Ciprian Foias

The Connection Between the Navier-Stokes Equations, Dynamical Systems, and Turbulence Theory

Publisher Summary There are three major distinct but not mutually contradictory theoretical views on turbulence—(a) the statistical view, in which the turbulence is considered as the observed behavior of the evolution of statistical distributions of flows instead of the evolution of one individual flow; (b) the viewpoint of regularity breakdown, in which the turbulence is considered to result from the blow-up of the vorticity in finite time, albeit necessarily on a set of small Hausdorff dimension; and (c) the dynamical systems view, in which the turbulence is a phenomenologic perception of the long time complicated behavior of the individual flows. The chapter discusses that it is reasonable to test them by trying to establish rigorous mathematical facts based on the Navier–Stokes equations, the interpretation of which are consistent with any of the three viewpoints mentioned above. The chapter presents the contribution by Ciprian Foias and Roger Temam to this program from the dynamical systems point of view. The chapter presents a proof of a regularity and backward uniqueness theorem on an open dense subset of the universal attractor of the 3D Navier–Stokes equations.

Finite Parameter Approximative Structure of Actual Flows

Publisher Summary This chapter discusses finite parameter approximative structure of actual flows. To explain the concept, it considers the Navier–Stokes equations, which are Galilean invariant.