- Francesca Bucci, Universita' di Firenze
- Daniel Toundykov, University of Nebraska at Lincoln
"Mathematical Modeling of Physical Phenomena"
It is our intention that the proposed Minisymposium (MS) should provide
a platform for those Applied Mathematicians and Control Theorists who
have an extensive experience in the mathematical modeling of various
physical and engineering henomena. Examples of such phenomena
would include fluid-structure and structural acoustic interactions,
quantum mechanical systems, communications networks, traffic
flows, wind storm and other climatological occurrences.
The respective modeling processes would eventually culminate in systems
of linear or nonlinear partial differential equation (PDE) models.
Moreover, it is conceivable that one or other of these PDE systems
would be under the action of so-called control terms, of either open
loop or feedback loop type.
The purpose of such control terms - typically placed on the boundary or
localized within the interior of the geometry - would be to influence
the solution of the controlled PDE dynamics in some pre-assigned
manner. (Such control actions would mathematically realize some
particular engineering control implementation.)
Of course, the particular physical/engineering circumstance under
consideration, as well as any associated control strategy, will dictate
the PDE model which would be analyzed. For example, we anticipate that
some of our speakers will showcase their expertise in the modeling and
analysis of fluid-structure phenomena.
Quite typically, the PDE's which describe such interactions of fluid
and structure comprise systems of elasticity (the structural hyperbolic
PDE component), coupled to a Navier-Stokes or Stokes dynamics - so in
the linear case, the corresponding fluid PDE component evinces
parabolic dynamics. Each respective hyperbolic and parabolic PDE
component will evolve within its own respective domain. Assuming the
structural displacements are small, relative to the scale of the
geometry, then typically the coupling between fluid and structure will
occur across a fixed boundary interface.
But in the most realistic modeling scenario for said fluid-structure
interactions, the boundary interface between the two distinct dynamics
would properly be moving. With such complications in mind, our MS will
also feature speakers who have a demonstrated insight into
mathematically capturing the complex phenomena of moving boundaries
between disparate PDE dynamics. We anticipate that these speakers will
apply, in some nonstandard way, techniques from the field of intrinsic
and Riemannian geometry: Such techniques have been successfully
employed in the recent past so as to provide physically relevant PDE
Conceivably, our speakers will discuss their geometric
methodology in the context of modeling fluid-structure phenomena, with
moving boundary interface.
As an accompaniment to coupled fluid-structure modeling issues, some of
our speakers might also address the mathematical modeling of those
physical systems which involve either electrically conductive fluids,
or else elastic structures that respond to magnetic forces. Such
phenomena give rise to PDE's of so-called magneto-hydrodynamics and
magneto-elasticity; these PDE systems might also account for further
interactions between the fluid and structural components, or for
additional forces; e.g., thermal e¤ects. Such magneto-hydrodynamical
and magneto-elastic PDE's are generally highly complex and nonlinear.
In order to obtain the most physically reliable - and yet also
analytically tractable - set of governing PDE equations, a careful and
nuanced modeling process must often be invoked.
Moreover, some of our invited speakers in our proposed MS could
conceivably discuss the modeling, analysis and control of traffic
flows. Such phenomena can be placed rigorously within the framework of
hyperbolic conservation laws. Subsequently, mathematical control
problems can be posed and considered for these nonlinear dynamics, with
an eventual view towards the implementation of physically realizable
In the course of discussing her or his recent research in the
mathematical modeling of physical phenomena, we anticipate that each of
our speakers will evince a far-ranging knowledge of topics which are
decidedly distinct from the mathematical and control theory sciences.
Such interdisciplinary training is generally indispensible here, in
order to provide a complete and useful solution to the particular
mathematical modeling problem under consideration: Indeed, the careful
derivation of the PDE model - controlled or uncontrolled - which is to
govern the physical phenomenon, the choice of mathematical realization
of any associated control action, the continuous and/or numerical
solution of the said PDE system, the interpretation of the analytical
results within the context of the original physical phenomenon being
studied; all these steps require, on the part of the mathematical
modeler, an exhaustive and in depth understanding of the science
the physical control application.
Moreover, it is also quite conceivable that a given speaker in our
proposed MS will provide, in his or her talk, details of numerical or
theoretical results, in support of the modeling research being