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，确定薄域中流体行为的有效二维模型），（2）对形状进行建模和分析增长的身体（如生物学中的各种情况），（3）具有弹性的材料的刚性和柔韧性（例如，由于不均匀的生长或肿胀或收缩而引起的），（4）弹性塑料中的能量缩放机制（对于以下的变形是不可逆的。 2）由于A. bressan而导致的一种新型的“受控生长”模型，该模型融合了偏微分方程，用于运输形态学的细胞（在细胞生物学中），并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Monge-Ampere约束的存在下，可预言弹性的变异模型，而没有任何A-Priori假设其最小化器的结构（例如凸度，较高的规律性或边界条件），使用凸集成，（4）维度降低。在弹性塑性的背景下通过伽马连接。