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）资助的。该项目旨在获得多维可压缩欧拉方程和势流方程的各种特解的存在性的建设性证明。构造性证明不仅提供了数学上的严密性，而且还提供了有关参数变化下的流动结构和行为的详细信息。这些信息使重要的工程问题取得了进展，例如从规则反射到马赫反射的过渡，或楔和斜坡处的弱激波与强激波。其他悬而未决的问题，如机翼上超音速气泡的性质或三相点的冯·诺依曼悖论，也需要新的技术来构建揭示结构信息的特定解决方案。更重要的是，最近的例子表明，特定的解决方案可以揭示非唯一性和不稳定的基本模型方程和数值方案来解决他们。该项目将研究更窄的函数类，其中适定性更有可能。欧拉方程和相关系统模拟气体或液体的流动，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比比皆是，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。许多应用依赖于计算机模拟流动。然而，计算结果并不完全可靠;有时会观察到自然界中不存在的虚假特征。该项目通过数值模拟将这些知之甚少的问题与基础数学模型中以前未知的缺陷联系起来，并研究可能的补救措施。

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.