Constructive Existence Proofs for PDE in Multi-Dimensional Compressible Inviscid Flow

多维可压缩无粘流中偏微分方程的构造性存在证明

• 批准号: 0907974
• 负责人: Volker Elling
• 金额: $ 14.01万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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: 2009-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907974&HistoricalAwards=false
• 关键词: Constructive; Existence; Proofs; PDE; Multi

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project is aimed at obtaining constructive proofs of existence of various particular solutions of the multi-dimensional compressible Euler and potential flow equations. Constructive proofs provide not only mathematical rigor, but also detailed information about flow structure and behavior under parameter changes. This information has allowed progress on important engineering questions, such as transition from regular to Mach reflection or weak versus strong shocks at wedges and ramps. Other open problems, like the nature of supersonic bubbles at wings or the von Neumann paradox for triple points, also require new techniques for constructing particular solutions revealing structural information. Even more importantly, recent examples show that particular solutions can reveal non-uniqueness and instability of the underlying model equations and the numerical schemes used to solve them. The project will study narrower function classes where well-posedness is more likely. The Euler equations and related systems model the flow of gas or liquid, a central problem in physics and engineering. Applications in science and technology abound, from the theory of flight over combustion engines, hydraulics, magneto-hydrodynamic plasma, atmospheric and oceanic motion, to stars and nebulae in cosmology. Many applications rely on computer simulations of flows. However, results of computations are not perfectly reliable; sometimes spurious features are observed that do not occur in nature. The project relates these poorly understood problems with numerical simulations to previously unknown flaws of the underlying mathematical models and investigates possible remedies.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目旨在获得多维可压缩欧拉方程和势流方程各种特解存在性的构造性证明。构造性证明不仅提供了数学上的严谨性，而且提供了参数变化下流动结构和行为的详细信息。这些信息有助于解决一些重要的工程问题，例如从常规反射到马赫反射的转换，或者在楔形和斜坡处弱冲击与强冲击的转换。其他尚未解决的问题，如机翼上超音速气泡的性质或三点的冯·诺伊曼悖论，也需要新的技术来构建揭示结构信息的特殊解。更重要的是，最近的例子表明，特解可以揭示潜在模型方程的非唯一性和不稳定性，以及用于解决它们的数值方案。该项目将研究更窄的函数类，其中适位性更有可能。欧拉方程和相关系统模拟了气体或液体的流动，这是物理学和工程学中的一个中心问题。在科学和技术方面的应用非常广泛，从内燃机飞行理论、水力学、磁流体动力学等离子体、大气和海洋运动，到宇宙学中的恒星和星云。许多应用程序依赖于流的计算机模拟。然而，计算结果并不完全可靠；有时观察到自然界中不存在的虚假特征。该项目将这些鲜为人知的问题与先前未知的基础数学模型缺陷的数值模拟联系起来，并研究可能的补救措施。