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On characteristic exponents in turbulence. (English) Zbl 0598.76054

Ruelle has found upper bounds to the magnitude and to the number of non- negative characteristic exponents for the Navier-Stokes flow of an impompressible fluid in a domain \(\Omega\). 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 deficiencies: (i) it relies on some unproved conjectures about certain constants: (ii) it is valid only in dimensions \(\geq 3\) and not 2; (iii) it is valid only in the limit \(\Omega\) \(\to \infty\). In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.

MSC:

76Fxx Turbulence
35Q30 Navier-Stokes equations
46N99 Miscellaneous applications of functional analysis
35B40 Asymptotic behavior of solutions to PDEs
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