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者正在进行的研究计划，该计划在工程和应用科学中引起的某些问题的数学分析，特别是涉及薄型域和薄弹性结构中的流体。 首席研究人员将结合数学几个领域的技术来研究某些工程设计问题。 这些问题与它们的“奇异限制”上下文相关联，其解决方案被调整为可变几何设置或A-posteriori获得的几何约束。 将要解决的主题分为四类：（1）流体动力学的尺寸降低（例如，确定有效的二维模型，以使液体在薄域中的行为），（2）对生长身体的形状进行建模和分析（在生物学中的各种情况下），（3），（3），（3），（3），例如，弹性材料的剧烈性和灵活性（例如，弹性的弹性，或者是弹性的，或者是弹性的，或者是在弹性的，或者是弹性的，或者是在弹性的，或者是弹性的，或者是弹性的， （4）弹性塑料材料中的能量缩放制度（对于某个点之外的变形是不可逆转的）。将研究以下分析技术：（1）使用相对熵的可压缩性熵和使用相对范围的可压缩性 - 纳维尔 - 长方形方程的尺寸和均质化，而Korn和Korn的缩放和较薄的植物中的较薄构造的损失（2）模型，（2）新颖的模型，（2）新颖的模型，（2）新颖的'（2）新颖的'（2）偏微分方程用于形态生产细胞（在细胞生物学中）的运输，具有两个约束的最小化问题，即形态学浓度和生长速度，（3）在蒙奇 - 集中约束的情况下，弹性的变化模型（3）在没有任何A弹性的情况下使用其最小情况下（例如，诸如较小的范围）的结构（或诸如contemizes的结构）（或诸如contemize）的结构（例如，较高的范围）（或诸如contemize）的结构（或诸如范围）的结构（或诸如contemize）的结构。 （4）在弹性塑性的背景下通过伽马连接的尺寸降低。