Abstract
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 Θ. 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 Θ → ∞. In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.
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Ruelle, D.: Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys.87, 287–302 (1982)
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 1976
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)
Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math.106, 93–100 (1977)
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]
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 1979
Rosenbljum, G.: Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR202, 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 transl. Sov. Math. (Iz. VUZ)20, 63–71 (1976)]
Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys.88, 309–318 (1983)
Aizenman, M., Lieb, E.: On semiclassical bounds for eigenvalues of Schrödinger operators. Phys. Lett.66 A, 427–429 (1978)
Glaser, V., Grosse, H., Martin, A.: Bounds on the number of eigenvalues of the Schrödinger operator. Commun. Math. Phys.59, 197–212 (1978)
Grosse, H.: Quasiclassical estimates on moments of the energy levels. Austr.52, 89–105 (1980)
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Communicated by A. Jaffe
Work partially supported by U.S. National Science Foundation grant No. PHY-8116101-A01
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Lieb, E.H. On characteristic exponents in turbulence. Commun.Math. Phys. 92, 473–480 (1984). https://doi.org/10.1007/BF01215277
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DOI: https://doi.org/10.1007/BF01215277