- 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 shell models.

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 control laws.

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 underlying

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 presented.

>>>Program