Growth and patterns: existence, stability, and dynamics

该项目的重点是如何在自然系统中出现的空间周期性模式，以及它们是如何被选择和控制域的增长，几何形状，更一般地说，时空异质性。模式通常指的是规则的、重复的几何结构，在许多不同的科学领域都有很多例子，例如流体中的条纹对流卷、向日葵上的螺旋种子排列以及哺乳动物胚胎在早期原肠胚形成中的分化。在自然和人造环境中，空间生长过程已被证明是组织和利用这种模式形成机制的强大工具，同时抑制了通常称为缺陷的缺陷的形成。该项目旨在严格理解在化学、材料科学、流体动力学和生物学等各个领域产生的数学模型中生长和模式的相互作用，并用于研究动物骨骼图案、共晶金属合金淬火、蒸发化学沉积和弹性表面晶体缺陷抑制等现象。这样的理解也有助于在宏观，微观和纳米长度尺度上制造新型和功能性材料，为创造所需的结构提供了一本食谱。该项目的一个方面是为本科生提供研究经验的机会。该项目旨在开发和实施无限维动力系统理论和泛函分析的新技术，以严格研究原型偏微分方程模型的增长模式形成。它还将通过开发新的数值方法来近似和继续有界和无界域中的空间图案结构来探索和阐明复杂的现象。为了表征生长速度和边界曲率对图案形成的影响，该项目将首先关注淬火不均匀性，这允许子域中的图案并在互补中抑制它们。在简单的平面淬火中，该项目将研究多维空间域中图案形成前沿的存在性、波数选择和稳定性。由于标准的空间动力学方法在这里只适用于有限的设置，功能分析方法将被开发来处理存在的连续谱的分叉，扰动，或连续计划，这样的图案化的解决方案。以研究非平面淬火几何图形为目标，该项目旨在将空间动力系统的工具扩展到具有多个无界空间方向的域。该项目还将研究其他类型的时空异质性的物理利益，如缓慢变化的参数斜坡，时间上缓慢，但空间上均匀的淬火，和空间局部源项的模式选择和缺陷形成。 除了考虑模式的存在性、稳定性和波数选择外，该项目还将使用调制技术，以更简单、更易处理的偏微分方程的动力学来近似选定模式的相互作用和缺陷。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project is focused on how spatially-periodic patterns arise in natural systems and how they are selected and controlled by domain growth, geometry, and more generally, spatio-temporal heterogeneities. A pattern typically refers to a regular, repeating geometric structure, and examples abound in many different scientific domains, such as striped convection rolls in fluids, spiral-seed arrangements on a sunflower, and differentiation in mammalian embryos in early stage gastrulation. In both natural and man-made settings, spatial growth processes have proven to be powerful tools for organizing and harnessing such pattern-forming mechanisms, while at the same time suppressing the formation of imperfections, commonly known as defects. This project seeks to rigorously understand the interaction of growth and patterns in mathematical models arising in various areas such as chemistry, material science, fluid dynamics, and biology, and used to study phenomena such as skeletal patterning in animals, quenching of eutectic metal alloys, evaporative chemical deposition, and defect suppression in elastic surface crystals. Such an understanding could also aid in the fabrication of novel and functional materials at macro-, micro, and nano-meter length scales, giving a cookbook for creating a desired structure. A facet of this project incorporates research experience opportunities for undergraduates.This project aims to develop and implement new techniques in infinite-dimensional dynamical systems theory and functional analysis to rigorously study growth in prototypical partial differential equation models for pattern-formation. It will also explore and illuminate complex phenomena by developing novel numerical approaches to approximate and continue spatially patterned structures in bounded and unbounded domains. To characterize the effect of growth speed and boundary curvature on pattern formation, the project will first focus on quenching inhomogeneities, which allow patterns in a sub-domain and suppress them in the complement. In simple planar quenching, the project will study existence, wavenumber selection, and stability of pattern-forming fronts in multi-dimensional spatial domains. As standard spatial dynamics methods apply here in only limited settings, functional analytic approaches will be developed to deal with the presence of continuous spectrum in bifurcation, perturbation, or continuation schemes for such patterned solutions. With the goal of studying patterns in non-planar quenching geometries, this project seeks to extend the tools of spatial dynamical systems to domains with more than one unbounded spatial direction. The project will also investigate pattern selection and defect formation for other types of spatio-temporal heterogeneity of physical interest, such as slowly-varying parameter ramps, temporally slow but spatially homogeneous quenches, and spatially localized source terms. In addition to considering existence, stability, and wavenumber selection of patterns, this project will also use modulational techniques to approximate the interactions and defects of selected patterns in terms of the dynamics of simpler, more tractable partial differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.