Mathematical Treatment of
Free Boundary Problems

This article has appeared in The Journal of the European Scientific Foundation (ESF Communications nº28, April 1993, pp.18-19) and consists of an Introduction, a tentative List of Problems, an indication of the Methodology and of the Objectives of this ESF Scientific Programme during 1993-1997.

INTRODUCTION

Free boundaries are curves or surfaces with a priori unknown locations which separate geometric regions with different characteristics. An example is the solidification of water, where the location of the free boundary between water and ice is nor given a priori and changes during the process. Free boundaries occur as natural features in the mathematical formulation of a great variety of very important scientific and technological processes. Such problems arise in materials processing (steel casting, crystal growth, epitaxy, superconductivity, etc), in biology (growth of bones and bacteria, etc), in combustion theory and other reaction–diffusion problems, in electrochemistry (electrochemical machining, etching processes, etc) or in fluid flow through porous media.

In spite of the international cooperation already existing, the rapid growth of this topic and the new and difficult mathematical challenges require a continuous combined effort by European FBP specialists in order to maintain their position and intercontinental role in the scientific development of this important area, as well as to incorporate important expertise from Eastern Europe.

The proposal for this Programme was favourably received by the ESRC Working Group on Basic Technical Sciences in 1991 since the applications indicated by the proposers put the subject very well at the interface between engineering and the fundamental and mathematical sciences. It was granted Year Zero status by the ESF General Assembly in 1991, so as to enlarge the number of scientists involved. A workshop was held in Portugal in February 1992 with 30 participants from 15 European countries, from which this first ESF Scientific Programme in the Technical Sciences was formulated.

Historically, the first work on this topic seems due to Lamé and Clapeyron, who in 1831 considered a simple model for the solidification of a liquid sphere. These FBP for the heat diffusion equation, also called Stefan problems, received little attention from mathematicians until recently, when in the sixties the modern functional approach to the theory of nonlinear partial differential equations brought new insight and new methods to their mathematical study.

Over the past two decades development in this area has established this research topic as an important disciplinary position between applied mathematical modelling and several subjects of fundamental mathematics such as partial differential equations, ordinary differential equations, numerical analysis, functional analysis (measure theory, operators and semigroup theory), the calculus of variations and optimal control, complex variable theory and differential geometry.

A LIST OF PROBLEMS

The objective of this Programme is not restricted to any particular class of FBP but to promote their mathematical treatment in order to improve our knowledge of the physical models. The following list of problems, in no way exhaustive, illustrates the potential for interaction between participating laboratories.

Change of Phase

"Stefan type problems" have an enormous number of technical applications in complex problems arising in the steel industry, silicon crystallisation, soil freezing, freezing of foodstuffs, injection moulding, crystallisation of polymers and in other problems in Material Science.

Problems in Continuum Mechanics

In non-Newtonian flows different regimes may coexist: for instance, in a Bingham fluid, rigid motions may arise in the flow. Viscoelastoplastic flows or elastoplastic deformations are examples of problems presenting many open mathematical questions. FBP in solid mechanics also arise in unilateral and contact problems (with or without friction), in damage and cracking of structures, in glueing models, spongy materials, woven structures, internal structures for composite materials with reinforcing, etc. Filtration and pollution problems have, in general, free boundaries, as for example, the dam problem or the saturated/unsaturated flow in porous media. Other fluid dynamics FBP arise in the flow of salt and fresh water in coastal aquifers, entry problems, fluidised beds, slow fluid flow with free surfaces, jets, etc.

Chemical, Diffusion and Combustion Problems

Typical examples of FBP arise in reaction–diffusion, degenerate diffusion and adsorption, percolation, flame fronts, electrochemical machining, etc. An interesting question is the study (appearance, shape, control) of the "dead core" in reaction–diffusion problems. A flame is characterised by a very thin reaction zone in the limit of high activation energy.

Other Free Boundary Problems

Some other FBP are now being studied because of their topicality: they arise in Nuclear Fusion and Global Change in the Earth. The equilibrium of a confined plasma in a Tokamak is described by the well-known Grad–Safranov equation, and the boundary of the region occupied by the plasma (separating it from the vacuum) is a very important free boundary.

METHODOLOGY

The mathematical treatment of FBP includes their modelling, the theoretical analysis of the qualitative behaviour of their models and the corresponding numerical approximation that allows quantitative study and application to concrete examples. A few illustrations where the mathematical treatment is used to study the viability of the model, its well-posedness, the error analysis in the numerical computations and the optimal control of solutions, are:

The Stefan Problem

Its history provides a helpful example of the interplay between free boundary problems and the real world. The Stefan problem is the simplest possible macroscopic model for phase changes in a pure material when they occur either by heat conduction or diffusion.

Contact Problems

Another example is that of predicting the geometry and mechanics of contact between elastic bodies and between solids and liquids. This problem is particularly amenable to mathematical development because it can be formulated in terms of variational inequalities and their numerical approximation is easy on modern computers. This illustrates the great potential for technology transfer.

Superconductivity Modelling

A phase change needs to be modelled and in the early days this was done assuming a smooth free boundary between the normal and superconducting phases. However, a closer inspection of the thermodynamics led to the consideration of certain models of superconductors in which the "free boundary" did not consist of a smooth surface but, rather, broke up into a much more complicated morphology.

OBJECTIVES

The methodology requires specialists at three levels — modelling, theoretical analysis and computation. These are seldom found together in a single laboratory or even in a single country. This Programme will allow contact between specialists at the most relevant levels, mainly through workshops and exchange visits, including younger researchers. There should be significant results, which have a mathematical importance of their own, with impact in different branches of science and technology with possible applications in industry.