Grant: Discovery Grants Program - Individual

Many modern physical processes in the natural sciences and engineering fields are the result of the unique set of circumstances that occurs when materials with differing properties and behaviours are brought into contact, and the resulting boundary is allowed to evolve in time. The primary objective of this program of research is to develop an understanding of the mathematical structure that underpins this class of problems in the effort to either impart some control over the associated processes, or achieve some form of optimality. Although the specifics of the underlying mechanisms depend strongly on the particular process, the evolution within a particular region is typically modelled with a partial differential equation (pde). What distinguishes the mathematical structure of interest of this proposal is that different regions are associated with different types of pdes, invalidating much of the classical theory. In other words, what characterizes the structure of these problems, known as mixed-type, is a decomposition into disjoint regions characterized by the type of pde (parabolic, hyperbolic, or elliptic), with different types in different regions. This contrasts sharply with more standard free boundary problems in solidification/melting processes where, for example, two parabolic equations are separated by a solidification front. There is a rich history of the separate study of pdes of each of the three distinct types (parabolic, hyperbolic, elliptic) and many mathematical tools have been developed to study the variety of solution properties. Having made these inroads, the scientific aim of this proposal is to initiate a comparable systematic attack on problems of mixed-type. Past progress in the study of these equations has been interdisciplinary combining experiments, computing and analysis, both asymptotic and rigorous. By examining the mathematical techniques, ideas and approaches to a unified set of tractable mixed-type problems, new ideas and new techniques can be identified and developed that apply to this important class that lies at the core of many innovative natural processes. The profound value of interpreting rigorous analysis within the context of particular applications has historically led to rapid advances by providing a framework to compare and contrast physical processes that are different on the surface, yet share the same underlying mathematical structure. To encourage this cross-pollination of ideas and techniques, this research will touch on many novel related processes that are only partially understood and as a result, no known necessary and sufficient conditions for controllability and optimality have been identified for them. A substantial, yet far from exhaustive list includes: i) Contrasting elastic/plastic behaviour in a growing crystal with the process of tunnelling through soil; ii) Flash sintering in which a ceramic powder becomes superplastic and densified by the passage of a large current; iii) Laser ablation and iv) laser polishing whereby a heat equation (parabolic) is intimately coupled with either a (hyperbolic) system of gas dynamics equations (ablation) or (elastic) thermoelastic equations (polishing). In many of these processes of particular interest is controllability and the optimal use of power. As evidenced by the involvement of the industries themselves, these processes of contemporary interest to Canadian industry and require a deep understanding of not only the physical processes, but also nonlinear pdes of mixed-type. While driving forward the research edge of the mathematics, the research is also important to industrial sectors that benefit from this innovation (cost savings, controllability) with progress benefiting Canada by impacting Canadian industries.