Inequalities pp 305-312 | Cite as

On Characteristic Exponents in Turbulence

  • Elliott H. Lieb


Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Ω. The latter is particularly important because it yields an upper bound to the Hausdorff dimension of attracting sets. However, Ruelle’s bound on the number has three deficiences : (i) it relies on some unproved conjectures about certain constants; (ii) it is valid only in dimensions ≧ 3 and not 2 ; (iii) it is valid only in the limit Ω-→ ∞. In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.


Incompressible Fluid Hausdorff Dimension Characteristic Exponent Sharp Constant Large Volume Limit 
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  1. 1.
    Ruelle, D.: Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys. 87, 287–302 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in mathematical physics: essays in honor of Valentine Bargmann, Lieb, E., Simon, B., Wightman, A. (eds.), pp. 269–303. Princeton, NJ: Princeton University Press 1976Google Scholar
  3. 3.
    Lieb, E., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975); 35, 1116 (1975) (Erratum)CrossRefGoogle Scholar
  4. 4.
    Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lieb, E.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Am. Math. Soc. 82, 751–753 (1976); the details appear in [6]MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. Proc. Am. Math. Soc. Symp. in Pure Math., Osserman, R., Weinstein, A. (eds.), Vol. 36, pp. 241–252 (1980). Much of this material is reviewed in Simon, B.: Functional integration and quantum physics, pp. 88-100. New York: Academic Press 1979Google Scholar
  7. 7.
    Rosenbljum, G.: Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR 202, 1012–1015 (1972) (MR45 No. 4216). The details are given in: Distribution of the discrete spectrum of singular differential operators. Izv. Vyss. Ucebn. Zaved. Matem. 164, 75–86 (1976) [English transi. Sov. Math. (Iz. VUZ) 20, 63-71 (1976)]MathSciNetGoogle Scholar
  8. 8.
    Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Aizenman, M., Lieb, E.: On semiclassical bounds for eigenvalues of Schrödinger operators. Phys. Lett. 66 A, 427–429 (1978)MathSciNetGoogle Scholar
  10. 10.
    Glaser, V., Grosse, H., Martin, A.: Bounds on the number of eigenvalues of the Schrödinger operator. Commun. Math. Phys. 59, 197–212 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Grosse, H.: Quasiclassical estimates on moments of the energy levels. Acta Phys. Austr. 52, 89–105 (1980)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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