Concentration Phenomena in Nonlinear PDEs and Elasto-plasticity Theory

非线性偏微分方程中的集中现象和弹塑性理论

• 批准号: EP/Z000297/1
• 负责人: Filip Rindler
• 金额: $ 221.82万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FZ000297%2F1
• 关键词: Concentration; Phenomena; Nonlinear; PDEs; Elasto

Numerous important open problems in Analysis, from such diverse areas as the compensated compactness theory of PDEs, the shape optimization of elastic materials, or the transport of geometric structures like vortex filaments in fluids and dislocation lines in crystalline materials, have at their core deep questions about "diffusely concentrating" sequences of maps, measures, or currents. Prototypical sequences of this kind display an increasing number of thin and repetitive structures as the typical length scale goes to zero. The challenge is to understand the asymptotic configurations that this "network" of structures can exhibit, which are usually highly restricted by the presence of a (linear) PDE constraint like divergence-freeness. Despite much progress in the related study of singularities in measures over the last decade, diffuse concentrations have remained shrouded in mystery. Building on the recent groundbreaking advances by the PI at the intersection of PDE Theory, Geometric Measure Theory, and the Calculus of Variations, the CONCENTRATE proposal aims at transformative progress in this highly active and rapidly evolving research area. As an application and guiding light to the theoretical investigation, the project will furthermore tackle the micro-to-macro homogenization of large-strain elasto-plasticity driven by the motion of dislocations, thus furnishing a rigorous and realistic model of plastic deformations. Often referred to as the "Holy Grail" of plasticity theory, such a homogenization result has so far proved elusive, despite much collective effort, since it requires a fine understanding of the diffuse concentrations encountered when passing from discrete dislocation lines to fields of dislocations. The PI's research leadership in these areas makes him uniquely placed to tackle the ambitious goals of this proposal through the development of novel mathematical tools and the solution of long-standing conjectures of both pure and applied character.

分析中的许多重要的开放问题，来自不同领域，如偏微分方程的补偿致密性理论、弹性材料的形状优化或几何结构的传输，如流体中的涡丝和晶体材料中的位错线，其核心都是深层次的问题关于地图、测量值或电流的“分散集中”序列。随着典型长度尺度变为零，这种原型序列显示出越来越多的细长和重复结构。挑战在于理解这种结构“网络”可以表现出的渐近配置，这些配置通常受到（线性）偏微分方程约束（如无散度）的存在的高度限制。尽管过去十年中测量奇点的相关研究取得了很大进展，但弥散浓度仍然笼罩在神秘之中。基于 PI 最近在偏微分方程理论、几何测度理论和变分演算交叉领域取得的突破性进展，CONCENTRATE 提案旨在在这个高度活跃和快速发展的研究领域取得变革性进展。作为理论研究的应用和指导，该项目还将进一步解决由位错运动驱动的大应变弹塑性的微观到宏观均匀化问题，从而提供严格而现实的塑性变形模型。这种均质化结果通常被称为塑性理论的“圣杯”，尽管付出了很多集体努力，但迄今为止仍难以实现，因为它需要对从离散位错线传递到位错场时遇到的扩散浓度有很好的理解。 PI 在这些领域的研究领导地位使他能够通过开发新颖的数学工具和解决长期存在的纯粹和应用猜想来实现该提案的雄心勃勃的目标。