On characteristic exponents in turbulence

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.