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

双曲守恒定律中可压缩流体流动的欧拉方程及相关问题的多维问题历史上，流体和固体力学研究不可压缩和可压缩物质的运动，无论是否有内部耗散。对于以内部耗散为二次效应的气体和固体，总波动力学由无粘、无热扩散的动力学所支配。在这些范畴中，固体的可压缩运动对应于对弹性波及其传播的研究；流体的可压缩运动通常与无粘性气体动力学有关。此外，可压缩固体和流体都有冲击波，因此我们必须寻找基本运动方程的间断解。另一方面，不可压缩运动本身涉及密度较大的流体的运动，其中不可压缩程度的理想化是有用的，例如水或油，以及某些固体的运动，如橡胶。虽然对于不可压缩流体，仍然有许多重要的数学问题需要解决，例如，三维空间中的Navier-Stokes方程的适定性。但在二维和三维空间中，可压缩固体(以非线性弹性动力学方程为代表)和流体(以无粘流的欧拉方程为代表)的数学研究更是不发达。这为提出者提供了三年的合作动机，以提高对不可压缩流体动力学的多维方程和弹性动力学相关问题的数学理解。我们计划的核心是安排小组成员之间和周围的持续互动，他们将(1)科学地合作重点是通过发展新的理论技术和使用和设计有效、稳健和可靠的数值方法来推进多维可压缩流动的分析；(2)在接下来的几年里共同努力，为多维可压缩欧拉方程和相关问题的研究创造必要的环境和人力；同时，(3)分担培养研究生和博士后研究员的责任。该项目致力于控制无粘性可压缩流体运动和相关问题的欧拉方程的数学研究。可压缩流体在自然界中无处不在，例如气体和等离子体，它们的研究对于理解空气动力学、大气科学、热力学等是至关重要的。虽然人们对一维流体流动已经有了很好的了解，但多维流动的一般理论在数学上却相对不发达。提出者将进行为期三年的合作，以促进对无粘性可压缩流体动力学多维方程的数学理解。该项目的成功将促进对这一数学和力学基本领域的了解，并将向该领域的突出问题介绍新一代研究人员。

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.