Abstract
For spatially extended conservative or dissipative physical systems, it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity. In the case of a turbulent viscous fluid, however, this simple idea is complicated by the phenomenon of intermittency. In the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation. These bounds have a reasonable large volume behavior. For two-dimensional fluids a particularly striking result is obtained: the total information creation is bounded above by a fixed multiple of the total energy dissipation (at fixed viscosity). The distribution of characteristic exponents is estimated in an intermittent model of turbulence (see [7]), and it is found that a change of behavior occurs at the valueD=2.6 of the self-similarity dimension.
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References
Betchov, R.: An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech.1, 497–504 (1956)
Brachet, M.: Intégration numérique des équations de Navier-Stokes en régime de turbulence développée. C.R. Acad. Sci. Paris294, Série II, 537–540 (1982)
Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys.83, 517–535 (1982)
Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C.R. Acad. Sci. Paris290 A, 1135–1138 (1980)
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics. Reading, Mass.: Addison-Wesley 1963(I), 1964(II), 1965(III)
Foias, C., Temam, R.: Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pure Appl.58, 339–368 (1979)
Frisch, U., Sulem, P.-L., Nelkin, M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech.87, 719–736 (1978)
Fujisaka, H., Mori, H.: A maximum principle for determining the intermittency exponent of fully developed steady turbulence. Progr. Theor. Phys.62, 54–60 (1979)
Girault, V., Raviart, P.-A.: Finite element approximation of the Navier-Stokes equation. In: Lecture Notes in Mathematics, Vol. 749. Berlin, Heidelberg, New York: Springer 1979
Kraichnan, R.H.: On Kolmogorov's inertial-range theories. J. Fluid Mech.62, 305–330 (1974)
Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow (2nd ed.). Moscow: Nauka, 1970 [2nd english ed. New York: Gordon and Breach 1969]
Ledrappier, F.: Some relations between dimension and Lyapounov exponents. Commun. Math. Phys.81, 229–238 (1981)
Leith, C.: Chaos and order in wheather prediction. Order in chaos (Los Alamos, 1982) North-Holland (to be published)
Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities. In: Studies in Mathematical Physics: Essays in honor of Valentine Bargman, Lieb, E., Simon, B., Wightman, A.S. (eds.), pp. 269–303. Princeton, NJ: Princeton University Press 1976
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires. Paris 1969
Mallet-Paret, J.: Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Diff. Eq.22, 331–348 (1976)
Mandelbrot, B.: Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3 +B. In: Turbulence and the Navier-Stokes equations. Lecture Notes in Mathematics, Vol. 565, pp. 121–145. Berlin, Heidelberg, New York: Springer 1976
Mañé, R.: On the dimension of the compact invariant sets of certain nonlinear maps (preprint)
Mañé, R.: Lyapounov exponents and stable manifolds for compact transformations (preprint)
Monin, A.S., Yaglom, A.M.: Statistical fluid mechanics: mechanics of turbulence. Moscow: Nauka, 1965 [english transl.: 2 Vol. (ed. by J. L. Lumley). Cambridge, Mass.: MIT Press 1971 and 1975]
Novikov, E.A., Steward, R.W.: Intermittency of turbulence and the spectrum of fluctuations of energy dissipation. Izv. Akad. Nauk SSSR Ser. Geofiz.3, 408–413 (1964)
Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Tr. Moskov Mat. Obšč.19, 179–210 (1968) [english transl.: Transl. Moscow Math. Soc.19, 197–221 (1968)]
Raghunathan, M.S.: A proof of Oseledec' multiplicative ergodic theorem. Israel J. Math.32, 356–362 (1979)
Reed, M., Simon, B.: Methods of modern mathematical physics. New York: Academic Press 1972(I), 1975(II), 1979(III), 1978(IV)
Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat.9, 83–87 (1978)
Ruelle, D.: What are the measures describing turbulence? Progr. Theor. Phys. Suppl.64, 339–345 (1978)
Ruelle, D.: Microscopic fluctuations and turbulence. Phys. Lett.72A, 81–82 (1979)
Ruelle, D.: Measures describing a turbulent flow. Ann. N.Y. Acad. Sci.357, 1–9 (1980)
Ruelle, D.: Characteristic exponents and invariant manifolds in Hilbert space. Ann. Math.115, 243–290 (1982)
Shaw, R.: Strange attractors, chaotic behavior, and information flow. Santa Cruz Preprint (1977)
Siggia, E.: Invariants for the one-point vorticity and strain rate correlation functions. Phys. Fluids24, 1934–1936 (1981)
Sinai, Ya.G.: On the entropy per particle for the dynamical system of hard spheres (preprint)
Temam, R.: Navier-Stokes equations. Revised edition. Amsterdam: North-Holland 1979
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Communicated by A. Jaffe
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Ruelle, D. Large volume limit of the distribution of characteristic exponents in turbulence. Commun.Math. Phys. 87, 287–302 (1982). https://doi.org/10.1007/BF01218566
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DOI: https://doi.org/10.1007/BF01218566