# Research

Most of my recent publications deal with shell models of turbulence and related models.

A shell model is an idealization of Navier-Stokes or Euler equations. It is an infinite systems of ODEs usually indexed by the positive integers or by a regular tree. The equations are nearest-neighbour coupled, have quadratic nonlinearities and coefficients growing exponentially fast.

The tipical derivation is as follows. Consider the velocity field
of some solutions of Navier-Stokes equations, write it in Fourier
components and then collect in a single component indexed by *n* the
energy of all the Fourier modes with wavenumber *k* in a
spherical shell of radius between 2^{n-1} and
2^{n}.

The equations for these shell components are not realistic, since the non-linear interaction beetween components can be arbitrarily chosen in several different ways, nevertheless they capture some of the statistical properties and features of three-dimensional turbulence, since they are usually taylored to model the cascade of energy to higher and higher wavenumbers.

The introduction of *Stochastic inviscid shell models:
well-posedness and anomalous dissipation* is a good place to
start.

Older research directions included statistical properties of random error correcting codes and branching process representations of non-linear ODEs.