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The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory. (English) Zbl 0673.35084
Directions in partial differential equations, Proc. Symp., Madison/Wis. 1985, Publ. Math. Res. Cent. Univ. Wis. Madison 54, 55-73 (1987).

[For the entire collection see Zbl 0643.00011.]

The authors present a survey on mathematical results and conjectures related to the concept of turbulence in viscous incompressible flow. First, they introduce the Navier-Stokes equations in abstract formulation and define and discuss the universal attractor X. Estimates of the fractal dimension of X, and the relation of these estimates to the Reynolds number are dealt with next. The final concept treated is the idea of an inertial manifold.

Reviewer: R.Illner

MSC:
35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE
76D05Navier-Stokes equations (fluid dynamics)
37C70Attractors and repellers, topological structure
35B30Dependence of solutions of PDE on initial and boundary data, parameters