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

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

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.
Citation: Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 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 (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. 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. 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. 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. 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, (2001).

[7]

R. Rosa, The global attractors for the 2D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71. 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. doi: 10.1007/BF02218613.

[9]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition,, Applied Mathematical Sciences, 68 (1997).

[10]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition,, Studies in Mathematics and its Applications, 2 (1984).

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (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. 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. 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. 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. 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, (2001).

[7]

R. Rosa, The global attractors for the 2D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71. 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. doi: 10.1007/BF02218613.

[9]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition,, Applied Mathematical Sciences, 68 (1997).

[10]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition,, Studies in Mathematics and its Applications, 2 (1984).

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