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Details of Grant 

EPSRC Reference: EP/L018934/1
Title: Singularities in Nonlinear PDEs
Principal Investigator: Rindler, Dr F
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Autonomous University of Madrid University of Cambridge University of Oxford
Weierstrass Institute for Applied Analys
Department: Mathematics
Organisation: University of Warwick
Scheme: EPSRC Fellowship
Starts: 01 October 2014 Ends: 31 December 2017 Value (£): 262,684
EPSRC Research Topic Classifications:
Continuum Mechanics Mathematical Analysis
Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
21 Jan 2014 EPSRC Mathematics Interviews - January 2014 Announced
27 Nov 2013 Mathematics Prioritisation Panel Meeting Nov 2013 Announced
Summary on Grant Application Form
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.
Key Findings
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Organisation Website: http://www.warwick.ac.uk