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We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier-Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao09]. From the generalized Fourier Navier-Stokes to the dyadic equationThis section contains one of the crucial steps in our approach. We show that the proof of Theorem 1.2 can be reduced to a proof of decay of solutions of a suitable

Abstract.A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved, and decay of energy like t −2 is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.

A stochastic infinite dimensional version of the GOY model is rigorously investigated. Well posedness of strong solutions, existence and p-integrability of invariant measures is proved. Existence of solutions to the zero viscosity equation is also proved. With these preliminary results, the asymptotic exponents ζ p of the structure function are investigated. Necessary and sufficient conditions for ζ 2 ≥ 2/3 and ζ 2 = 2/3 are given and discussed on the basis of numerical simulations.