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

LewickaDMS-0846996 在这个项目中的一个统一的主题是渐近行为的给定方程作为域接近一个“退化”regionin的限制。 所考虑的问题是非线性变分或偏微分方程，退化问题可以有多种形式：失维、失正则或无界。 第一组问题涉及非线性弹性力学的数学理论，它研究三维弹性体的大变形。 该研究员进行了一项长期的研究计划，其范围是：通过伽马收敛方法推导出低维壳理论，并理解它们与中表面几何形状的联系，分析表面的（无穷小）等距和刚度对推导出的理论的影响，研究非线性现象，例如给定壳在压缩下的屈曲和起泡（一个应用与植物生长有关）。 这个专题的第二组问题与流体力学有关。 当边界性质退化时，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燃烧的行进锋，以及三维薄壳中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（或其他“退化”）物体在物理世界中无处不在，因此精确理解它们的平衡定律具有许多潜在的应用。 例如，许多生长组织（叶、花或海洋无脊椎动物）在自由生长期间表现出复杂的构型，人们希望用人造手段来复制它们。 在弹性力学的数学理论中，一个相关的长期存在的问题是，从三维物体的非线性理论出发，严格预测这种低维物体的理论。 对于板，最近的一项努力已经导致了严格的证明，这种理论的层次结构，这取决于所施加的力的大小，并导致拉伸，起皱，弯曲，或这些组合。 对于壳（当半曲面为曲面时），尽管在文献和工程应用中广泛使用了它们的自适应推广，但从数学观点来看，所知甚少。 研究者确定了固体和流体动力学的连续力学中的几个非线性问题，自然地在特定的退化域中提出，目的是严格理解基于一般原理的系统行为。 该课程的核心是变分法、微分方程、几何、材料科学、流体动力学、数值分析，甚至生物学之间有趣的联系。 他们有潜力提供有用的观察，如结构力学，同时整合的目标，暴露科学成果，以更广泛的社区在不同的教育水平。

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.