Mathematical Sciences: Problems in Conservation Laws

9404384 Zumbrun这个项目是关于研究非线性偏微分方程组。它由两个主要程序组成，分别是抛物线守恒律和双曲型守恒律。第一个项目涉及研究粘性守恒定律中的波的稳定性。这个项目与双曲可采性和无粘性极限这两个核心问题密切相关。目前基于能量方法的理论仍然是即兴的和不完整的。本文提出了一种新的逐点稳定性分析方法，用于处理几个未解决的问题，包括：勒贝格可积行为、稀疏性、多波型、欠压激波和“假松弛”激波、弱爆燃波、MHD中的多维锋面以及激波捕捉格式的非一致收敛。第二个项目涉及双曲型守恒律的改进的波跟踪方法。波跟踪提供了大量关于Glimm随机选择方案所获得的近似解的信息。有人建议，通过改进会计技术，可以在限制过程中提取更多的此类信息。以前，关于非凸系统的衰变和收敛到N波已被建立。最近，nxn非共振系统的周期解的存在性和衰减性已被证明，推广了Glimm和Lax关于2x2系统的工作。它计划研究非凸系统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的方程式的。特别是对连续介质力学方程进行了研究。重点研究粘性激波和稀疏波的稳定性和收敛问题。这种分析可以应用于现实世界中遇到的问题，例如在气体动力学中。***

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. ***