CAREER: Nonlinear phenomena in evolution PDE

对现实生活过程的许多描述导致了方程的公式化，其解随着时间的推移而变化。例如，波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规定的初始条件构造解的可能性。在一些进化过程中，随着时间的推移，给出了一个规定的限制(或有界性)，那么问题是有界性属性是否永远存在，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或激光光学中的自聚焦燃烧)。这些进化过程在本质上是基本的，但我们仍然处于对它们的全面分析描述的婴儿期。本项目旨在揭示非线性发展偏微分方程解的整体性态，特别是色散偏微分方程解的整体性态，其中非线性导致整体性态与线性发展有显著差异。一个特别的重点将放在奇点形成的研究上，例如崩溃和爆炸。主要研究人员的研究计划从提出非线性波动方程中最简单模型的崩溃理论开始，将该理论扩展到更一般的模型和方程，然后继续探索简单色散系统的解的动力学，这些系统到目前为止研究得很少，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异性理论。这些方程产生于描述水波或声波的物理背景，以及激光光学、流体和空气动力学等各个领域。本项目将推动非线性演化方程理论的前沿，特别是提高我们对坍塌现象的理解。它将为现实生活中的物理应用提供对集中度和聚集度的严格解析描述。这项工作将通过国家和国际层面的合作得到加强，并将有助于加强机构间的联系，例如乔治华盛顿大学(GWU)和霍华德大学之间的联系。此外，该项目将在所有指导级别提供数学培训和教育经验，增强美国未来的竞争力。它将侧重于吸引、培养和留住数学和理科领域的优秀女学生和来自代表性不足群体的学生。从中学到研究生及以后的教育活动的垂直一体化将有助于加强项目议程，并有助于招聘和留住STEM领域的熟练劳动力。这些活动将通过华盛顿特区数学圈方案在中学一级进行；在高中一级通过华盛顿特区大学暑期预科方案(每年在全国招收200多名学生)进行；在本科一级通过聘请首席调查员监督高级研究论文，并在华盛顿特区大学“数学女性暑期方案”担任导师和客座讲师；最后在研究生一级，通过组织特区研究生营，重点是吸引女性和代表性不足的少数族裔学生。

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.