CAREER: Thin shells - problems in nonlinear elasticity and fluid dynamics

在这个项目中，一个统一的主题是当区域接近极限的“退化”区域时，给定方程的渐近行为。所考虑的问题是非线性变分或偏微分方程，所讨论的退化可以采取各种形式：丢失维度、丢失正则性或无界性。第一组问题与非线性弹性数学理论有关，该理论研究三维弹性体的大机械变形。研究人员承担了一个长期的研究计划，范围包括：通过Gamma收敛的方法推导低维壳体理论，了解它们与中面几何的联系，分析曲面的(无穷小)等距以及刚度对所得理论的影响，研究给定壳体在压缩下的屈曲和起泡等非线性现象(一个应用与植物生长有关)。这个项目中的第二组问题与流体动力学有关。当边界行为退化时，在极限中考虑了各种结构和尺度的边界不规则性。其他问题涉及无界通道中的流动前锋燃烧，以及三维薄壳在Navier边界条件下具有大初始数据的解的动力学。由于各种几何形状的弹性薄(或“简并”)物体在物理世界中无处不在，精确理解支配它们平衡的规律有许多潜在的应用。例如，许多正在生长的组织(叶子、花或海洋无脊椎动物)在自由生长过程中表现出复杂的结构，人们希望用人工手段来繁殖它们。在弹性力学的数学理论中，一个长期存在的问题是从全三维物体的非线性理论出发，精确地预测这种低维物体的理论。在平板上，最近的一项努力导致了对这类理论层级的严格证明，这取决于所施加的力的大小，并导致拉伸、皱缩、弯曲或这些理论的组合。对于壳(当中间曲面是曲面时)，尽管它们的特殊推广在文献和工程应用中得到了广泛的应用，但从数学的角度来看，人们所知的要少得多。研究者确定了固体和流体动力学连续力学中的几个非线性问题，自然地提出了特定的简并域，目的是基于一般原理严格理解系统的行为。该程序的核心是变分法、微分方程式、几何学、材料科学、流体力学、数值分析甚至生物学之间有趣的联系。它们有潜力在结构力学等方面提供有用的观察，同时整合向不同教育水平的更广泛社区公布科学成果的目标。

LewickaDMS-0846996 A unifying theme in this project is the asymptotic behaviorof given equations as the domain approaches a "degenerate" regionin the limit. The considered problems are nonlinear variationalor partial differential equations, and the degeneracy in questioncan take various forms: loss of dimension, loss of regularity, orunboundedness. The first set of questions relates to themathematical theory of nonlinear elasticity, which studies largemechanical deformations of three-dimensional elastic bodies. Theinvestigator undertakes a long-term research program, with thescope of: deriving lower-dimensional shell theories through themethods of Gamma-convergence and understanding their connectionswith the geometry of the mid-surface, analyzing the(infinitesimal) isometries of surfaces and the effects ofrigidity on the derived theories, studying nonlinear phenomenasuch as buckling and blistering for a given shell undercompression (one application is related to plant growth). Thesecond set of questions in this project relates to fluiddynamics. Boundary irregularities of various structures andscales are considered in the limit when the boundary behaviorbecomes degenerate. Other problems concern traveling fronts incombustion in unbounded channels, and the dynamics of solutionswith large initial data under the Navier boundary conditions inthin three-dimensional shells. Because elastic thin (or otherwise "degenerate") objects ofvarious geometries are ubiquitous in the physical world, theprecise understanding of laws governing their equilibria has manypotential applications. For example, many growing tissues(leaves, flowers, or marine invertebrates) exhibit complicatedconfigurations during their free growth and one would like toreproduce them with man-made means. A related long-standingproblem in the mathematical theory of elasticity is to rigorouslypredict theories of such lower-dimensional objects starting fromthe nonlinear theory of full three-dimensional objects. Forplates, a very recent effort has lead to rigorous justificationof a hierarchy of such theories, depending on the magnitude ofthe applied forces and resulting in stretching, crumpling,bending, or a combination of these. For shells (when themid-surface is curved), despite extensive use of their ad hocgeneralizations in the literature and engineering applications,much less is known from the mathematical point of view. Theinvestigator identifies several nonlinear problems in continuummechanics of solids and fluid dynamics, naturally posed inspecific degenerate domains, with the intention of rigorouslyunderstanding the behavior of the system based on generalprinciples. At the heart of the program are interestingconnections between calculus of variations, differentialequations, geometry, material science, fluid dynamics, numericalanalysis, and even biology. They have a potential to deliveruseful observations in e.g. structural mechanics, whileintegrating the goal of exposing scientific results to a broadercommunity at various education levels.