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