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Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. (English) Zbl 1406.35232
Summary: The main objective of this paper is to study the existence of a finite dimensional global attractor for the three dimensional Navier-Stokes equations with nonlinear damping for $$r>4$$. Motivated by the idea of [1], even though we can obtain the existence of a global attractor for $$r\geq 2$$ by the multi-valued semi-flow, it is very difficult to provide any information about its fractal dimension. Therefore, we prove the existence of a global attractor in $$H$$ and provide the upper bound of its fractal dimension by the methods of $$\ell$$-trajectories in this paper.

##### MSC:
 35Q30 Navier-Stokes equations 35B41 Attractors 37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents 76D05 Navier-Stokes equations for incompressible viscous fluids 28A80 Fractals
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