Singularities in Nonlinear PDEs

非线性偏微分方程中的奇点

• 批准号: EP/L018934/1
• 负责人: Filip Rindler
• 金额: $ 33.47万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL018934%2F1
• 关键词: Singularities; 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.

该提案旨在开发一个框架，用于理论上理解非线性偏微分方程解中的奇点，从而连接理论和应用数学。在科学和技术中，奇点通常对应于物理、工程或经济模型的限制行为，因此对于理解其行为至关重要。例如，某些材料（例如 CuAlNi 晶体）将尝试通过发展无限精细的内部振荡（即所谓的微观结构）来适应规定的边界变形。这种材料有许多重要的应用，例如形状记忆合金，即使在变形后也会记住它们的形状，并且一旦加热到一定温度以上就会恢复到原来的形状。自然界中遇到的其他高度振荡的情况是湍流。实际上，这种振荡的最精细尺度受到低于某个阈值的原子效应的出现的限制，但通常这种原子到连续体的长度尺度非常小，以至于宏观上我们可以假设频率几乎是无限的，因此，通常的连续体力学模型达到了建模有效性的边界。特别是，无限快的振荡无法表达为函数，需要切换到更高级的框架。其他示例是描述损坏和分层的模型。在这里，人们想要推断一种遭受了一些结构损坏或磨损的材料的行为，然而，这可能在宏观上不可见。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型空客 A380 机翼肋骨破裂的案例）。人们对偏微分方程中出现的奇点的兴趣从未如此强烈。由于许多技术应用依赖于可预测性和对奇点的洞察，因此必须推动对底层机制的更深入理解。目前的知识状况并不令人满意，对许多效应知之甚少。在本提案概述的研究中，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效应有更深入的了解。从技术上讲，我们将基于最近开发的工具进行开发，即所谓的“微局域紧致形式”，该工具允许以统一的方式捕获和研究各种奇异效应。在该项目的过程中，我们将特别考虑以下问题：-我们将考虑双曲守恒定律中的奇异性，并旨在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以及如何利用这种描述 均质化理论以及损伤和分层过程的建模。我们还将探讨这些新结果对变分法中一些基本问题的影响（例如莫里猜想）。 - 我们将进一步加深对补偿紧性作为偏微分方程分析工具的理论理解。最后，与工程师合作，我们将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补偿紧性的实际理解 科学、技术和工程应用中的奇点。

项目成果

Piecewise affine approximations for functions of bounded variation

有界变分函数的分段仿射近似

• DOI: 10.1007/s00211-015-0721-x
• 发表时间: 2015
• 期刊: Numerische Mathematik
• 影响因子: 2.1
• 作者: Kristensen J

ORIENTATION-PRESERVING YOUNG MEASURES

保持青少年定向的措施

• DOI: 10.1093/qmath/haw019
• 发表时间: 2016
• 期刊: The Quarterly Journal of Mathematics
• 作者: Koumatos K

On the structure of measures constrained by linear PDEs

关于线性偏微分方程约束的测度结构

• 发表时间: 2018
• 期刊: Proceedings of the International Congress of Mathematicians, ICM 2018
• 作者: De Philippis G.

Characterization of Generalized Young Measures Generated by Symmetric Gradients

对称梯度生成的广义年轻测度的表征

• DOI: 10.1007/s00205-017-1096-1
• 发表时间: 2017
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: De Philippis G

Differential Inclusions and Young Measures Involving Prescribed Jacobians

涉及规定雅可比行列式的微分包含和年轻测度

• DOI: 10.1137/140968860
• 发表时间: 2015
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Koumatos K