Two topics on regularity and blow-up for the dyadic model.

The talk starts with a presentation of the dyadic model, a system of infinitely many differential equations that should capture the essential features of the evolution and interaction of energy in the motion of incompressible fluids. So far the model has provided heuristics insights on the original PDEs. Recent results [Tao2014] have shown that the connection is more robust and some insight can be made partially rigorous.

We present two results. In the first part we prove global existence of smooth solutions for a slightly supercritical dyadic model. This catches for the dyadic a conjecture that for Navier-Stokes equations was formulated in [Tao2009]. In the second part, we consider the model with non-linearity of strong intensity and driven by Gaussian noise. We prove the emergence of blow--up, that is loss of regularity after a random time. The contribution of the noise is essential to prove that blow-up occurs with full probability, by means of conditional recurrence.