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

FRG：合作研究：可压缩流体流动欧拉方程的多维问题及双曲守恒定律中的相关问题

• 批准号: 0244487
• 负责人: Dehua Wang
• 金额: $ 12.38万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244487&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Multi; Dimensional

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.

AbstractFRG：可压缩流体流动流的Euler方程的多维问题以及双曲线保护的相关问题，从法则上，流体和固体力学研究，具有不可压缩和可压缩材料的运动，有或没有内部隔离。对于具有内部耗散效应的气体和固体，总波动力学由无粘性，热扩散无动力学控制。在这些类别中，固体的压缩词对应于弹性波的研究及其传播。流体的可压缩运动通常与Viscid气体动力学有关。此外，可压缩的固体和流体都表现出冲击波，因此我们必须搜索对基础运动方程式的不受欢迎的解决方案。另一方面，不可压缩的运动涉及浓缩流体的运动，在这些运动中，在这些运动中的运动理想化是有用的，例如。水或油，以及诸如橡胶之类的固体的运动。 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这为提议者提供了在Athe Year努力中进行协作的动机，以推进对弹性动力学中可压缩的流体动态和相关问题的数学理解，并在弹性动力学中进行的相关问题。我们计划的核心是安排在Andarounds的成员（1）范围内（1）跨越（1）跨越的跨度，该组织将（1）跨越（1）跨越（1）跨越（1）跨越的跨度（1）跨度的跨度（1）跨越（1）的跨度。新理论技术，并通过使用和设计有效的，可靠的数值方法；（2）在接下来的几年中共同努力，以创造有关多维压缩器方程和相关问题的研究所必需的环境和人力，以繁荣发展；与此同时，（3）分享培训研究生和Postdoctoral Fellows的责任。该项目致力于对Euler方程进行的数学研究，该研究促进了Indiscid可压缩的液体和相关问题的运动。可压缩的流体在我们周围发生，例如气体和等离子体的研究对于理解空气疗法，大气科学，热力学等至关重要。虽然一维流体流得到充分了解，但多维流的统治理论是相比于数学的。提议者将进行三年的努力，以提高对无粘性可压缩流体动力学方程的数学理解。该项目中的核心将促进对这一基本的MATHEMATICS和机械师的了解，并将引入新一代研究人员的新一代研究人员，以了解该领域的杰出问题。