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.
A) In the framework of the problems of electrical conduction in
biological tissues and thermal diffusion in composite media there are
still many open problems in need of a better understanding which we
shall tackle. Our research could provide a crucial advancement in
this field, where often the models in use are in general only
phenomenological models with no mathematical or physical validation. With
this purpose, we propose
a refinement of the results present in the literature, and
the study of open problems in the same lines of
investigation, where microstructures involving active interfaces are
present and for which homogenization techniques are essential to
derive the macroscopic models. In the case of
electroporation, due to the novelty of the problem we propose to
investigate the viability of a homogenization approach in this field.
B1) It is interesting to connect the effective diffusion
coefficient in the membrane (the macroscopic obstacle region) to the
geometry of the obstacles. Similar results have been proven
for the standard diffusion equation. Our challenge
is to derive the effective equation for a Smoluchowski kind of equation.
In our problem a drift term with derivative of the field and of its square
is present. Our goal is that of treating a general equation in any dimension
larger than two in which the drift
term is the space derivative, along one space direction, of a polynomial
of the unknown field.
B2) The geometry of our domain requires the development of some specific
tools. The presence of elastically reflective conditions at the upper and lower
sides of the strip and at the obstacles boundaries suggest to search the
limit problem as a diffusive equation with mixed Dirichlet and Neumann boundary
conditions. It will be necessary
to study also how the obstacles influence the configuration of small disks
that prevent the convergence of the Lorentz gas to the linear Boltzmann
equation dynamic to find the right diffusive limit to
characterize the stationary state of the Lorentz model. The numerical
simulations of the linear Boltzmann dynamic would be
a useful tool in the identification of stationary state, mass flow and typical
crossing time in different obstacles configurations in the strip.
It will be very interesting to understand if, even in this kinetic theory
context, the non-monotonic behavior [20] of the crossing time with respect
to the size of the obstacles will be found.
C1) We aim at results connecting the asymptotic behavior of solutions for
large times to the density function and to the geometry of the manifold, which
is going to be characterized in terms of Faber-Krahn type inequalities.
C2) We also are interested in universal (i.e., independent of initial data)
type estimates for solutions of the mentioned problem when the density
function decays fast enough at infinity. They should include, when it appears,
the interface blow-up rate. Again, this is expected to depend on the interplay
of geometric and diffusive properties in the problem.
D) In the scientific literature the question about the correct
diffusion equation found when the diffusion coefficient depend on the
space variable is posed very often [31].
We think that such a question is too naive and that
a more detailed knowledge of the microscopic system is necessary
to model correctly the macroscopic behavior. We propose to attack the
problem from a microscopic point of view: we shall consider particles
performing a random walk on a two dimensional lattice and assume
that the rates at which particles jump are not spatially homogeneous.
Our goal is to prove rigorously, using the Varadhan's approach to the
hydrodynamic limit, that the Fick and the Fokker-Planck laws are
recovered in the hydrodynamic limit provided the inhomogeneity is on the
site or on the bond of the lattice. This would be
a breakthrough in the understanding of the problem.
E) Recently, new viscoelastic materials, such as viscoelastic gels, have been
described by virtue of convolution integral with singular kernels
(fractional and hypergeometric kernels). When the
natural aging of the viscoelastic material has to be taken into account, time
dependent kernels are needed. The behavior of some new material
(ferrogel and magneto rheological elastomers) can be determined
coupling viscoelastic and magnetic effects.
The aim of this project is to study
asymptotic behavior of solutions via prolongation of weak solutions whose
existence is proved locally in time both with or without magnetization.
F) The innovative aspect of the present project is represented by the
combination of asymptotic methods with algebraic properties
(invariance obtained via Baecklund transformations) to construct new soliton
solutions. The recursion operator admitted by all such equations as well as
further non-Abelian equations allows to investigate the multisoliton manifold
which is well known in the Abelian case, but never studied in the non-Abelian
setting.