Analysis of Nonlinear Partial Differential Equations in Conservation Laws and Related Applications

守恒定律中非线性偏微分方程的分析及相关应用

• 批准号: 0906160
• 负责人: Dehua Wang
• 金额: $ 15.49万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906160&HistoricalAwards=false
• 关键词: Analysis; Nonlinear; Partial; Differential; Equations

This project is devoted to the study of multi-dimensional conservation laws arising in fluid dynamics and related applications. In particular, the study focuses on the following topics from the theory of two-dimensional compressible isentropic Euler equations and related partial differential equations: (a) the steady and unsteady transonic flows past an obstacle, (b) a fluid dynamic approach for the Gauss-Codazzi equations of isometric embedding/immersion, (c) two-dimensional solutions with special data of the Euler equations, and (d) the two-dimensional problems arising in magnetohydrodynamics, elastodynamics, and radiation hydrodynamics. The goals of the research are: (a) developing novel analytic methods and numerical schemes to construct multi-dimensional solutions, (b) exploring global structure of solutions, (c) understanding long-time behavior, for example, evolution of singularities and stability, and (d) gaining insights into the multi-dimensional problems, in particular identifying the functional spaces of multi-dimensional solutions.The project is devoted to a mathematical study of some nonlinear partial differential equations governing the motion of compressible fluid flows and related applications. Compressible fluids are ubiquitous in nature, the most common examples being gases. Their study is crucial for understanding aerodynamics, atmospheric science, astrophysics, plasma physics, etc. While the one-dimensional problems are rather well understood, the general theory for the multi-dimensional case is mathematically underdeveloped. The project is to advance the mathematical understanding of the multi-dimensional equations of compressible flows. The aim of the project is to advance knowledge in this fundamental area of mathematics and mechanics, and to provide education and training to students in this important field.

该项目致力于研究流体动力和相关应用中产生的多维保护定律。特别是，该研究侧重于以下主题，该主题来自二维可压缩的等递质欧拉方程和相关的偏微分方程：（a）稳定且不稳定的跨性别者流过障碍物，（b）与等值范围/接触的eSuse（c）dimension（c）dimensions（c）的稳定动力学方法（c）在磁流失动力学，弹性动力学和辐射流体动力学中引起的二维问题。该研究的目标是：（a）开发新的分析方法和数值方案来构建多维解决方案，（b）探索解决方案的全球结构，（c）理解长期行为，例如，例如，奇异和稳定性的演变，以及（d）对多维问题的洞察力，尤其是确定了某种方法，以确定某种功能，以确定某种功能的功能。非线性部分微分方程，负责可压缩流体流动和相关应用的运动。可压缩的流体本质上是普遍存在的，最常见的例子是气体。他们的研究对于理解空气动力学，大气科学，天体物理学，血浆物理学等至关重要。虽然一维问题却相当理解，但多维案例的一般理论在数学上是欠发达的。该项目是为了提高对可压缩流的多维方程的数学理解。该项目的目的是提高数学和力学基本领域的知识，并向这个重要领域的学生提供教育和培训。