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.

分析中的许多重要开放问题，来自诸如PDE的补偿紧凑理论，弹性材料的形状优化或结晶材料中流体和错位线中的涡流丝等几何结构的运输，在其核心深处有关“扩散浓缩”的核心问题。随着典型的长度尺度为零，此类原型序列显示出越来越多的薄和重复结构。面临的挑战是了解该结构“网络”可以表现出的渐近配置，通常受（线性）PDE约束（如发散范围）的存在很大程度上受到限制。尽管在过去十年中措施的奇异性研究方面取得了很大进展，但弥漫性浓度仍然笼罩在神秘状态。 PI在PDE理论的交集，几何测量理论和变化的计算的基础上，基于PI的最新开创性进步，浓缩提案旨在在这个高度活跃且迅速发展的研究领域的变革性进步。作为对理论研究的应用和指导光，该项目将通过位错运动驱动的大型弹性塑性性的微型弹性塑性性的微质量均匀化，从而提供了严格且现实的塑性变形模型。迄今为止，这种均质化结果通常被称为可塑性理论的“圣杯”，尽管大量的集体努力，但事实证明，这是难以捉摸的，因为它需要对从离散脱位线到位错领域的弥散浓度有充分的理解。 PI在这些领域的研究领导使他通过开发新颖的数学工具以及解决纯粹和应用特征的长期猜想的解决方案，使他独一无二地解决该提案的雄心勃勃的目标。