CAREER: Nonlinear phenomena in evolution PDE

职业：演化偏微分方程中的非线性现象

• 批准号: 1929029
• 负责人: Svetlana Roudenko
• 金额: $ 2.26万
• 依托单位: Florida International University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-10-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1929029&HistoricalAwards=false
• 关键词: CAREER; Nonlinear; phenomena; evolution; PDE

Many descriptions of real-life processes lead to the formulation of equations whose solutions change over time. For example, wave- or dispersive-type equations display this kind of behavior. These evolution equations provide the possibility of constructing solutions from prescribed initial conditions. In some evolution processes, a prescribed restriction (or boundedness) over time is given and the question is then whether the boundedness property persists forever or whether some sort of "unboundedness" can arise (e.g., a freak wave suddenly appearing in the ocean or a self-focusing burn in laser optics). These evolution processes are fundamental in nature, yet we are still in the infancy of their full analytical description. This project seeks to shed new light on the global behavior of solutions to nonlinear evolution partial differential equations (PDE), in particular, of dispersive PDE for which the nonlinearities cause a significant difference in global behavior compared with linear evolution. A special emphasis will be placed on the study of singularity formation, such as collapse and blow-up. The principal investigator's research program starts from advancing the theory of collapse for the simplest models in nonlinear wave equations, proceeds by expanding this theory to more generalized models and equations, and then goes on to explore the dynamics of solutions for simple dispersive systems, which have hitherto been studied little but which describe important models in the physical world. The ultimate goal of the research is to develop singularity theory for dispersive systems like the Davey-Stewartson system. These equations arise in physical contexts such as the description of water waves or acoustic waves, and in various fields such as laser optics and fluid and air dynamics.This project will advance the frontiers in the theory of nonlinear evolution equations, in particular, improving our understanding of collapse phenomena. It will produce a rigorous analytical description of concentration and aggregation for physical applications in real life. This work will be enhanced by collaborations at the national and international levels and will help strengthen interinstitutional ties, for example, between the George Washington University (GWU) and Howard University. In addition, the project will provide mathematical training and educational experiences at all mentoring levels, reinforcing future US competitiveness. It will focus on attracting, training, and retaining in mathematics and science outstanding female students and students from underrepresented groups. Vertical integration of educational activities, from middle school to the graduate level and beyond, will help to reinforce the project's agenda and contribute to the recruitment and retention of skillful workforce in the STEM fields. These activities will occur at the middle school level through the DC Math Circle program; at the high school level through the GWU summer precollege program (which annually recruits over two hundred students nationally); at the undergraduate level by engaging the principal investigator in the supervision of senior research theses and her serving as a mentor and guest lecturer at the GWU "Summer Program for Women in Math" ; and, finally, at the graduate level by organizing the DC Grad Camp, with the emphasis on attracting female and underrepresented minority students.

许多对现实生活过程的描述导致方程的公式化，其解随时间而变化。例如，波动型或色散型方程就表现出这种行为。这些演化方程提供了从给定的初始条件构造解的可能性。在一些演化过程中，给定了随时间的规定限制（或有界性），然后问题是有界性属性是否永远持续，或者是否会出现某种“无界性”（例如，海洋中突然出现的反常波浪或激光光学中的自聚焦灼伤）。这些演化过程在本质上是基本的，但我们仍然处于对其进行全面分析描述的初期阶段。该项目旨在揭示非线性演化偏微分方程（PDE）解的全局行为，特别是色散偏微分方程，其中非线性导致全局行为与线性演化相比有显着差异。一个特别的重点将放在奇点的形成，如崩溃和爆破的研究。首席研究员的研究计划从推进非线性波动方程中最简单模型的崩溃理论开始，通过将该理论扩展到更广义的模型和方程，然后继续探索简单色散系统的解的动力学，迄今为止研究很少，但描述了物理世界中的重要模型。研究的最终目标是发展像Davey-Stewartson系统这样的色散系统的奇点理论。这些方程出现在物理背景下，如水波或声波的描述，并在各种领域，如激光光学和流体和空气动力学。本项目将推进非线性演化方程理论的前沿，特别是提高我们对塌陷现象的理解。它将为真实的生活中的物理应用产生对浓度和聚集的严格分析描述。这项工作将通过国家和国际两级的合作得到加强，并将有助于加强机构间的联系，例如乔治华盛顿大学和霍华德大学之间的联系。此外，该项目还将在所有指导级别提供数学培训和教育经验，加强美国未来的竞争力。它将侧重于吸引、培训和留住数学和科学领域的优秀女学生和来自代表性不足群体的学生。从中学到研究生及更高层次的教育活动的纵向整合将有助于加强该项目的议程，并有助于招聘和留住STEM领域的熟练劳动力。这些活动将通过DC数学圈计划在中学一级进行;通过GWU夏季学前班计划在高中一级进行（每年在全国招收超过两百名学生）;在本科阶段，主要研究员负责高级研究论文的监督，并担任GWU“女性数学暑期项目”的导师和客座讲师;最后，在研究生一级，组织哥伦比亚特区格拉德营，重点是吸引女性和代表性不足的少数民族学生。