Long Time Behaviors of Large Solutions of Hyperbolic Balance Laws

本文研究了具有低阶耗散的非线性双曲平衡律大解的长时间动力学问题，这是目前理论上最缺乏的。研究将集中在大的解决方案和强波的问题。这常常会遇到对弱解的定性行为的研究，而不需要任何规律性。首先，我们将证明多孔介质流动的Darcy定律是阻尼Euler方程的长时间渐近解，其次，我们计划建立交通流的可压缩松弛Euler方程的基本波的非线性稳定性，然后将所得结果推广到更一般的耗散双曲平衡律。研究这些问题需要新的技术，将在本项目中发展。工作涉及一类描述非线性流动，波和物质的偏微分方程的数学分析。研究旨在更好地理解系统解的定性行为，并在多孔介质流动中有重要应用。本文的研究结果对航空航天科学中可压缩流的研究，交通控制系统，半导体器件行业和能源行业。

The object of this research is to investigate the long timedynamics of large solutions for nonlinear hyperbolic balancelaws with lower order dissipations where the theoryis most lacking. The research will concentrate on problems forlarge solutions and strong waves. This often encounters theinvestigation of qualitative behavior of weak solutions withoutany regularity. First, we will verify the Darcy's lawfor porous media flows as long time asymptotic of damped Euler equations.Next, we plan to establish nonlinear stability ofelementary waves for compressible Euler equations with relaxationfor traffic flows and then extend the results obtained that far to moregeneral dissipative hyperbolic balance laws. The study of these problemsrequires new techniques that will be developed in this project.The work deals with mathematical analysis of a class of partialdifferential equations describing nonlinear flows, waves and materials.The research seeks better understanding on qualitative behaviors ofsolutions to systems with important applications toporous media flows, traffic flows and semi-conductor devices.The results of this research are expected to benefit thestudy of compressible flows in areospace sciences, traffic controlsystems, semi-conductor devices industries and the energy industries.