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 2n-1 and 2n.
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.