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On the fractal dimension of attractors for viscous incompressible fluid flows. (English) Zbl 0626.35078

Generalizing previous work of C. Foias and R. Temam [J. Math. Pures Appl., IX. Ser. 58, 339-368 (1979; Zbl 0454.35073)] on Navier- Stokes equations the author formulates an abstract initial value problem on a Hilbert space and proves that every functional invariant subset (roughly speaking: every subset invariant for all time under the flow described by the differential equation) has finite fractal dimension. In particular this is true for attractors. In a second part this result is applied to Navier-Stokes equations with nonhomogeneous boundary conditions, to N.S.E. on a Riemannian manifold, to thermohydraulic equations and to magnetohydrodynamic equations.
Reviewer: G.Keller

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
76D05 Navier-Stokes equations for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics

Citations:

Zbl 0454.35073
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