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close this bookThe Impact of Chaos on Science and Society (UNU, 1997, 415 pages)
close this folder4. The impact of chaos on mathematics
View the document(introduction...)
View the documentAbstract
View the documentI. Introduction: A historical view
View the documentII. The Lorenz attractor
View the documentIII. The Feigenbaum bifurcation
View the documentIV. Hydrodynamic turbulence
View the documentV. Ergodic theory of differentiable dynamical systems: Axiom A systems
View the documentVI. Ergodic theory of differentiable dynamical systems: General systems
View the documentVII. Quadratic maps of the interval and the Hénon attractor
View the documentVIII. Zeta functions
View the documentIX. Conclusion
View the documentReferences

IV. Hydrodynamic turbulence

The phenomenon called turbulence is complex and difficult to analyse. What we know about it results from experimental, numerical, and mathematical studies. One important feature of turbulence is that, as claimed by Ruelle and Takens [4], it is chaotic: this has been proved experimentally by the observation of strange attractors, Feigenbaum bifurcations, etc. at the onset of turbulence. A mathematical proof of chaos in the Navier - Stokes equation does not exist at this time. (And when one is obtained, it will no longer create much excitement.) We have thus here a very interesting situation from the epistemological viewpoint, where we are firmly convinced of a certain mathematical fact (the existence of chaos in the solutions of the Navier - Stokes equations) but our belief is based on experimental evidence.

Another conceptually important question arises in the theory of turbulence in relation with chaos: this is the question of natural measures or SRB measures. Consider a hydrodynamical system kept in turbulent motion by the action of some time-independent external forces. One expects that time averages can be defined for the turbulent motion. Time averages correspond to "ensemble averages", i.e., to a probability measure invariant by time evolution (in fact ergodic). The relevant ergodic measure has support on the attractor describing the motion. In the framework of the Hopf - Landau theory of turbulence, the attractor is quasiperiodic and carries only one invariant measure. But a strange attractor carries uncountably many distinct ergodic measures! Which one should we use to represent time averages? As it happens we have a reasonable answer to this question for certain dynamical systems in terms of so-called natural or SRB measures (see next section). But independently of the answer, it is conceptually important to recognize that there is a problem. For many years, people have tried to determine time averages by "solving the Hopf equation" (this equation only expresses invariance of a probability measure under the Navier - Stokes time evolution). It is now clear that an essential ingredient beyond the Hopf equation is needed to determine time averages. (Ad hoc truncations of the equations, and assumptions of gaussianity are unlikely to be the answer, see Ruelle [13]).