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

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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

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

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

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