Construction and Physicality of Compressible Euler Flows

本项目旨在通过更好地理解理论模型的有效性范围，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，这是一种在许多应用中相关的状态，如高速飞行，燃烧，内爆和惯性约束聚变。即使对于标准模型，也很少有人严格了解它们的有效性范围。这一努力突出了理论见解在流体力学中的关键作用。这样的理解与解决气体动力学方程的计算工作密切相关。其中一个研究方向是研究内爆球形冲击波，现在是通过强大的激光诱导和控制。分析结果对这两个方面都很重要：它们可以提供不可观测量的估计，并提供可用于基准数值代码的精确解。该项目的重点是提供抽象数学结果之外的信息的方法，从而帮助评估在实践中经常使用的模型的相关性和局限性。该项目解决了描述可压缩流体流动的非线性方程的基本的，长期存在的开放问题。总体目标是建立可能存在的奇异流远离平衡，并使用这样的解决方案来界定标准的可压缩欧拉方程的有效性范围。重点是具有超越抽象存在结果的预测能力的方法。该项目寻求的结果将扩展目前的近平衡理论在一个空间维度，并适用于径向的解决方案与崩溃的冲击和空腔。该项目的主要动机是可压缩欧拉系统，其中许多基本问题仍然是开放的。所采用的方法应该给出局部和全局解的行为，如局部波的相互作用和渐近行为的描述，并提供可靠的计算工具的设计有用的见解。重点放在理解零压力区域的作用，由于消失的温度或消失的密度（真空）。所考虑的问题似乎是必须克服的重要障碍，以获得对可压缩流中非线性现象的正确理解。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.