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

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。该项目的目的是获得多维可压缩欧拉方程和势流方程各种特解存在的构造性证明。构造性证明不仅提供了数学上的严谨性，而且还提供了有关参数变化下流动结构和行为的详细信息。这些信息有助于在重要的工程问题上取得进展，如从常规反射过渡到马赫反射或楔形和坡道处的弱与强激波。其他悬而未决的问题，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定解。更重要的是，最近的例子表明，特定的解可以揭示基本模型方程和用于求解它们的数值格式的不唯一性和不稳定性。该项目将研究更窄的函数类，这些函数类更有可能是适定性的。欧拉方程和相关系统对气体或液体的流动进行建模，这是物理和工程中的一个中心问题。科学和技术方面的应用很多，从内燃机飞行理论、水力学、磁流体动力等离子体、大气和海洋运动，到宇宙学中的恒星和星云。许多应用依赖于流动的计算机模拟。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。该项目将这些知之甚少的问题与数值模拟与基础数学模型以前未知的缺陷联系起来，并调查了可能的补救措施。

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.