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

NSF Award Abstract-DMS-0203858 Mathematical Sciences：$L^1$ stability of b曲守恒律with geographic source and kinetic equations摘要0203858 Ha本研究致力于双曲守恒律弱解的稳定性以及动力学理论中的相关问题。 稳定性将通过构造显式的李雅普诺夫泛函来研究。 具体目标是：（i）建立具有几何源项的双曲守恒律方程和具有碰撞项的动力学模型弱解的稳定性;（ii）研究具有边界效应的Boltzmann方程冲击波的非线性稳定性，以及某些碰撞动力学方程的流体力学极限。具有几何源项的双曲守恒律出现在许多物理情况下，例如浅水流过渠道，通过管道的喷嘴流，以及多维欧拉方程中的自相似气体流。 解的稳定性问题在由这些方程建模的系统设计中很重要，包括飞机和航天飞机发动机。 玻尔兹曼方程和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数值模拟的精确方法。

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.