Construction and Physicality of Compressible Euler Flows

该项目旨在通过更好地了解理论模型的有效性范围来缩小流体动力学的实际应用与其理论基础之间的差距。它特别关注远离平衡的可压缩流动，这是一种与许多应用相关的制度，如高速飞行、燃烧、内爆和惯性约束聚变。即使是标准模型，人们对它们的有效范围也知之甚少。这一努力突显了理论见解在流体力学中的关键作用。这样的理解伴随着求解气体动力学方程的计算努力。一条这样的研究路线研究球面冲击波的内爆，现在球面冲击波是通过强大的激光诱导和控制的。分析结果对这两个方面都很重要：它们可以提供对不可观测数量的估计，并提供可用于基准数字代码的准确解决方案。这个项目专注于提供抽象数学结果以外的信息的方法，从而帮助评估在实践中常规使用的模型的相关性和局限性。本项目解决描述可压缩流体流动的非线性方程的基本、长期悬而未决的问题。首要目标是确定远离平衡的可能奇异流动的存在性，并使用这样的解来界定标准可压缩欧拉方程的有效范围。重点放在具有超越抽象存在结果的预测能力的方法上。该项目寻求的结果将把目前的近平衡理论扩展到一个空间维度，并同样适用于具有坍塌冲击和空洞的径向解决方案。该项目的主要动机是可压缩欧拉系统，该系统的许多基本问题仍然悬而未决。所使用的方法应该描述局部和全局解的行为，例如局域波相互作用和渐近行为，并为设计可靠的计算工具提供有用的见解。重点是了解零压区的作用，这是由于温度消失或密度(真空)消失所致。所考虑的问题似乎是必须克服的基本障碍，必须克服这些障碍才能正确理解可压缩流动中的非线性现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，认为值得支持。

This project aims at narrowing the gap between practical applications of fluid dynamics and its theoretical underpinnings through a better understanding of the range of validity of the theoretical models. It is particularly concerned with compressible flows far from equilibrium, a regime of relevance in many applications such as high-speed flight, combustion, implosions, and inertial confinement fusion. Even for standard models little is known rigorously about their range of validity. This effort highlights the critical role of theoretical insights in hydrodynamics. Such understanding goes hand in hand with the computational effort to solve the equations of gas dynamics. One such line of research studies imploding spherical shock waves, now induced and controlled via powerful lasers. Analytic results are important for both aspects: they can provide estimates for non-observable quantities and provide exact solutions that can be used to benchmark numerical codes. This project focuses on methods that provide information beyond abstract mathematical results, thereby helping to evaluate the relevance and limits of models routinely used in practice.This project addresses fundamental, long standing open problems for nonlinear equations describing compressible fluid flow. The overarching goals are to establish existence of possibly singular flows far from equilibrium, and to use such solutions to delimit the range of validity of the standard compressible Euler equations. The focus is on methods with predictive power beyond abstract existence results. The project seeks results that will extend the current near-equilibrium theory in one space dimension, and apply also to radial solutions with collapsing shocks and cavities. The project is primarily motivated by the compressible Euler system, for which many fundamental questions remain open. The methods utilized should give a description of local and global solution behavior, such as local wave interactions and asymptotic behavior, and provide useful insights for the design of reliable computational tools. Emphasis is placed on understanding the role of zero-pressure regions, due either to vanishing temperatures or to vanishing densities (vacuums). The issues considered appear to be essential road blocks that must be overcome to gain a proper understanding of non-linear phenomena in compressible flows.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.