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

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

NSF Award Abstract-DMS-0203858 Mathematical Sciences: $L^1$ stability of hyperbolic conservation laws with geometrical sources and kinetic equations Abstract 0203858 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.