Ricerc@Sapienza

Evolution equations can be derived in many different contexts and
can be used to model the behavior of different systems.
For instance, in diffusive systems
the basic phenomenon is that individuals move in an environment
spreading from areas of high concentration to areas of low
concentration. Depending on the system to be modelled,
the individuals can be atoms, ions, molecules or even macroscopic
entities such as pedestrians.
The diffusion equation describing the evolution of the particle density
can be derived with different techniques and approaches, for instance
assuming the validity of a continuity equation associated with some
constitutive relations such as Fick's law
or performing a scaling limit in a discrete
space model on which particles perform a symmetric random walk.
The aim of this project is to investigate evolution phenomena, with a
particular attention to diffusion, in the
context of heterogeneous environments, namely, when the properties
governing the microscopic motion of the particles depend on the
space coordinates. The problems we are interested in are essentially threefold:
i) the study of diffusive systems described by the standard diffusion
equation in which, due to environment heterogeneity,
for instance the presence of obstacles, the diffusion coefficient is not
constant.
ii) Derivation of non-standard diffusion equations governing the macroscopic
motion of particles in presence of not constant diffusion coefficients.
This problem will be approached
deriving the macroscopic equation governing the motion of
particles undergoing a symmetric random microscopic motion in
an heterogeneous environment which induces different rates of motion
in different regions of space.
iii) Memory effects in evolutive systems.
Different applications, ranging from biological systems to pedestrian
motions, will be investigated.