Mathematical Sciences: Problems in Conservation Laws

数学科学：守恒定律问题

• 批准号: 9404384
• 负责人: Kevin Zumbrun
• 金额: $ 4万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-15 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404384&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Conservation; Laws

9404384 Zumbrun This project concerns the study of nonlinear systems of partial differential equations. It consists of two main programs, in parabolic and hyperbolic conservation laws, respectively. The first project involves the study of stability of waves in viscous conservation laws. This project is intimately connected with the central problems of hyperbolic admissibility and the inviscid limit. Current theory, based on energy methods, remains ad hoc and incomplete. A new, pointwise stability analysis is proposed for the treatment of several open problems, including: Lebesgue integrability behavior, rarefactions, multiple wave patterns, undercompressive and "fake Lax" shocks, weak deflagration waves, multi-dimensional fronts in MHD, and nonuniform convergence of shock capturing schemes. The second project involves refined wave tracing methods for hyperbolic conservation laws. Wave tracing gives a great deal of information about approximate solutions obtained by the Glimm random choice scheme. It is proposed that, by refined accounting techniques, more of this information can be extracted in the limiting process. Previously, decay and convergence to N-waves have been established for nonconvex systems. More recently, existence and decay have been shown for periodic solutions of nxn, nonresonant systems, generalizing the work of Glimm and Lax for 2x2 systems. It is planned to study periodic solutions of nonconvex and of resonant systems, and, ultimately, continuous dependence on initial data. This project deals with equations of applied mathematics. In particular, the equations of continuum mechanics will be study. Emphasis will be placed on the study of stability and convergence of viscous shock and rarefaction waves. The analysis can be applied to real world problems encountered, for example, in gas dynamics. ***

9404384 Zumbrun 该项目涉及偏微分方程非线性系统的研究。它由两个主要程序组成，分别是抛物线和双曲守恒定律。第一个项目涉及粘性守恒定律中波的稳定性研究。该项目与双曲可接纳性和无粘极限的核心问题密切相关。当前基于能量方法的理论仍然是临时的且不完整的。提出了一种新的逐点稳定性分析来处理几个开放问题，包括：勒贝格可积行为、稀疏、多波模式、欠压和“假松弛”冲击、弱爆燃波、MHD 中的多维前沿和非均匀收敛冲击捕捉方案。第二个项目涉及双曲守恒定律的改进波追踪方法。波浪追踪提供了大量关于通过 Glimm 随机选择方案获得的近似解的信息。建议通过改进的会计技术，可以在限制过程中提取更多此类信息。此前，非凸系统的衰减和收敛到 N 波已经被建立。最近，nxn、非谐振系统的周期解的存在和衰变已被证明，概括了 Glimm 和 Lax 对于 2x2 系统的工作。计划研究非凸和谐振系统的周期解，并最终研究对初始数据的连续依赖性。 该项目涉及应用数学方程。特别是，将研究连续介质力学方程。重点研究粘性激波和稀疏波的稳定性和收敛性。该分析可以应用于现实世界中遇到的问题，例如气体动力学。 ***