Construction and Physicality of Compressible Euler Flows

点击翻译按钮获取中文摘要

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.