Foias, Ciprian; Témam, Roger 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 Cited in 1 ReviewCited in 13 Documents MSC: 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs Keywords:survey; turbulence; viscous incompressible flow; Navier-Stokes equations in abstract formulation; universal attractor; Estimates; fractal dimension; Reynolds number; inertial manifold Citations:Zbl 0643.00011 PDFBibTeX XML