Singular limits with geometric effects

该奖项支持首席研究员正在进行的研究计划，对工程和应用科学中出现的一些问题进行数学分析，特别是涉及薄域和薄弹性结构中的流体。 首席研究员将联合收割机技术从几个领域的数学研究某些工程设计问题。 这些问题是由他们的“奇异极限”上下文和事实，他们的解决方案是调整到一个可变的几何设置或后验获得的几何约束。 将要解决的主题分为四类：（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）生长体形状的建模和分析（如在生物学的各种背景下），（3）弹性预应变材料的刚性和柔性（例如，由于不均匀生长或膨胀或收缩而引起的），（4）弹塑性材料中的能量标度体系（在超过某一点时，变形是不可逆的）。将研究下列分析技术：（1）利用相对熵和Korn不等式的标度以及薄区域上的共形Korn不等式对可压缩Navier-Stokes方程进行降维和均匀化; Bressan，该方程耦合了形态生成细胞运输的偏微分方程（3）在Monge-Ampere约束下的预应变弹性变分模型，其中对最小解的结构没有任何先验假设（4）在弹塑性问题中通过Gamma收敛降维。

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.