CAREER: Nonlinear PDE Models in Mathematical Physics and Experiment

职业：数学物理和实验中的非线性偏微分方程模型

基本信息

批准号：
1352353
负责人：
Jeremy Marzuola
金额：
$ 44万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352353&HistoricalAwards=false
关键词：
CAREER Nonlinear PDE Models Mathematical

项目摘要

Abstract Understanding interesting physical phenomena often requires classifying potentially physical observable solutions as attractive critical points (or semi-stable long-lived orbits) of infinite dimensional dynamical systems through partial differential equation theory and numerical experiments, which provide a rich set of problems that can be accessible at all levels of research and training. Such solutions and their stability can be studied on long time scales in relation to small scale models for light in optical media in nonlinear Schrodinger and Dirac models, surface waves on a surface tension scale in the gravity-capillary equations, or molecular dynamics in terms of both large scale nonlinear diffusions of crystals through thermodynamic fluctuations as well as Lagrangian mechanics for smaller systems of electrons trapped in various nuclear potentials. They can also be studied in macroscopic systems such as interaction of surface waves and internal waves in coupled fluids models, vortex formation in fluid flow around biological objects in the Navier-Stokes equations with boundary, and dark matter formation in models from general relativity using Einstein-Scalar Field equations and their reductions as Schrödinger-Poisson models. In applications that include large systems, complicated nonlinearities, and/or interesting geometric settings, the analysis and numerics can become increasingly difficult. The quest to understand complexity in partial differential equation models has led to the development of dramatically new analytic and computational techniques to explore questions of symmetry, phase transitions, uniqueness for the attractive states, as well as generalizations of Fourier transform methods using scattering theory, spectral theory and microlocal analysis to understand their stability. These techniques can be applied for instance on spaces with curvature, boundary, metric singularities or other difficult features such as noise in a sample or trapping due to external potential wells. This proposal will involve research in partial differential equations directly related to optics and electronic structure with relevant boundary conditions and potentials, as well as other equations from fluid dynamics, general relativity and thermodynamic fluctuations on crystal surfaces. The PI will also work with postdoctoral fellows, graduate students and undergraduates on integrated research into models, computation and experiment, especially though collaboration with members of the UNC Fluids Lab and International Mathematics Climate Network. An important aspect of that training will be to develop graduate courses and undergraduate courses in dynamics and computation to prepare trainees for a variety of careers in science. From a human resources standpoint, models like those in this proposal provide a large pool of problems that give a strong background in computation and geometry, as well as some applied statistics, which can be used for training purposes in work with researchers of all levels, from undergraduate to postdoctoral, then applied in many scientific fields. In addition, the PI will continue to support the mathematics department role in the University of North Carolina Science Expo to work towards broader outreach goals of making mathematical sciences more accessible to the public.

摘要：理解有趣的物理现象通常需要通过偏微分方程理论和数值实验将潜在的物理可观测解分类为无限维动力系统的有吸引力的临界点（或半稳定长寿命轨道），这提供了丰富的问题集在各个级别的研究和培训中，可以在与非线性薛定谔和狄拉克模型中的光学介质中的光、重力中的表面张力尺度上的表面波相关的长时间尺度上研究此类解决方案及其稳定性。毛细管它们还可以在宏观系统中进行研究，例如表面波和内部波的相互作用，这些方程或分子动力学通过热力学涨落进行晶体的大规模非线性扩散，以及用于捕获在各种核势中的较小电子系统的拉格朗日力学。耦合流体模型中的波、具有边界的纳维-斯托克斯方程中生物物体周围流体流动中的涡流形成，以及使用爱因斯坦标量场方程及其简化的广义相对论模型中的暗物质形成在包括大型系统、复杂非线性和/或有趣的几何设置的应用中，分析和数值可能变得越来越困难，对理解偏微分方程模型的复杂性的探索导致了全新的发展。可以应用分析和计算技术来探索对称性、相变、吸引态的唯一性问题，以及使用散射理论、谱理论和微局域分析来推广傅里叶变换方法来了解它们的稳定性。例如，具有曲率、边界、度量奇点或其他困难特征的空间，例如样本中的噪声或由于外部势阱而引起的捕获，该提案将涉及与具有相关边界条件的光学和电子结构直接相关的偏微分方程的研究。 PI还将与博士后、研究生和本科生合作，特别是通过与成员的合作，对模型、计算和实验进行综合研究。北卡罗来纳大学流体实验室和国际数学气候网络的一个重要方面是开发动力学和计算方面的研究生课程和本科生课程，从人力资源的角度来看，为各种科学职业做好准备。在该提案中，提供了大量问题，这些问题在计算和几何方面提供了强大的背景，以及一些应用统计数据，可用于与从本科生到博士后的各个级别的研究人员一起工作的培训目的，然后应用于许多领域此外，PI。将继续支持数学系在北卡罗来纳大学科学博览会中的作用，努力实现更广泛的推广目标，使公众更容易接触到数学科学。