Singular limits with geometric effects

具有几何效应的奇异极限

• 批准号: 1613153
• 负责人: Marta Lewicka
• 金额: $ 28万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613153&HistoricalAwards=false
• 关键词: Singular; limits; geometric; effects

This award supports the ongoing research program of the Principal Investigator on the mathematical analysis of some problems arising in engineering and applied science, in particular involving fluids in thin domains and thin elastic structures. The Principal Investigator will combine techniques from several areas of mathematics to study certain engineering design problems. These problems are linked by their "singular limits" context and the fact that their solutions are tuned to a variable geometry setting or to a-posteriori obtained geometrical constraints. The topics that will be tackled fall into four categories: (1) dimension reduction in fluid dynamics (e.g., determining effective two-dimensional models for the behavior of a fluid in a thin domain), (2) modeling and analysis of the shape of a growing body (as in various contexts in biology), (3) rigidity and flexibility of elastic prestrained materials (as caused, for example, by inhomogeneous growth or swelling or shrinkage), (4) energy scaling regimes in elasto-plastic materials (for which deformations are irreversible beyond a certain point).The following analytical techniques will be investigated: (1) dimension reduction and homogenization of compressible Navier-Stokes equations using relative entropy and scaling of the Korn and the conformal Korn inequalities in thin domains, (2) a novel "controlled growth" model, due to A. Bressan, that couples the partial differential equation for the transport of morphogen-producing cells (in cellular biology) with two constrained minimization problems for the morphogen concentration and the growth velocity, (3) variational models in prestrained elasticity in the presence of the Monge-Ampere constraint without any a-priori assumption on the structure of its minimizers (such as convexity, higher regularity, or boundary conditions) studied using convex integration, (4) dimension reduction via Gamma-convergence in the context of elasto-plasticity.

该奖项支持首席研究员对工程和应用科学中出现的一些问题进行数学分析的持续研究项目，特别是涉及薄域和薄弹性结构中的流体。首席研究员将结合几个数学领域的技术来研究某些工程设计问题。这些问题是由它们的“奇异极限”上下文和它们的解决方案被调整到可变几何设置或后验获得的几何约束这一事实联系在一起的。将讨论的主题分为四类：(1)流体动力学中的降维（例如，确定薄域中流体行为的有效二维模型），(2)对生长体形状的建模和分析（如在生物学的各种环境中），(3)弹性预紧材料的刚性和柔韧性（例如，由不均匀生长或膨胀或收缩引起），(4)弹塑性材料中的能量缩放机制（变形超过某一点是不可逆的）。以下分析技术将被研究：(1)利用相对熵和Korn和保形Korn不等式在薄域内的尺度对可压缩Navier-Stokes方程进行降维和均匀化；(2)a . Bressan提出的一种新的“受控生长”模型，该模型将（细胞生物学中）产生形态细胞的运输偏微分方程与形态细胞浓度和生长速度的两个约束最小化问题耦合在一起；(3)使用凸积分研究了存在蒙日-安培约束的预约束弹性变分模型，该模型对其最小值的结构（如凸性、高正则性或边界条件）没有任何先验假设；(4)在弹塑性背景下通过伽马收敛进行降维。