Singularities in Nonlinear PDEs

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.

As science and technology in the UK depend upon a sound theoretical foundation to remain at the cutting-edge of progress, I anticipate the research outlined within this proposal to have far-reaching and wide-ranging impact. First and foremost, this is a mathematics proposal and as such falls within the realm of basic research. However, it is the aim of the research described here to further our understanding of singularities in nonlinear PDEs occurring in science, technology, and engineering. Thus, in the medium to long term the proposal also directly relates to important industrial applications, for example in the simulation of turbomachinery, which is highly relevant in the aerospace and defense sectors.

The main beneficiaries of this research will be:
- The academic disciplines of mathematics and engineering, mostly through the publication of high-quality research output in the relevant journals.
- Research-oriented industries, such as aerospace and defense, by new contributions to the theoretical foundations of the mathematics involved in their activities, and also through more concrete applications facilitated by industrial contacts.
- The wider public through the increased economic competitiveness of the UK stemming from a strong theoretical foundation of the UK's science landscape.
- Finally, my own academic and professional development would greatly benefit from the focused research activity and the organisational aspects of this proposal; this would allow me to take the next steps on the path to becoming a leading permanent researcher in Mathematical Analysis.

All resulting publications will be available on my website together with additional material such as presentations, short explanatory notes, teaching materials, and links to further resources.

There are two institutions which will contribute a direct link to applications: First, the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) in Berlin, Germany, conducts project-oriented applied research into real-world questions from economics, science and technology, and I plan to use my contacts with this institution to shape the applied side of the present research.

Second, I have been invited to spend an extended amount of time at the Whittle Laboratory within the engineering department of Cambridge University. I intend to investigate singularities occurring in the air flows through turbofan jet engines there.

While advanced mathematics forms the foundation of much of recent technological advances, the general public's understanding of mathematics is still rather limited. Part of the reason for this is that theoretical mathematics is difficult to explain to a lay audience without giving meaningful applications that "everyone" can understand. Since this proposal has both a theoretical and an applied side, it is tailor-made for public engagement activities: Starting from the exciting applications, for example in the aerospace industry, one can explain how advanced mathematics is necessary to understand and predict the fundamental processes.

Concretely, I plan to participate in the annual Warwick Mathematics Open Days, explaining these applications to prospective students and the wider public. Also, I intend to set up a website explaining some of the research outcomes of this proposal in a way that is understandable to an interested lay audience.