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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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