Ricerc@Sapienza

Spatial heterogeneities play a special role in different contexts.
In some situations the presence of regular microstructures
modifies the effective values of some physical constants,
such as diffusion coefficients, thermal conduction coefficients, and
so on.
Macroscopic heterogeneities, on the other hand, can play the
role of obstacles, modifying locally the dynamics of the systems
and yielding in some cases not intuitive results, such as dynamics
acceleration.
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. This results in space dependent parameters in PDEs
modelling and space dependent jump rates in stochastic processes.
The problems we are interested in
range from biological systems to pedestrian
motions and 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 a
constant.
ii) Derivation of non-standard diffusion equations governing the macroscopic
motion of particles in presence of not constant diffusion coefficients
starting from a system of particles undergoing
a random microscopic walk in
an heterogeneous environment which induces different rates of motion
in different regions of space.
iii) Investigation on memory effects in evolution phenomena in heat
conductors, viscoelastic and magneto-viscoelastic solids.
Study of nonlinear evolution operator equations and determination of
admitted special solutions. A particular case of applicative interest
in quantum mechanics, i.e., finite dimensional (matrix) soliton
solutions are constructed.