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On the dimension of the attractors in two-dimensional turbulence. (English) Zbl 0658.58030

In conventional turbulence theory a heuristical estimate of the number of degrees of freedom of a turbulent flow is given by \(N\sim (l_ 0/l_ c)^ d\), where \(l_ 0\) denotes the linear size of the region occupied by the fluid and the length \(l_ c\) is a small scale determine by the physical properties of turbulence below which viscosity effects determine entirely the motion. In the three dimensional space the length \(l_ c\) is defined through dimensional analysis by the energy dissipation flux \(\epsilon\) as \(\nu^{3/4}/\epsilon^{1/4}\). In this paper the authors prove a rigorous estimate of the above relation for the two dimensional space by identifying the number of degrees of freedom with the dimension of the universal attractor of the d-dimensional Navier-Stokes equations. The estimate is optional up to a logarithmic correction. The relevance of this estimate to turbulence and related results are briefly discussed.
Reviewer: Y.Kozai

MSC:

76F99 Turbulence
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Batchelor, G. K., Computation of the energy spectrum in homogeneous, two dimensional turbulence, Phys. Fluids, 12, suppl. 2, 233-239 (1969) · Zbl 0217.25801
[2] Babin, A. V.; Vishik, M. I., Les attracteurs des équations d’évolution et les estimations de leurs dimensions, Usp. Math. Nauk, 38, 4, 133-187 (1983), [232]
[3] Brézis, H.; Gallouet, T., Nonlinear Schrödinger evolution equations, Nonlinear Analysis Theory, Methods and Applications, 4, 677-681 (1980) · Zbl 0451.35023
[4] Constantin, P., Collective \(L^∞\) estimates for families of functions with orthonormal derivatives, Indiana University Math. J., 36, 603-615 (1987) · Zbl 0658.35011
[5] Constantin, P.; Foias, C., Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Comm. Pure Appl. Math., XXXVIII, 1-27 (1985) · Zbl 0582.35092
[6] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Memoirs of A.M.S., 53, No. 314 (January, 1985)
[7] Constantin, P.; Foias, C.; Manley, O.; Temam, R., Determining modes and fractal dimensions of turbulent flows, J. Fluid Mech., 150, 427-440 (1985) · Zbl 0607.76054
[9] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, 339-368 (1979) · Zbl 0454.35073
[11] Foias, C.; Manley, O.; Temam, R.; Treve, Y., Asymptotic analysis of the Navier-Stokes equations, Physica, 9D, 157-188 (1983) · Zbl 0584.35007
[12] Kraichnan, R. H., Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10, 1417-1423 (1967)
[14] Landau, L.; Lifschitz, I. M., Fluid Mechanics (1953), Addison-Wesley: Addison-Wesley New York
[15] Lieb, E., On characteristic exponents in turbulence, Comm. Math. Phys., 92, 473-480 (1984) · Zbl 0598.76054
[16] Lieb, E., An \(L^p\) bound for the Riesz and Bessel potentials of orthonormal functions, J. Funct. Analysis, 51, No. 2, 159-165 (1983) · Zbl 0517.46025
[17] Lieb, E.; Thirring, W., Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities, (Lieb, E.; Simon, B.; Wightman, A., Studies in Mathematical Physics: Essays in Honor of Valentine Bargman (1976), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ) · Zbl 0342.35044
[18] Mandelbrot, B., Fractals: Form, Chance and Dimension (1977), Freeman: Freeman San Francisco · Zbl 0376.28020
[19] Marchioro, C., An example of absence of turbulence for any Reynolds number, Comm. Math. Phys., 105, 99-106 (1986) · Zbl 0607.76052
[20] Minea, Gh., Remarques sur l’uncité de la solution stationnaire d’une équation de type Navier-Stokes, Revue Roumaine Math. Pures et Appl., 21, 1071-1075 (1976) · Zbl 0365.76027
[21] Ruelle, D., Large volume limit distribution of characteristic exponents in turbulence, Comm. Math. Phys., 87, 287-301 (1982) · Zbl 0546.76083
[22] Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1979), North-Holland: North-Holland Amsterdam · Zbl 0426.35003
[23] Temam, R., Navier-Stokes equations and nonlinear functional analysis, (NSF/CBMS Regional Conference Series in Appl. Math. (1983), SIAM: SIAM Philadelphia) · Zbl 0522.35002
[25] Temam, R., Infinite dimensional dynamical systems in fluid mechanics, (Browder, F., Nonlinear Functional Analysis and Applications, Proc. AMS Summer Research Institute. Nonlinear Functional Analysis and Applications, Proc. AMS Summer Research Institute, Berkeley, CA (1983))
[26] Tennekes, H., A comparative pathology of atmospheric turbulence in two and three dimensions, (Ghil, M., Turbulence and Predictability in Geophysical Fluid Dynamics and in Climate Dynamics, Proc. Int. School of Physics. Turbulence and Predictability in Geophysical Fluid Dynamics and in Climate Dynamics, Proc. Int. School of Physics, Varenna (1983))
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