Hyperbolic Conservation Laws and Applications

双曲守恒定律及其应用

• 批准号: 1613213
• 负责人: Dehua Wang
• 金额: $ 27.5万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613213&HistoricalAwards=false
• 关键词: Hyperbolic; Conservation; Laws; Applications

The aim of this research program is to develop new methods of mathematical analysis and techniques for studying some nonlinear partial differential equations governing the motion of compressible flows and related applications. Compressible fluids such as gases are ubiquitous in nature. Understanding of dynamics of compressible fluids is a crucial ingredient to prediction and control of important physical processes arising in aerodynamics, atmospheric science, astrophysics, plasma physics, biology and medicine, material science, and others. While the highly idealized one-dimensional problems are rather well understood, the general theory for the real-life multi-dimensional case is mathematically underdeveloped. The project will advance the mathematical understanding of the multi-dimensional equations of compressible flows and related problems in emerging applications. The research program will advance knowledge of the fundamental areas of mathematics and mechanics as well as applications. Graduate students will be involved in this research and trained on the outstanding problems in the related research fields.This project is devoted to a mathematical study of some nonlinear partial differential equations in multi-dimensional conservation laws and related applications. In particular, the study focuses on the following topics from the theory of inviscid and viscous compressible flows and related applications:(a) the existence and stability of vortex sheets in multi-dimensional compressible elastodynamics,(b) the global smooth isometric embedding of surfaces of negative curvature,(c) the stochastic partial differential equations of compressible flows, and(d) the active liquid crystal systems in biology/biophysics. The goal of the research is to develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous conservation laws and applications, to explore new phenomena of the motion of compressible flows, and to gain insights into the general multi-dimensional problems and emerging real-world applications.

本研究计划的目的是发展新的数学分析方法和技术，用于研究一些控制可压缩流运动的非线性偏微分方程及其相关应用。 可压缩流体，如气体，在自然界中无处不在。对可压缩流体动力学的理解是预测和控制空气动力学、大气科学、天体物理学、等离子体物理学、生物学和医学、材料科学等领域中出现的重要物理过程的关键因素。虽然高度理想化的一维问题是相当好的理解，为现实生活中的多维情况下的一般理论是数学欠发达。该项目将推进可压缩流的多维方程和新兴应用中的相关问题的数学理解。该研究计划将推进数学和力学以及应用的基本领域的知识。研究生将参与本研究，并就相关研究领域的突出问题进行培训。本项目致力于多维守恒律中某些非线性偏微分方程的数学研究及其相关应用。特别是，该研究侧重于从无粘和粘性可压缩流理论和相关应用的以下主题：（a）在多维可压缩弹性动力学涡面的存在性和稳定性，（B）的整体光滑等距嵌入的表面的负曲率，（c）可压缩流的随机偏微分方程，和（d）在生物/生物物理学的主动液晶系统。该研究的目标是开发新的分析方法和有效的技术来解决多维无粘和粘性守恒定律和应用中的一些重要问题，探索可压缩流运动的新现象，并深入了解一般的多维问题和新兴的现实世界的应用。

项目成果

Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients

• DOI: 10.1007/s00208-018-01798-w
• 发表时间: 2019-01
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: R. Chen;Jilong Hu;Dehua Wang