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March  2011, 31(1): 239-252. doi: 10.3934/dcds.2011.31.239

Attractors for the three-dimensional incompressible Navier-Stokes equations with damping

Xue-Li Song 1, and  Yan-Ren Hou 1,

1. 

College of science, Xi’an Jiaotong University, Xi’an, 710049, China, China

Received  March 2010 Revised  October 2010 Published  June 2011

In this paper, we show that the strong solution of the three-dimensional Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u\ (\alpha>0, \frac{7}{2}\leq \beta\leq 5)$ has global attractors in $V$ and $H^2(\Omega)$ when initial data $u_0\in V$, where $\Omega\subset \mathbb{R}^3$ is bounded.
Keywords: Global attractor, Damping, Weak solution, Navier-Stokes equation, Strong solution..
Mathematics Subject Classification: Primary: 35B40, 35Q30; Secondary: 37L3.
Citation: Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809. doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[3]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Diff. Eqns., 231 (2006), 714-754. doi: 10.1016/j.jde.2006.08.021.  Google Scholar

[4]

N. J. Cutland, Global attractors for small samples and germs of 3D Navier-Stokes equations, Nonlinear Anal., 62 (2005), 265-281. doi: 10.1016/j.na.2005.02.114.  Google Scholar

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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, Cambridge, 2001.  Google Scholar

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G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613.  Google Scholar

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R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809. doi: 10.1016/j.jmaa.2008.01.041.  Google Scholar

[3]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Diff. Eqns., 231 (2006), 714-754. doi: 10.1016/j.jde.2006.08.021.  Google Scholar

[4]

N. J. Cutland, Global attractors for small samples and germs of 3D Navier-Stokes equations, Nonlinear Anal., 62 (2005), 265-281. doi: 10.1016/j.na.2005.02.114.  Google Scholar

[5]

A. V. Kapustyan and J. Valero, Weak and srong attractors for the 3D Navier-Stokes system, J. Diff. Eqns., 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008.  Google Scholar

[6]

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, Cambridge, 2001.  Google Scholar

[7]

R. Rosa, The global attractors for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[8]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613.  Google Scholar

[9]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[10]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

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