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Global attractors for the three-dimensional Navier-Stokes equations. (English) Zbl 0855.35100

The main purpose of this paper is to show that the weak solutions of the Navier-Stokes equations on any bounded, smooth three-dimensional domain ${\Omega }\subset {ℝ}^{3}$ have a global attractor for any positive value of viscosity. The proof of this result is based on a new point of view for the construction of the semiflow generated by these equations.

The paper first presents the essentials for the theory of semiflows on a perfect space. Then it turns out that for a suitable forcing function $f$, the weak solutions of the Navier-Stokes equations can be identified with the restriction of a semiflow on a Fréchet space to an appropriate invariant subset $W$. By using the general theory of global attractors for semiflows on metric spaces, general sufficient conditions are derived for the semiflows to have a global attractor. Thus, two issues of possible nonuniqueness of the weak solutions and the possible lack of global regularity of the strong solutions are bypassed. It is also shown that, under additional assumptions, this global attractor consists entirely of strong solutions.

The time dependent issue (the forcing function $f$ is time dependent) is addressed finally. In order to develop a dynamical theory to handle this situation, the traditional approach of skew product flows is made use of.

##### MSC:
 35B41 Attractors (PDE) 35Q30 Stokes and Navier-Stokes equations 37L30 Attractors and their dimensions, Lyapunov exponents 35D05 Existence of generalized solutions of PDE (MSC2000)
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