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. ***

小行星9404384 这个项目是关于非线性偏微分方程组的研究。它包括两个主要的程序，分别在抛物线和双曲守恒律。第一个项目涉及粘性守恒律中波动稳定性的研究。这个项目是密切相关的双曲容许性和无粘极限的中心问题。目前的理论，基于能源的方法，仍然是特设和不完整的。一个新的，逐点稳定性分析提出了几个开放的问题，包括治疗：勒贝格可积性行为，稀疏，多波图案，欠压缩和“假拉克斯”冲击，弱爆燃波，多维锋MHD，和非一致收敛的冲击捕获计划。第二个项目涉及双曲守恒律的精细波追踪方法。波追踪给出了大量的信息近似解所获得的格里姆随机选择计划。有人建议，通过完善的会计技术，可以提取更多的这种信息的限制过程。以前，衰减和收敛到N-波已经建立了非凸系统。最近，nxn非共振系统的周期解的存在性和衰减性已经被证明，推广了Glimm和Lax对2x2系统的工作。计划研究非凸和共振系统的周期解，并最终研究对初始数据的连续依赖性。 这个项目是关于应用数学方程的.特别是连续介质力学方程的研究。重点将放在研究粘性激波和稀疏波的稳定性和收敛性。该分析可以应用于真实的世界遇到的问题，例如，在气体动力学。 ***