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.

现实生活过程的许多描述导致了方程的提法，其解决方案随着时间的流逝而变化。例如，波浪或色散型方程显示了这种行为。这些演化方程提供了从规定的初始条件中构建解决方案的可能性。在某些进化过程中，会随着时间的推移给出规定的限制（或有限），然后问题是界限财产是否永远存在，还是某种“无限制”会出现（例如，海洋中突然出现的怪胎波还是在激光器中出现自我关注的燃烧）。这些进化过程本质上是基本的，但是我们仍处于其完整的分析描述中。该项目试图对非线性进化偏微分方程（PDE）的全球行为（尤其是分散性PDE）的全球行为进行新的启示，而非线性与线性进化相比会导致全球行为有显着差异。将特别重点放在奇异性形成的研究上，例如崩溃和爆炸。首席研究者的研究计划从推进非线性波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笼统的模型和方程式来进行，然后继续探索简单分散系统解决方案的动态，这些解决方案的动力学是迄今为止研究的，但很少研究过物理世界中重要模型。该研究的最终目标是为Davey-Stewartson系统等分散系统开发奇异理论。这些方程式出现在物理环境中，例如对水波或声波的描述，以及在激光光学和流体和空气动力学等各个领域中出现。该项目将在非线性演化方程式的理论中推进边界，尤其是在提高我们对崩溃现象的理解。它将对现实生活中物理应用的集中和聚集进行严格的分析描述。在国家和国际层面的合作将增强这项工作，并将有助于加强乔治华盛顿大学（GWU）和霍华德大学之间的机构间关系。此外，该项目将在各个指导水平上提供数学培训和教育经验，从而增强未来的美国竞争力。它将专注于吸引，培训和保留数学和科学杰出的女学生以及来自代表性不足的小组的学生。从中学到研究生层面及以后的教育活动的垂直整合将有助于加强该项目的议程，并有助于招募和保留STEM领域的熟练劳动力。这些活动将通过DC Math Circle计划在中学级别进行；通过GWU夏季预科计划（每年招募全国200多名学生）在高中级别上；在本科阶段，通过与首席研究人员参与高级研究论文的监督，并在GWU“数学夏季女性计划”中担任导师和客座讲师；最后，在研究生层面，通过组织DC Grad Camp，重点是吸引女性和代表性不足的少数民族学生。