Ricerc@Sapienza

The analysis of nonlocal porous medium equations, i.e. nonlinear diffusion equations with fractional operators, is a recent research topic in the theory of the partial differential equations. Many papers appeared in the last decade are devoted to the study of such equations but only in the functional analysis framework (i.e. existence and uniqueness of the solution, hypercontractivity estimates and so on). The classical porous medium equation has been introduced to overcome the paradox of infinte propagation of the heat flow. A special solution is the so-called Kompaneets-Zel'dovich-Barenblatt solution which is a compactly supported parabolic function.

The goal of this research project is to study stochastic models associated to the fractional porous medium equations. Therefore, the heat diffusion equations with spatial pseudo-differential operators represent our main object of interest. We will focus our attention on the following issues.
1) We are interested to the study of the nonlocal operators and how they affect the long-range memory of the stochastic processes linked to the fractional equations. The mathematical tools could require the use of the Lévy subordinators.
2) Furthermore, we want introduce reasonable random walks moving in a lattice in order to describe the microscopic dynamic of the fractional porous medium equations.
3) Another research topic concerns the relationship between some nonlocal equations and the random flight processes. These random models describe a particle starting at the origin with a randomly chosen direction and with finite speed. The direction of the particle changes at each collision with some scattered obstacles where a new direction of motions is taken.