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.

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

Abstract. For the deterministic dyadic model of turbulence, there are examples of initial conditions in l 2 which have more than one solution. The aim of this paper is to prove that uniqueness, for all l 2 -initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.

A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process