论文信息 - MEASURES DESCRIBING A TURBULENT FLOW

MEASURES DESCRIBING A TURBULENT FLOW

INTRODUCTION The motion of a fluid in a region R of R' or R' is defined by a function t : , ( t ) , where u ( t ) belongs to some functional space, %, of velocity fields in R. In a turbulent regime, one expects r , ( t ) to be distributed according to some probability law. This probability law is defined by a measure p on 7f . invariant under the deterministic time evolution of the system. We have reached some understanding of the invariant measures for a time evolution in a finite-dimensional space 7 f . The notions of strange attractor, sensitive dependence on initial condition, and characteristic exponents have been useful in this respect. Also, it is possible to define stable and unstable manifolds almost everywhere, and one can, in a number of cases, identify those measures which are stable under small stochastic perturbations. In this paper, we discuss the extension of results obtained for finite-dimensional dynamical systems to the more realistic case of the time evolution defined in a Hilbert space by the Navier-Stokes equation.

David Ruelle | D. Ruelle

[1] Hiroshi Fujita,et al. On the Navier-Stokes initial value problem. I , 1964 .

[2] R. Temam,et al. Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[3] R. Bowen. Ergodic theory of Axiom A flows , 1975 .

[4] V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[5] Ju I Kifer. ON SMALL RANDOM PERTURBATIONS OF SOME SMOOTH DYNAMICAL SYSTEMS , 1974 .

[6] Sur la deuxième bifurcation d'une solution stationnaire de systèmes du type Navier-Stokes , 1977 .

[7] David Ruelle,et al. An inequality for the entropy of differentiable maps , 1978 .

[8] D. Ruelle. Microscopic fluctuations and turbulence , 1979 .

[9] J. Mallet-Paret. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright , 1976 .

[10] D. Ruelle. SENSITIVE DEPENDENCE ON INITIAL CONDITION AND TURBULENT BEHAVIOR OF DYNAMICAL SYSTEMS , 1979 .

[11] D. Ruelle,et al. The ergodic theory of AxiomA flows , 1975 .

[12] J. Lions. Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[13] David Ruelle,et al. Characteristic Exponents and Invariant Manifolds in Hilbert Space , 1982 .

[14] David Ruelle,et al. A MEASURE ASSOCIATED WITH AXIOM-A ATTRACTORS. , 1976 .

[15] Hiroko Morimoto,et al. On the Navier-Stokes initial value problem , 1974 .