Singularities in Nonlinear PDEs

这一建议旨在为从理论上理解非线性偏微分方程解的奇异性建立一个框架，并以此搭建理论数学和应用数学之间的桥梁。在科学和技术领域，奇点通常对应于物理、工程或经济模型的极限行为，因此对于理解其行为至关重要。例如，某些材料(如CuAlNi晶体)将试图通过发展无限精细的内部振荡，即所谓的微结构，来适应指定的边界变形。这种材料有许多重要的应用，例如在形状记忆合金中，这种合金即使变形后仍能记住其形状，一旦加热到一定温度以上，就会恢复形状。自然界中遇到的其他高度振荡的情况是湍流。在现实中，这种振荡的最好的尺度是以低于某个阈值的原子效应的出现为界的，但通常这种原子到连续介质的长度尺度是如此之小，以至于从宏观上我们可以假设频率几乎是无限的，因此，通常的连续介质力学模型达到了它们的建模有效性的边界。特别是，无限快的振荡不能表达为函数，需要切换到更高级的框架。其他的例子是描述损伤和分层的模型。这里，人们想要推断一种材料的行为，这种材料遭受了一些结构损伤或磨损，然而，这在宏观上可能是不可见的。现代技术中的许多工程挑战都可以归因于这种效应(例如，最近广为人知的新空客A380翼肋破裂的案例)。人们对PDE中发生的奇点的兴趣从未像现在这样强烈。由于如此多的技术应用依赖于对奇点的可预测性和洞察力，因此必须推动对潜在机制的更深入理解。目前的知识状况并不令人满意，许多影响只被很好地理解。在这项提案中概述的研究中，我们的目标是提供一套工具来解决奇点理论中一些最紧迫的问题，并将推动对导致奇点形成的潜在影响的更好理解。从技术上讲，我们将基于最近开发的一种工具，即所谓的“微局部致密性形式”，它允许以统一的方式捕捉和研究各种奇异效应。在项目过程中，我们将具体考虑以下问题：-我们将考虑双曲守恒定律中的奇异性，并致力于在该领域的重要开放问题上取得进展。-我们将研究如何有效地描述微观结构的层次结构，并将这种描述用于均匀化理论和损伤和分层过程的建模。我们还将探索这些新结果对变分中一些基本问题(例如Morrey猜想)的影响。-我们将进一步从理论上理解补偿紧性作为偏微分方程组分析工具的作用。最后，我们将与工程师合作，考虑对现实世界应用的影响，并将利用在这项工作过程中获得的理论见解来提高对科学、技术和工程应用中奇点的实际理解。

This proposal aims to develop a framework for the theoretical understanding of singularities in solutions to nonlinear partial differential equations and as such bridges theoretical and applied mathematics. In science and technology, singularities often correspond to the limiting behaviour of a physics, engineering or economics model and hence are of paramount importance in understanding its behaviour. For example, certain materials (such as CuAlNi crystals) will try to accommodate prescribed boundary deformations by developing infinitely fine internal oscillations, so-called microstructure. Such materials have many important applications, for example in shape-memory alloys, which remember their shape even after being deformed, and will return to it once they are heated above a certain temperature. Other highly oscillatory situations encountered in nature are turbulent flows. In reality, the finest scale for such oscillations is bounded by the emergence of atomistic effects below a certain threshold, but often this atomistic-to-continuum length scale is so small that macroscopically we can assume that the frequency is nearly infinite and thus, the usual continuum mechanics models hit their boundary of modelling validity. In particular, infinitely fast oscillations are not expressible as functions and one needs to switch to a more advanced framework. Other examples are models describing damage and delamination. Here, one wants to infer the behaviour of a material that has suffered some structural damage or attrition, which, however, might not be macroscopically visible. Many engineering challenges in modern technologies can be attributed to such effects (for example in the recent widely-publicised case of cracking in the wing ribs of the new Airbus A380).Interest in singularities occurring in PDEs has never been greater. As so many technological applications depend on predictability and insight into singularities, it is imperative to push towards a greater understanding of the underlying mechanisms. The state of knowledge at the moment is unsatisfactory and many effects are only poorly understood.In the research outlined in this proposal we aim to provide a set of tools to tackle some of the most pressing problems in the theory of singularities and will push for a greater understanding of the underlying effects causing the formation of singularities. Technically, we will base the development on a recently developed tool, the so-called "microlocal compactness form" that allows to capture and investigate a variety of singular effects in a unified way.In the course of the project we will specifically consider the following questions:- We will consider singularities in hyperbolic conservation laws and aim to make progress on the important open questions in the field.- We will investigate how the hierarchy of microstructure can be efficiently described and this description harnessed in homogenisation theory and the modelling of damage and delamination processes. We will also explore the ramifications of such new results on some fundamental questions in the Calculus of Variations (e.g. Morrey's conjecture).- We will further the theoretical understanding of compensated compactness as a tool in the analysis of PDEs.Finally, in collaboration with engineers, we will consider the implications for real-world applications and will use the theoretical insights gained in the course of this work to improve the practical understanding of singularities in applications of science, technology, and engineering.