Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35B40, 35Q30; Secondary: 37L30.

 Citation:

•  [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.