Mathematical Sciences: Problems in Conservation Laws

这个项目涉及非线性偏微分方程组的研究。它由两个主要程序组成，分别采用抛物线和双曲守恒律。第一个项目涉及粘性守恒定律下波的稳定性研究。这个项目与双曲可容许性和不粘极限的中心问题密切相关。目前的理论，以能量方法为基础，仍然是临时的和不完整的。针对Lebesgue可积性、稀疏性、多波型、欠压缩和“假Lax”激波、弱爆波、MHD中的多维锋面以及激波捕获方案的非均匀收敛等开放问题，提出了一种新的点态稳定性分析方法。第二个项目涉及双曲守恒定律的精细波追踪方法。波动跟踪给出了大量关于由随机选择方案获得的近似解的信息。有人建议，通过完善的会计技术，可以在限制过程中提取更多的这些信息。以前，对于非凸系统，已经建立了n波的衰减和收敛。最近，证明了nxn非共振系统周期解的存在性和衰减性，推广了2x2系统的Glimm和Lax的工作。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。这个项目涉及应用数学方程。特别地，我们将研究连续介质力学的方程。重点研究粘性激波和稀薄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。* * *

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. ***