FRG: Collaborative Research: Multi-Dimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation Laws

多维欧拉方程问题 可压缩流体流动及相关问题 历史上，流体和固体力学研究不可压缩和可压缩材料的运动，有或没有内部耗散。对于内部耗散作为对流效应的气体和固体，总的波动动力学由无粘、无热扩散的动力学控制。在这些类别中，固体的可压缩运动对应于弹性波及其传播的研究;流体的可压缩运动通常与无粘气体动力学有关。此外，可压缩固体和流体都表现出冲击波，因此我们必须寻找基本运动方程的不连续解。另一方面，不可压缩运动涉及密度较大的流体的运动，其中理想化的不可压缩性是有用的，例如水或油，以及某些固体如橡胶的运动。虽然对于不可压缩流体还有许多重要的数学问题需要解决，例如，三维空间中Navier-Stokes方程的适定性，压缩固体的数学研究（如非线性弹性动力学方程所示）和流体（用无粘流的欧拉方程表示）在二维和三维空间的发展甚至更少。这提供了动力的提议者合作，在三年的努力，以促进数学的理解，多-无粘可压缩流体动力学的三维方程和弹性动力学中的相关问题。我们计划的核心是在小组成员之间和周围安排一个持续的互动，他们将（1）科学合作，通过发展新的理论技术，使用和设计有效的、鲁棒的和可靠的数值方法，专注于多维可压缩流分析的进步;（2）在未来几年内共同努力，为多维可压缩欧拉方程及其相关问题的研究创造必要的环境和人力;同时（三）该项目致力于对控制无粘可压缩流体运动的欧拉方程进行数学研究和相关问题。可压缩流体在自然界中无处不在，如气体和等离子体，对它们的研究对于理解空气动力学、大气科学、热力学等都是至关重要的。这两个项目的提出者将在三年的时间里共同努力，以推进对无粘可压缩流体动力学的多维方程的数学理解。该项目的成功将推进数学和力学这一基础领域的知识，并将为该领域的突出问题引入新一代研究人员。

ABSTRACTFRG: Multi-Dimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation LawsHistorically, fluid and solid mechanics study the motion ofincompressible and compressible materials, with or without internaldissipation. For gases and solids with internal dissipation as asecondary effect, the gross wave dynamics is governed by inviscid,thermal diffusionless, dynamics. Within these categories, compressiblemotion for solids corresponds to the study of elastic waves and theirpropagation; compressible motion for fluids is usually associated withinviscid gas dynamics. Furthermore both compressible solids and fluids exhibit shock waves and hence we must search for discontinous solutions to the underlying equations of motion.Incompressible motion on the other hand concernsitself with the motion of denser fluids where the idealization ofincompressibility is useful, e.g. water or oil, as well as the motion ofcertain solids like rubber. While there are still many importantmathematical issues to be resolved for incompressible fluids, for example,the well-posedness of the Navier-Stokes equations in three spacedimensions, the mathematical study of compressiblesolids (as represented by the equations of nonlinear elastodynamics) andfluids (as represented by the Euler equations of inviscid flows)in two and three space dimensions is even less developed.This provides the motivation to the proposers to collaborate in athree year effort to advance the mathematical understanding of themulti-dimensional equations of inviscid compressible fluid dynamicsand related problems in elastodynamics.The core of our plan is to arrange a sustained interaction between andaround the members of the group, who will(1) collaborate scientifically, focusing on the advancement of theanalysis of multi-dimensional compressible flows by developing newtheoretical techniques and by using and designing effective, robust andreliable numerical methods;(2) work together over the next several years to create the environmentand manpower necessary for the research on multi-dimensional compressibleEuler equations and related problems to flourish; and in the meantime,(3) share the responsibility of training graduate students andpostdoctoral fellows.The project is devoted to a mathematical study of the Euler equationsgoverning the motion of an inviscid compressible fluid and relatedproblems. Compressible fluids occur all around us in nature, e.g. gasesand plasmas, whose study is crucial to understanding aerodyanmics,atmospheric sciences, thermodynamics, etc.While the one-dimensional fluid flows are rather well understood, thegeneral theory for multi-dimensional flows is comparatively mathematicallyunderdeveloped. The proposers will collaborate in a threeyear effort to advance the mathematical understanding of themulti-dimensional equations of inviscid compressible fluid dynamics.Success in this project will advance knowledge of this fundamental area ofmathematics and mechanics and will introduce a new generation ofresearchers to the outstanding problems in the field.