The modelling, analysis and optimization of natural processes characterized by a confluence of partial differential equations of differing type

自然科学和工程领域的许多现代物理过程都是由于具有不同性质和行为的材料接触而产生的独特环境，并且允许产生的边界随时间演变。这一研究项目的主要目标是发展对支撑这类问题的数学结构的理解，以便对相关过程进行某种控制，或实现某种形式的最优性。尽管潜在机制的细节在很大程度上取决于特定的过程，但特定区域内的进化通常用偏微分方程（pde）来建模。这一提议的数学结构的不同之处在于，不同的区域与不同类型的pde相关联，这使许多经典理论失效。换句话说，这些被称为混合型的问题的结构特征是分解成以pde类型（抛物线型、双曲型或椭圆型）为特征的不相交区域，不同的区域有不同的类型。这与凝固/熔化过程中更标准的自由边界问题形成鲜明对比，例如，两个抛物线方程被凝固前沿分开。对三种不同类型（抛物线型、双曲型、椭圆型）中的每一种解的单独研究都有丰富的历史，并且已经开发了许多数学工具来研究各种解的性质。在取得这些进展之后，本建议的科学目标是对混合类型的问题发起类似的系统攻击。过去对这些方程的研究进展是跨学科的，结合了实验、计算和分析，既渐进又严格。通过研究一组可处理的混合类型问题的数学技术、思想和方法，可以识别和开发适用于这一重要类别的新思想和新技术，这是许多创新自然过程的核心。在特定应用的背景下解释严格分析的深刻价值，通过提供一个框架来比较和对比表面上不同的物理过程，但共享相同的底层数学结构，在历史上导致了快速的进步。为了鼓励这种思想和技术的交叉授粉，本研究将涉及许多新的相关过程，这些过程只被部分理解，因此，没有已知的可控性和最优性的必要和充分条件。一个实质性的，但远非详尽的清单包括：i)对比生长晶体的弹性/塑性行为与隧道穿过土壤的过程；ii)闪速烧结，通过大电流使陶瓷粉末变得超塑性和致密化；iii)激光烧蚀和iv)激光抛光，其中热方程（抛物线）与气体动力学方程（烧蚀）或（弹性）热弹性方程（抛光）的（双曲）系统密切耦合。在许多这些过程中，特别感兴趣的是可控性和最佳的功率使用。正如工业本身的参与所证明的那样，这些过程对加拿大工业具有当代意义，不仅需要对物理过程有深刻的理解，而且需要对混合类型的非线性过程有深刻的理解。在推动数学研究前沿的同时，这项研究对工业部门也很重要，这些部门受益于这项创新（节约成本，可控性），通过影响加拿大的工业，使加拿大受益。

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.