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

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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

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

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

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