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On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations. (English. Russian original) Zbl 0687.35072

Russ. Math. Surv. 42, No. 6, 27-73 (1987); translation from Usp. Mat. Nauk 42, No. 6, 25-60 (1987).
This is a survey article presenting the author’s contributions in the field of dissipative infinite dimensional dynamical systems. Although the author studies nonlinear evolution equations of both parabolic and hyperbolic type, the main object centers around the initial boundary value problem for the Navier-Stokes equations. The phase space of the dynamical system is assumed to be a complete metric space, in which it is defined a semigroup of completely continuous or asymptotically compact operators. There is a discussion of pointwise dissipativity, absorbing and invariant sets, leading to the existence of minimal global attractors of bounded sets and of points. The attractors are constructed as the \(\omega\)-limit sets of the dynamical system. Moreover, the question of the finiteness of the Hausdorff dimension of the attractor is answered. The author also tries to reveal the connections and differences in the results carried out by various groups working in the field.
Reviewer: M.A.Boudourides

MSC:

35Q30 Navier-Stokes equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
47H20 Semigroups of nonlinear operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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