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.
A)The proposed models are technically difficult, due to the intricate
microscopic geometry together and to the presence of active interfaces.
The common denominator is the periodic
microstructure with interfaces playing an active role.
This topic has been the object of intensive investigation
[02m-07m], but crucial points needing a better
understanding are still there. In particular, the different formula used
for the macroscopic conductivity or diffusivity of the medium, given
the electrical/thermal properties of its components, seem to be the result of
experiments with little or no mathematical understanding.
We will consider different transmission conditions, i.e.,
different interface modeling, describing how
the surrounding material interacts with the inclusions via the
interface. We hope to obtain macroscopic models and overall
conductivities/diffusivities, providing a mathematical motivation
of the experimental formula and to underline their possible limitations
and fields of applications.
Another breakthrough is connected to existence and uniqueness
problems. The models are described by systems of PDEs involving
elliptic, parabolic, pseudo-parabolic or abstract parabolic equations and,
due to the presence of active interface, also tangential operators can
appear. In some situations, also the underlying geometrical
structure plays a crucial role. These PDE systems are not
standard and need a delicate well-posedness investigation.
Some results in this direction have
been obtained [08m,09m], but the variety of different situations
calls for a further deepening.
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
with a drift term. 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 1D problem has been treated in [07e] developing ad-hoc arguments.
We are now developing a new techniques, based on generating functions,
pointing out the similarities between our problem and the classical
gambler ruin problem. Our hope is to use such a strategy to attack the
marginal process associated with the 2D dimensional walk.
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. In this relation, some thresholds appear discriminating
between different behaviors of solutions.
C2) Methods similar to the ones effective in the study of diffusion in
Riemannian manifolds can be applied in discrete graphs, which are a
well-known subject of theoretical and applicative interest
[05d,06d]. We aim to investigate a common approach to fast
diffusion both in continuous and discrete metrics.
D) The question about the correct diffusion equation for not constant
diffusion coefficient is posed very often [13e]. This point of view
is too naive: a detailed knowledge of the microscopic system is necessary
to model correctly the macroscopic behavior and performing a
breakthrough in the understanding of the problem. We attack the
problem from a microscopic point of view, considering particles
performing a 2D random walk and assuming that the jumping rates
are not spatially homogeneous. The goal is proving
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.
Considering not regular diffusion coefficient
will need to develope specific techniques known as substitution lemmas.
E) Recently, new viscoelastic materials, such as viscoelastic gels, have been
described by convolution integral with singular kernels
(fractional and hypergeometric). When the
natural aging of the viscoelastic material is taken into account, time
dependent kernels are used. The behavior of some new material
(ferrogel and magneto rheological elastomers) can be determined
coupling viscoelastic and magnetic effects.
We study the 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 multi-soliton manifold
which is well known in the Abelian case, but never studied in the non-Abelian
setting. The connected Hamiltonian, and, if it is admitted, bi-Hamiltonian
structure is also not yet known: we aim to construct it.