July  2021, 41(7): 3343-3366. doi: 10.3934/dcds.2020408

Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains

1. 

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

2. 

College of Mathematics and Science, Shanghai Normal University, Shanghai, China

3. 

Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, China

4. 

Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348, Site du Futuroscope - Téléport 2

5. 

11 Boulevard Marie et Pierre Curie, Bâtiment H3, TSA 61125, 86073 Poitiers Cedex 9, France

* Corresponding author: Alain Miranville

Received  June 2020 Published  July 2021 Early access  December 2020

Fund Project: Research was partly supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039)

This paper is concerned with the tempered pullback dynamics of the 2D Navier-Stokes equations with sublinear time delay operators subject to non-homogeneous boundary conditions in Lipschitz-like domains. By virtue of the estimates of background flow in Lipschitz-like domain and a new retarded Gronwall inequality, we establish the existence of pullback attractors in a general setting involving tempered universes.

Citation: Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
References:
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J. M. Ball, Global attractors for damped semiliear wave equations, Disc. Cont. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

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E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.  doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar

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[15]

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

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J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Comm. Pure Appl. Anal., 14 (2015), 1603-1621.  doi: 10.3934/cpaa.2015.14.1603.  Google Scholar

[17]

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

[18]

J. K. Hale, History of Delay Equations, Conference Proceedings of Delay Differential Equations and Applications, NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, 2006, 1–28. doi: 10.1007/1-4020-3647-7_1.  Google Scholar

[19]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

O. A. Ladyzhenskaya, The Mathematical Theory if Viscous Incompressible Flow, NGordon and Breach Science Publishers, New York-London 1963.  Google Scholar

[21]

O. Ladyzhenskaya, Attractors for Semigroup and Evolution Equations, Cambridge Uni. Press, Cambridge, 1991. Springer, second editon, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[22]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[23]

D. LiQ. 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

[24]

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

[25]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[26]

P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delay in continuous and sub-linear operators, Disc. Cont. Dyn. Syst., 26 (2010), 989-1006.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[27]

A. Miranville and X. Wang, Upper bounded on the dimension of the attractor for non-homogeneous Navier-Stokes equations, Disc. Cont. Dyn. Syst., 2 (1996), 95-110.  doi: 10.3934/dcds.1996.2.95.  Google Scholar

[28]

A. Miranville and X. Wang, Attractors for non-autonomous non-homogenerous Navier-Stokes equations, Nonlinearity, 10 (1997), 1047-1061.  doi: 10.1088/0951-7715/10/5/003.  Google Scholar

[29]

I. MoiseR. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

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J. C. Robinson, Attractors and finite-dimensional behaviour in the 2d Navier-Stokes equations, ISRN Math Anal., 203 (2013), 291823. doi: 10.1155/2013/291823.  Google Scholar

[32]

Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc., 123 (1995), 801-811.  doi: 10.1090/S0002-9939-1995-1223521-9.  Google Scholar

[33]

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

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[35]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[36]

Y. Wang, X-G. Yang and X. Yan, Dynamics of 2D Navier-Stokes equations with Rayleigh's friction and distributed delay, Electronic J. Differential Equations, 2019 (2019), Paper No. 80, 18 pp.  Google Scholar

[37]

X.-G. YangB. FengT. Maier de Souza and T. Wang, Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equations in Lipschitz domain, Disc. Cont. Dyn. Syst. B, 24 (2019), 363-386.  doi: 10.3934/dcdsb.2018084.  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, Y. Qin, Y. Lu and T. F. Ma, Dynamics of 2D incompressible non-autonomous Navier-Stokes equations on Lipschitz-like domains, Appl. Math. & Optimization, 2019, 1–55. doi: 10.1007/s00245-019-09622-w.  Google Scholar

[40]

X.-G. Yang and S. Wang, Well-posedness for the 2D non-autonomous incompressible fluid flow in Lipschitz-like domain, J. Partial Differential Equations, 32 (2019), 77-92.  doi: 10.4208/jpde.v32.n1.6.  Google Scholar

show all references

References:
[1]

J. M. Ball, Global attractors for damped semiliear wave equations, Disc. Cont. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[2]

V. Barbu and S. S. Sritharan, Navier-Stokes equations with hereditary viscosity, Z. Angew Math. Phys., 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.  Google Scholar

[3]

R. M. BrownP. A. Perry and Z. Shen, On the dimension of the attractor of the non-homogeneous Navier-Stokes equations in non-smooth domains, Inidian University Math. J., 49 (2000), 81-112.  doi: 10.1512/iumj.2000.49.1603.  Google Scholar

[4]

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

[5]

T. Caraballo and G. Kiss, Attractors for differential equations with multiple variable delays, Disc. Cont. Dyn. Syst., 33 (2013), 1365-1374.  doi: 10.3934/dcds.2013.33.1365.  Google Scholar

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438.  doi: 10.1006/jmaa.2000.7464.  Google Scholar

[7]

T. CaraballoP. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.  Google Scholar

[8]

T. CaraballoP. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342.  doi: 10.1016/j.jde.2007.05.015.  Google Scholar

[9]

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

[10]

T. Caraballo and J. Real, Asymptotic behavior for 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

[11]

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

[12]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York–Heidelberg–Dordrecht–London, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[13]

E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.  doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar

[14] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[15]

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

[16]

J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Comm. Pure Appl. Anal., 14 (2015), 1603-1621.  doi: 10.3934/cpaa.2015.14.1603.  Google Scholar

[17]

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

[18]

J. K. Hale, History of Delay Equations, Conference Proceedings of Delay Differential Equations and Applications, NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, 2006, 1–28. doi: 10.1007/1-4020-3647-7_1.  Google Scholar

[19]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

O. A. Ladyzhenskaya, The Mathematical Theory if Viscous Incompressible Flow, NGordon and Breach Science Publishers, New York-London 1963.  Google Scholar

[21]

O. Ladyzhenskaya, Attractors for Semigroup and Evolution Equations, Cambridge Uni. Press, Cambridge, 1991. Springer, second editon, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[22]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[23]

D. LiQ. 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

[24]

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

[25]

P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.  doi: 10.1016/j.na.2006.09.035.  Google Scholar

[26]

P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delay in continuous and sub-linear operators, Disc. Cont. Dyn. Syst., 26 (2010), 989-1006.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[27]

A. Miranville and X. Wang, Upper bounded on the dimension of the attractor for non-homogeneous Navier-Stokes equations, Disc. Cont. Dyn. Syst., 2 (1996), 95-110.  doi: 10.3934/dcds.1996.2.95.  Google Scholar

[28]

A. Miranville and X. Wang, Attractors for non-autonomous non-homogenerous Navier-Stokes equations, Nonlinearity, 10 (1997), 1047-1061.  doi: 10.1088/0951-7715/10/5/003.  Google Scholar

[29]

I. MoiseR. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar

[30] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[31]

J. C. Robinson, Attractors and finite-dimensional behaviour in the 2d Navier-Stokes equations, ISRN Math Anal., 203 (2013), 291823. doi: 10.1155/2013/291823.  Google Scholar

[32]

Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc., 123 (1995), 801-811.  doi: 10.1090/S0002-9939-1995-1223521-9.  Google Scholar

[33]

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

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[35]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[36]

Y. Wang, X-G. Yang and X. Yan, Dynamics of 2D Navier-Stokes equations with Rayleigh's friction and distributed delay, Electronic J. Differential Equations, 2019 (2019), Paper No. 80, 18 pp.  Google Scholar

[37]

X.-G. YangB. FengT. Maier de Souza and T. Wang, Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equations in Lipschitz domain, Disc. Cont. Dyn. Syst. B, 24 (2019), 363-386.  doi: 10.3934/dcdsb.2018084.  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, Y. Qin, Y. Lu and T. F. Ma, Dynamics of 2D incompressible non-autonomous Navier-Stokes equations on Lipschitz-like domains, Appl. Math. & Optimization, 2019, 1–55. doi: 10.1007/s00245-019-09622-w.  Google Scholar

[40]

X.-G. Yang and S. Wang, Well-posedness for the 2D non-autonomous incompressible fluid flow in Lipschitz-like domain, J. Partial Differential Equations, 32 (2019), 77-92.  doi: 10.4208/jpde.v32.n1.6.  Google Scholar

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