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 and Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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