PECASE: Systems of Conservation Laws and Related Models in Applied Sciences - Math Awareness and Outreach

提案标题：PECASE：应用科学中的守恒定律和简化模型系统-数学意识和外展机构：马里兰州大学帕克研究员的研究计划在于连续介质物理和应用偏微分方程之间的接口，重点是守恒定律的非线性系统。这些发散形式的拟线性方程组控制着可压缩流体动力学、非线性材料科学、粒子物理、半导体、燃烧、多相流、天体物理以及其他应用领域中广泛的物理现象，近年来在理论和数值方面都取得了重大进展。本次调查的主要目的是建立一个桥梁之间的最新发展，在一般数学理论和显着的应用领域，在过去几年中已经发展起来。该计划的主要重点将是重要的，直到最近，未开发的研究领域，包括：双曲系统的守恒律与大的初始数据;多维系统的非线性弹性，流体动力学和燃烧理论;消失粘性解决方案的可压缩流动模型;稳定性的边界层的可压缩流体的现实模型;双曲型守恒律的数值格式分析，这一跨学科研究的进展将主要依赖于非线性偏微分方程新分析技术的发展。在这一领域的分析和数值方法一起发展;分析的理解有助于准确和有效的数值方案的建设，而数值实验往往导致理论分析。本研究计划的进展将有助于（a）成功调查各种重要的物理系统模型的守恒定律和（B）的高性能计算算法的设计。调查员融入她的工作教育活动，证明应用数学的重要性，在广泛的科学。特别强调材料科学，生物学，金融和尖端技术的应用。计划中的教育活动包括高中生、本科生和研究生项目。该研究员致力于增加数学科学的多样性，鼓励代表性不足的群体学习应用数学，并选择它作为职业。该项目最初作为CAREER奖资助，并于2004年9月转换为总统工程师和科学家早期职业奖（PECASE）奖。

Proposal Title: PECASE: SYSTEMS OF CONSERVATION LAWS AND RELATEDMODELS IN APPLIED SCIENCES - MATH AWARENESS AND OUTREACHInstitution: University of Maryland College ParkThe research program of the investigator lies on the interface between continuum physics and applied partial differential equations, with emphasis on nonlinear systems of conservation laws. These quasilinear systems in divergence form govern a broad spectrum of physical phenomena in compressible fluid dynamics, nonlinear materials science, particle physics, semiconductors, combustion, multi-phase flows, astrophysics, and other applied areas.In recent years, major progress has been made in both the theoretical and the numerical aspects of this field. The main objective of this investigation is to build a bridge between the most recent developments in the general mathematical theory and significant areas of application that have developed in the last few years. The main focus of this program will be given to significant and, until recently, unexplored areas of research, including: hyperbolic systems of conservation laws with large initial data; multi-dimensional systems in nonlinear elasticity, fluid dynamics, and combustion theory; vanishing viscosity solutions to models of compressible flows; stability of boundary layers for realistic models of compressible fluids; and analysis of numerical schemes for hyperbolic conservation laws.Advances in this interdisciplinary research will rely substantially on the development of new analytical techniques in nonlinear partial differential equations. Analytical and numerical methods in this field have developed together; analytical understanding contributes to the construction of accurate and efficient numerical schemes, while numerical experiments often lead the theoretical analysis. Progress in this research program will contribute (a) to the successful investigation of a wide variety of important physical systems modeled by conservation laws and (b) to the design of high performance computational algorithms.The investigator integrates into her work educational activities that demonstrate the importance of applied mathematics in a broad spectrum of sciences. Special emphasis is given to applications in materials sciences, biology, finance, and cutting edge technologies. The planned educational activities include programs for high school students, undergraduates, and graduate students. The investigator works to increase the diversity in the mathematical sciences by encouraging under-represented groups to study applied mathematics and choose it as a career.This project was originally funded as a CAREER award, and was converted to a Presidential Early Career Award for Engineers and Scientists (PECASE) award in September 2004.