Dear Colleagues, we would like to invite you to the following Probability seminar that will take place at the Department of Mathematics of Padova, on February 18. Speaker: Francesco Morandin Title: Turbulence, shell models and critical exponents for dissipation Abstract: Shell models of turbulence are nonlinear dynamical systems inspired by fluid dynamics. They are idealized and simplified, but tailored to exhibit the same energy cascade behaviour of three dimensional Euler and Navier-Stokes equations. A typical feature of these models is in fact anomalous dissipation of energy, which in finite time "escapes" to infinity, yielding a blow-up and instantaneous loss in regularity. A dissipative term corresponding to viscosity can recover regularity, for some of these models, but in total generality one needs hyper-dissipation, with an exponent larger than the physical one. Recent results hint that in the more refined framework of tree (hierarchical) models the required exponent may be actually lower. Speaker: Alessandro Montagnani Title: A stationary solution for turbulence shell models. Abstract The focus of the talk is to study well-posedness, with respect to generic Gaussian distributed initial data, in turbulence shell models. In the of state-of-the-art results we have existence of solution for any finite energy initial conditions. Here we show the generic existence of solutions with respect to initial data distributed as Gaussian invariant measures, in "mixed" dyadic and tree-like shell models, extending the classical deterministic results. The existence is given thanks to compactness argument and techniques similar to the ones used by Albeverio and Cruzeiro for Euler equation (and more recently with a different approach by F. Flandoli), adapted to our model. Uniqueness is not provided, and the natural oscillating behaviour of the solutions obtained may suggests that it doesn't hold at all All of you are very welcome. Best regards, David Barbato