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

提案标题：PECASE：守恒定律系统和相关模型应用科学-数学意识和OUTREACH研究所：马里兰大学帕克研究人员的研究项目位于连续介质物理和应用偏微分方程组之间的界面上，重点是非线性守恒定律系统。这些散度形式的拟线性系统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、天体物理等应用领域的广泛物理现象。近年来，该领域的理论和数值方面都取得了重大进展。这项调查的主要目的是在一般数学理论的最新发展和过去几年发展的重要应用领域之间建立一座桥梁。该计划的主要重点将放在重要的和直到最近仍未探索的研究领域，包括：具有大量初始数据的双曲型守恒律系统；非线性弹性、流体动力学和燃烧理论中的多维系统；可压缩流动模型的零粘性解；可压缩流体的实际模型的边界层稳定性；以及双曲型守恒律的数值格式分析。这一跨学科研究的进展将在很大程度上依赖于非线性偏微分方程新分析技术的发展。这一领域的解析方法和数值方法是共同发展的，理解解析有助于构造准确有效的数值格式，而数值实验往往引领理论分析。这一研究计划的进展将有助于(A)成功地研究由守恒定律建模的各种重要物理系统，以及(B)设计高性能计算算法。研究人员将证明应用数学在广泛科学中的重要性的教育活动融入到她的工作中。特别强调在材料科学、生物、金融和尖端技术方面的应用。计划的教育活动包括面向高中生、本科生和研究生的项目。这位研究人员致力于通过鼓励代表不足的群体学习应用数学并将其作为职业来增加数学科学的多样性。该项目最初是作为职业奖而资助的，并于2004年9月转变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.