CAREER: Nonlinear phenomena in evolution PDE

许多对现实生活过程的描述导致了其解随时间变化的方程的形成。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，给定了规定的限制（或有界性），然后问题是有界性是否永远存在，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烧伤）。这些进化过程在本质上是基本的，然而我们仍然处于对它们进行全面分析描述的初级阶段。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，其非线性与线性演化相比在全局行为上存在显著差异。将特别强调奇点形成的研究，如坍缩和爆炸。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坍缩理论开始，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。这项研究的最终目标是发展像戴维-斯图尔特逊系统这样的色散系统的奇点理论。这些方程出现在物理环境中，如对水波或声波的描述，以及在激光光学、流体和空气动力学等各个领域。该项目将推进非线性演化方程理论的前沿，特别是提高我们对崩塌现象的理解。它将为现实生活中的物理应用产生对集中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，该项目还将提供各级数学培训和教育经验，以加强美国未来的竞争力。它将重点吸引、培养和留住数学和科学领域的优秀女学生和弱势群体的学生。教育活动的垂直整合，从中学到研究生水平甚至更高，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DC数学圈计划在中学阶段进行；在高中阶段，通过GWU夏季大学预科项目（每年在全国招收200多名学生）；在本科阶段，聘请首席研究员指导高级研究论文，并担任GWU“女性数学暑期项目”的导师和客座讲师；最后，在研究生层面，通过组织华盛顿特区研究生营，重点吸引女性和少数族裔学生。

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.