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

数学科学：具有几何源和动力学方程的双曲型守恒律的L^1稳定性摘要本文研究了双曲型守恒律弱解的稳定性及动力学中的相关问题。稳定性将通过构造显式李雅普诺夫泛函来研究。具体目标是：(1)建立具有几何源项的双曲守恒律弱解和具有碰撞项的某些动力学模型的稳定性；（ii）研究了具有边界效应的玻尔兹曼方程激波的非线性稳定性，以及一些碰撞动力学方程的流体动力极限。具有几何源项的双曲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、多维欧拉方程中的自相似气体流动等。在用这些方程建模的系统的设计中，解的稳定性问题是很重要的，其中包括飞机和航天飞机发动机。玻尔兹曼方程和斯摩鲁霍夫斯基方程是运动论中的基本方程。这些方程的稳定性分析可用于开发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.