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

摘要：可压缩流体流动欧拉方程的多维问题以及双曲守恒定律中的相关问题历史上，流体和固体力学研究不可压缩和可压缩材料的运动，无论有或没有内部耗散。对于具有内部耗散作为二次效应的气体和固体，总波动力学由无粘性、无热扩散动力学控制。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粘性气体动力学相关。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，不可压缩运动涉及较稠密流体的运动，其中不可压缩性的理想化是有用的，例如水或油，以及某些固体（如橡胶）的运动。虽然不可压缩流体还有许多重要的数学问题有待解决，例如三个空间维度上的纳维-斯托克斯方程的适定性，但二维和三个空间维度上的可压缩固体（以非线性弹性动力学方程为代表）和流体（以无粘流欧拉方程为代表）的数学研究还很不发达。这为提议者提供了三年内合作的动力努力推进对无粘可压缩流体动力学的多维方程和弹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）进行科学合作，通过开发新的理论技术以及使用和设计有效、稳健和可靠的数值方法，重点关注多维可压缩流动分析的进展；（2）在未来几年内共同努力创造必要的环境和人力促进多维可压缩欧拉方程及相关问题的研究蓬勃发展；同时，（3）承担培养研究生和博士后的责任。该项目致力于非粘性可压缩流体运动的欧拉方程及相关问题的数学研究。可压缩流体在自然界中随处可见，例如气体和等离子体，其研究对于理解空气动力学、大气科学、热力学等至关重要。虽然一维流体流动已被很好地理解，但多维流动的一般理论在数学上相对不发达。提议者将在三年的时间内合作，推进对无粘可压缩流体动力学多维方程的数学理解。该项目的成功将增进对数学和力学这一基础领域的了解，并将为新一代研究人员介绍该领域的突出问题。

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.