Piermarco CANNARSA, Università di Roma Tor Vergata, ITALY

  Decay estimates for second order evolution equations with memory damping

    It is well-known that viscoelastic models enjoy stability properties of various kinds. In
mathematical terminology, the addition of a convolution term to a conservative system may
force the system energy to decay at infinity at a rate which depends on the integral kernel
(relaxation function) itself. This talk will discuss the analysis of such a damping effect
developed in recent years in collaboration with F. Alabau-Boussuoira and D. Sforza. We will
consider both abstract evolution equations and concrete examples of partial differential
equations, addressing absolutely continuous as well as merely integrable relaxation
functions under suitable structural conditions.

  Pierluigi COLLI, Università di Pavia, ITALY

 Solutions to certain phase segregation problems of the Allen-Cahn type

    This talk will be a sequel to Podio-Guidugli's lecture, based on joint  work with him,
G. Gilardi, and J. Sprekels.Firstly, I shall present an existence theorem for a model of phase 
segregation of Allen-Cahn type, diffusion-free and isothermal,  consisting of a system of two
nonlinear differential equations, the  one ordinary the other partial. Next, I shall consider a
variant of  this system, where thermal effects are taken into account. Finally, I  shall address
to some analytical work of ours which is still in  progress, namely, a suitable regularization of
a system of  Cahn-Hilliard type, isothermal but with diffusion.

 Antonio FASANO, Università di Firenze, ITALY

 A multiscale approach to erosion of porous media

    We consider the flow of a Newtonian fluid through a solid porous matrix, supposing that
solid particles can be detached from the matrix owing to the stress exerted by the flow. As a
result of erosion the matrix pores are enlarged and the flowing fluid becomes a suspension
with evolving physical properties. In order to formulate a mathematically treatable model a
special 2-D geometry is considered in which the medium consists of an array of symmetric
lamellae, with the macroscopic flow occurring in one direction. Thus we have two space
coordinates: the longitudinal coordinate, in the direction of the macroscopic flow, and the
transversal coordinate. The unknowns are the velocity and pressure fields of the flow, and
the profile of the solid/fluid interface. The system is studied first at the microscopic scale
(the one of the channels between the solid lamellae) and then all the relevant quantities are
expanded as power series of a small parameter e, representing the ratio between the
average inter-lamellar distance and the size of the whole medium. By matching the terms
with the same power of e the governing equations are derived at the various orders.
Averaging over cross sections the macroscopic governing equations are eventually
obtained. The resulting free boundary problem has a quite peculiar structure, since it
exhibits a nonlocal condition. Indeed it turns out that the interface evolution is affected by
the whole pressure field along the medium. Existence and uniqueness are proved and
some numerical simulations are presented.

 Daniela GIACHETTI, SAPIENZA Università di Roma, ITALY

  Quasilinear singular elliptic equations

   We will present some recent results dealing with Dirichlet problems for elliptic equations
involving lower order terms h(u)|Du| r depending on the solution u and on its gradient, in
the case that the function h(s) is singular somewhere and 1 < r ≤ 2.
We will first focus the attention on the case where the singularity lies at zero, so that the
lower order term is singular at least at each point of the boundary. In this case we will
assume that the datum f(x) is non negative and, for simplicity, bounded, proving existence
of distributional solutions with different regulariry depending on the growth of the singularity
and on the sign of the term h(u).
Then we will consider the case where the lower order term is singular at a point different
from zero, for example at u = 1, and the real function h(s) is integrable near the singularity.
In this case we will prove existence of solutions for every datum belonging to a suitable
Lebesgue space. Moreover we will show that the solution pass through the singularity when
data are big enough.

 Claudio GIORGI, Università di Brescia, ITALY

  A new theoretical scheme for the analysis of equations with memory

 Murrough GOLDEN, Dublin Institute of Technology, IRELAND

 Phase Transitions in Materials with Thermal Memory and a Fourier term

    A model for thermally induced phase transitions in rigid materials with thermalmemory was
recently proposed, both for the case where the phases have the sameconductivity
properties and where they are different. The model predicts disconti-nuities in the
temperature and temperature gradient, which depend on velocities ofthermal disturbances
in the media. In this presentation, the model is generalizedto the case of heatflow relations
which include instantaneous contributions of theFourier type as well as memory terms. The
temperature gradient is decomposedinto two parts, each zero on one phase and equal to
the temperature gradient on theother. However, they vary smoothly over the transition zone.
Asymptotic analysisis carried out which show that, to leading order, temperature is
continuous acrossthe transition zone and the normal derivatives of the temperature on each
phaseboundary obey a condition of the classical form with no explicit dependence on
thememory terms. This latter result emerges out of first order terms in the
asymptoticanalysis. Effects explicitly related to thermal memory only begin to play a role
inthe analysis of second order terms. These results contrast sharply with those formaterials
without the instantaneous terms.

 Maurizio GRASSELLI, Politecnico di MILANO, ITALY

  Cahn-Hilliard equations with memory and dynamic boundary conditions

    We consider a Cahn-Hilliard equation where the velocity of the orderparameter depends on
the past history of the laplacian of the chemicalpotential. This dependence is expressed
through a time convolutionintegral characterized by a smooth nonnegative
exponentiallydecreasing memory kernel. The chemical potential is subject to theno-flux
condition, while the order parameter satisfies a (nonlinear)dynamic boundary condition. The
latter accounts for possibleinteractions with the container walls. This equation has already
beenanalyzed in the case of standard boundary conditions (e.g. Dirichlettype). However, the
present case requires a different approach since,for instance, higher-order estimates cannot
be obtained by multiplyingthe equation with some power of a suitable linear operator. We
intend to illustrate some new results on this initial and boundary value problem.

 Miguel HERRERO, Universidad Complutense de Madrid, SPAIN

  Modelling  Problems In Blood Coagulation

    Blood coagulation is a robust security mechanism of human organisms, which prevents
bleeding from minor injuries to occur.  Any disruption in such a system may have significant
consequences . For instance, an impaired ability of blood to coagulate is cause of
haemophilia, a serious hereditary disorder . On the other hand, an inordinate increase in the
activation of the blood coagulation system may lead to abnormal thrombi formation, and
consequently to a number of thrombotic pathologies .The process of blood coagulation
makes use of a complex array of interdependent, and finely tuned, biochemical reactions
( the so-called biochemical cascade), of which many details are known by now.

 Vilmos KOMORNIK, University of Strasbourg, FRANCE;

  Fourier series in control theory
  
   We discuss some recent variants of the classical theorems of Ingham and Beurling on
nonharmonic Fourier series. We explain how to use them to obtain observability theorems
for some linear evolutionary systems.

 Alfredo LORENZI, Università degli Studi di Milano, ITALY

Control and identification problems for materials with memory

  Let Ω ⊂ ⁿ be an open set with a C²-boundary and let ω ⊂⊂ Ω be an open subset. Let be a second-order uniformly elliptic operator.           

We consider the integro-differential equation

subject to the initial condition

and to the homogeneous boundary conditions

We deal with the following two different problems:

(P1) Determine suitable conditions on the behaviour of a near 0, for given T > 0, for the the closure in L²(Ω) of the set {y^u(T) : u } to be the whole space, i.e., problem (P1) is approximately controllable.

(P2) Determine function u under the additional condition

μ being a scalar finite Borel measure on [0,T].  

 Jaime Edilberto MUNÒZ RIVERA,  National Laboratory for Scientific
Computation, Rio de Janeiro, BRAZIL
 
  Qualitative properties to magnetoelastic plates

We consider the magneto elastic plate model. We prove that the model is not of analytic type and also that there is no exponential stability. But under suitable conditions we show that the solution decays polynomially to zero.
On the other hand by introducing a viscosity in "some direction" we show that the resulting system is
of analytic type.

 Luciano PANDOLFI, Politecnico di Torino, ITALY

 Boundary flux identification for thermal systems with memory

    In this paper (joint work with A. Favini) we consider a linear version of the heat equation
with memory, affected by a distributed input signal u. The signal is finite dimensional and
affects the system through a known source operator.
We assume that it is possible to observe the flux of heat at the boundary of the system. In
fact we assume that we have as many independents measurements as the number of
independent inputs. We present an algorithm which permits the approximate reconstruction
of the input signal in “real time”; i.e. an estimate of the value of u at time t is constructed
using measures taken at previous times.
The algorithm depends on a penalization parameter α and we prove L²(0,T) convergence of
the estimate to u(t) when α → 0+, for every T > 0. The proof requires suitable (natural)
assumptions on the system and uses frequency domain techniques.

 Paolo PODIO GUIDUGLI, Università di Roma Tor Vergata, ITALY

 Models of phase segregation with and without diffusion and temperature effects

    Three new models of phase segregation will be discussed, based on two  balance
equations for, respectively, microforces and microenergy, and  on fairly nonstandard but
thermodynamically compatible constitutive  choices. The first model is akin to the classical
Allen-Cahn (for  others, Ginzburg-Landau) equation for an order parameter whose  space-
time evolution may lead to phase segregation by atomic  rearrangement in the absence of
diffusion; the second model  generalizes the Cahn-Hilliard equation, that takes diffusion into 
account; the third model is a version of the first that allows for  temperature effects. The
main difference with the classical A-C and  C-H mathematical models is that, instead of
dealing with one equation  for one unknown, we deal with three systems of equations, for
the  order parameter and the chemical potential in the first two cases, for  the order
parameter, the chemical potential, and the temperature, in  the third. The study of those
systems has been made the subject of an  ongoing joint research program with P. Colli, G.
Gilardi, and J.  Sprekels. My talk will concentrate on the derivation of the  mathematical
models; at this meeting, a companion talk by P. Colli  will delineate peculiar features of their
mathematical analysis.

 Colin ROGERS, Hong Kong Polytechnic University, CHINA

 On Ermakov Structure in 2+1-Dimensional Magnetogasdynamics

    An elliptic vortex-type ansatz introduced into a 2+1-dimensional system governing
rotating homentropic magnetogasdynamics with a parabolic gas law is shown to lead to an
eight-dimensional nonlinear dynamical system which admits exact analytical solution in
terms of an elliptic integral representation. A novel magnetogasdynamic analogue of the
pulsrodon of shallow water f-plane theory is isolated thereby. In the case of a purely
transverse magnetic field, the general dynamical system is shown to have underlying
Hamiltonian structure of Ermakov-type. Pulsrodon-type solutions may again be
constructed.

 Maria Agostina VIVALDI, SAPIENZA Università di Roma, ITALY

 Homogenization models with fractal Strings

   Geometry and energy contribute to define most physical characteristics of a
body. Changes in geometry and energy of a body subjected to complicated
modifications may follow diferent patterns. Geometry may undergo disruptive
alterations while the energy behaviour displays more consevative features.
In this talk  some illustrative examples are described. The approach is of
variational nature and relies on Hilbert space convergence of quadratic energy
forms.