L^1 Stability of Hyperbolic Coservation Laws with Geometrical Sources and Kinetic Equations

美国国家科学基金会奖摘要-dms-0203858数学科学奖：$L^1$双曲型守恒律的几何来源和动力学方程的稳定性这项研究讨论了双曲型守恒律的弱解的稳定性和动力学理论中的相关问题。稳定性将通过构造显式Lyapunov泛函来研究。具体目标是：(I)建立具有几何源项的双曲型守恒律的弱解的稳定性；(Ii)研究具有边界效应的Boltzmann方程激波的非线性稳定性，以及某些碰撞动力学方程的流体动力学极限。具有几何源项的双曲型守恒律出现在许多物理情况中，如浅水流过通道、喷嘴流过管道以及多维欧拉方程中的自相似气体流动。在以这些方程为模型的系统设计中，解的稳定性问题很重要，包括飞机和航天飞机发动机。Boltzmann方程和Smoluchowski方程是动力学理论中的基本方程。这些方程的稳定性分析可用于发展相应物理系统的精确数值模拟方法。

NSF Award Abstract-DMS-0203858 Mathematical Sciences: $L^1$ stability of hyperbolic conservation laws with geometrical sources and kinetic equationsAbstract0203858 Ha This research addresses the stability of weak solutions of hyperbolic conservation laws and related problems in kinetic theory. Stability will be studied by constructing explicit Lyapunov functionals. Specific goals are: (i) establish the stability of weak solutions to hyperbolic conservation laws with geometric source terms and certain kinetic models with collision terms; (ii) study the nonlinear stability of shock waves of the Boltzmann equation with boundary effects, and hydrodynamic limits of some collisional kinetic equations.Hyperbolic conservation laws with geometric source terms appear in many physical situations, such as shallow water flow through a channel, nozzle flow through a duct, and self-similar gas flow in multi-dimensional Euler equations. The issue of stability of solutions is important in the design of systems modeled by these equations, which include aircraft and space shuttle engines. The Boltzmann equation and the Smoluchowski equation are fundamental equations in kinetic theory. Stability analysis for these equations can be used in development of accurate methods for numerical simulation of the corresponding physical systems.