Mathematical models of some diffusive processes whith free boundaries

Abstract

These notes have been inspired by a series of lectures I have taught during the last years at the Scuola Normale Superiore in Pisa. The lectures were addressed to the students of the Ph.D. Programme "Mathematics for Industrial Technologies". Their general purpose was to provide several examples of how a mathematical model for a given process can be formulated, starting from the raw material, i.e. some basic physical information. Thus particular attention was devoted to the modelling stage, still keeping the necessary level of mathematical rigour. Topics like rescaling, usually not a main concern for mathematicians, find here some emphasis to point out that solving real world problems requires quite often more than the knowledge of fundamental theorems. Not only one has to understand the physical (or biological, etc.) nature of the problem, but even once a sensible set of equations is obtained, then practical questions arise that are not in mathematical books; for instance: how to understand if some effects are dominant and some are negligible. This is normally crucial, because modelling is in many cases the result of a compromise and it is very important to know the relative weight of the various ingredients entering the final equations. Also it may happen that the process can have concurrent phenomena taking place at time and space scales very different from each other and that it goes through several stages in which the relative importance of the various simultaneous mechanisms changes, sometimes producing a radical modification of the mathematical structure of the model. All these circumstances contribute to make the technomathematician (so to speak) a very peculiar character that needs also a very peculiar training. This motivates the many digressions the reader will find. In this notes I have selected some topics having diffusion as a common denominator. Even more specifically, for obvious reasons of space I further restricted my attention to problems exhibiting free boundaries, which occur with great frequency in a variety of applications. Diffusion is ubiquitous in nature and plays a fundamental role in innumerable processes quite relevant to many applied sciences and to industrial technologies. Diffusion is also responsible for evolution of biological organisms, since it intervenes in transmitting the chemical signals regulating e.g. growth mechanism and the formation of coloured patterns in animals coat.