American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 225-238. doi: 10.3934/dcdsb.2011.16.225

On a class of three dimensional Navier-Stokes equations with bounded delay

Sandro M. Guzzo 1, and  Gabriela Planas 2,

1. 

Universidade Estadual do Oeste do Paraná - UNIOESTE, Colegiado do curso de Matemática, Rua Universitária, 2069. Cx.P. 711, 85819-110 Cascavel, PR, Brazil
2. 

Departamento de Matemática, IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859 Campinas, SP, Brazil

Received  June 2010 Revised  September 2010 Published  April 2011

In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence of weak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitable assumptions, we show the exponential stability of stationary solutions.
Keywords: bounded delay, Navier-Stokes equations, asymptotic behavior..
Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 35B4.
Citation: Sandro M. Guzzo, Gabriela Planas. On a class of three dimensional Navier-Stokes equations with bounded delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 225-238. doi: 10.3934/dcdsb.2011.16.225
References:
[1]

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

[2]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, Proc. R. Soc. Lond. A, 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.  Google Scholar

[3]

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

[4]

M. J. 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

[5]

W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B., 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47.  Google Scholar

[6]

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

[7]

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

[8]

G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245.  Google Scholar

[9]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge texts in applied mathematics, Cambridge University Press, 2001.  Google Scholar

[10]

Y. Tang and M. Wan, A remark on exponential stability of time-delayed Burgers equation, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 219-225. doi: 10.3934/dcdsb.2009.12.219.  Google Scholar

[11]

T. Taniguchi, The exponential 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

[12]

R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Studies in Mathematics and its applications. Volume 2, The Netherlands, 1984.  Google Scholar

show all references

References:
[1]

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

[2]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, Proc. R. Soc. Lond. A, 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.  Google Scholar

[3]

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

[4]

M. J. 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

[5]

W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B., 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47.  Google Scholar

[6]

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

[7]

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

[8]

G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245.  Google Scholar

[9]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge texts in applied mathematics, Cambridge University Press, 2001.  Google Scholar

[10]

Y. Tang and M. Wan, A remark on exponential stability of time-delayed Burgers equation, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 219-225. doi: 10.3934/dcdsb.2009.12.219.  Google Scholar

[11]

T. Taniguchi, The exponential 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

[12]

R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Studies in Mathematics and its applications. Volume 2, The Netherlands, 1984.  Google Scholar

[1]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[2]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[3]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[4]

Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
[5]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[6]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[7]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[8]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
[9]

Oleg Imanuvilov. On the asymptotic properties for stationary solutions to the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2301-2340. doi: 10.3934/dcds.2020366
[10]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[11]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[12]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[13]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[14]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[15]

Claude W. Bardos, Trinh T. Nguyen, Toan T. Nguyen, Edriss S. Titi. The inviscid limit for the 2D Navier-Stokes equations in bounded domains. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2022004
[16]

Hakima Bessaih, María J. Garrido-Atienza. Longtime behavior for 3D Navier-Stokes equations with constant delays. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1931-1948. doi: 10.3934/cpaa.2020085
[17]

Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2495-2532. doi: 10.3934/dcdsb.2020020
[18]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
[19]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[20]

Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy