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》中许多重要的开放性问题，来自不同的领域，如偏微分方程的补偿紧致性理论、弹性材料的形状优化，或流体中的涡旋细丝和晶体材料中的位错线等几何结构的传输，其核心都是关于“扩散集中”序列的地图、测量或电流的深层问题。这类序列的原型显示越来越多的薄和重复的结构，因为典型的长度尺度趋于零。挑战在于理解这种结构“网络”可能呈现的渐近构型，这通常受到（线性）PDE约束（如发散性）的高度限制。尽管在过去的十年中，测量中奇点的相关研究取得了很大的进展，但弥漫浓度仍然笼罩在神秘之中。在PI在PDE理论、几何测量理论和变分法的交叉领域取得突破性进展的基础上，CONCENTRATE的提案旨在在这个高度活跃和快速发展的研究领域取得革命性进展。作为理论研究的应用和指导，本项目将进一步解决位错运动驱动的大应变弹塑性微观到宏观的均匀化问题，从而提供一个严谨、真实的塑性变形模型。通常被称为塑性理论的“圣杯”，这种均质化结果迄今为止被证明是难以捉摸的，尽管有很多集体的努力，因为它需要对从离散位错线到位错场传递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性质的猜想来解决这一提议的雄心勃勃的目标。