenskaya, On some nonlinear problems in the theory of continuous media, Am. Math. Soc. Transl., 70 (1968), 73-89.   Google Scholar [23] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York: Gordon and Breach, 1969.  Google Scholar [24] J. Leray, Essai sur les mouvements plans d'un liquide visqueux que limitent des parois, J. Math. Pure Appl., 13 (1934), 331-418.   Google Scholar [25] D. Li, Q. Liu and X. Ju, Uniform decay estimates for solutions of a class of retarded integral inequalities, J. Differential Equations, 271 (2021), 1-38.  doi: 10.1016/j.jde.2020.08.017.  Google Scholar [26] L. Li, X.-G. Yang, X. Li, X. Yan and Y. Lu, Dynamics and stability of the 3D Brinkman-Forchheimer equation with variable delay (Ⅰ), Asymptot. Anal., 113 (2019), 167-194.  doi: 10.3233/ASY-181512.  Google Scholar [27] J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [28] J.-L. Lions and G. Prodi, Une théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.   Google Scholar [29] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996.  Google Scholar [30] 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.  Google Scholar [31] P. Marín-Rubio, A. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Disc. Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.  Google Scholar [32] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Disc. Contin. Dyn. Syst., 21 (2008), 1245-1258.  doi: 10.3934/dcds.2008.21.1245.  Google Scholar [33] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Revised edition, North Holland Publishing Company-Amsterdam, New York, 1979.  Google Scholar [34] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [35] B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differential Equations, (2013), No. 191, 25 pp.  Google Scholar [36] J. Wang, C. Zhao and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Comm. Nonl. Sci. Numer. Simul., 91 (2020), 105459, 14 pp. doi: 10.1016/j.cnsns.2020.105459.  Google Scholar [37] X.-G. Yang, B. Feng, S. Wang, Y. Lu and T. F. Ma, Pullback dynamics of 3D Navier-Stokes equations with nonlinear viscosity, Nonlinear Anal. RWA, 48 (2019), 337-361.  doi: 10.1016/j.nonrwa.2019.01.013.  Google Scholar [38] X.-G. Yang, B. Guo, C. Guo and D. Li, The fractal dimension of pullback attractors for the 2D Navier-Stokes equations with delay, Math. Meth. Appl. Sci., 43 (2020), 9637-9653.  doi: 10.1002/mma.6634.  Google Scholar [39] X.-G. Yang, L. Li, X. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1395-1418.  doi: 10.3934/era.2020074.  Google Scholar [40] X.-G. Yang, R.-N. Wang, X. Yan and A. Miranville, Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domain, Disc. Contin. Dyn. Syst., 41 (2021), 3343-3366.  doi: 10.3934/dcds.2020408.  Google Scholar [41] S. Zheng, Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics, 2004. doi: 10.1201/9780203492222.  Google Scholar
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