Large and multidimensional solutions of hyperbolic balance laws

本课题探讨了带耗散的双曲平衡律的三个研究课题。首先，我们计划建立一般物理耗散系统的大时间行为和基本波的稳定性，包括带阻尼的欧拉方程、带松弛的欧拉方程和一维欧拉-泊松系统。先前的结果适用于小光滑解；建议的研究将集中在大溶液和强波问题上。其次，我们打算建立一类耗散双曲平衡律的BV理论，包括阻尼p系统、松弛p系统和欧拉-泊松方程。最后，我们将研究在平面扩散波附近或有界域上初始数据具有阻尼的三维可压缩欧拉方程。研究结果将有助于更好地理解非线性双曲偏微分方程中基本方程的行为。本研究的目的是探讨耗散双曲平衡律的几个开放问题。双曲平衡定律是模拟流体、气体和波浪运动的重要偏微分方程。研究将集中在大解问题、强波问题和多空间维度的现实模型问题上。这些情况往往涉及由非线性和共振引起的复杂而有趣的现象。研究结果有望应用于可压缩流体动力学、弹性材料力学、交通控制系统、半导体器件建模、多孔介质流动和地球物理动力学。本文的研究将有助于设计有效的数值格式进行科学计算。

This project investigates three research topics in hyperbolic balance law with dissipation. First, we plan to establish the large time behavior and the stability of elementary waves for general physical dissipative systems, including Euler equations with damping, Euler equations with relaxation, and Euler-Poisson systems in one dimension. Previous results are available for the small smooth solutions; the proposed research will concentrate on problems for large solutions and strong waves. Second, we intend to establish BV theory for a class of dissipative hyperbolic balance laws including damped p-system, p-system with relaxation, and Euler-Poisson equations. Last, we will study the three dimensional compressible Euler equations with damping with initial data near a planar diffusive wave or on a bounded domain. The results of the research will lead to a better understanding of the behavior of basic equations in nonlinear hyperbolic PDEs. The object of this research is to investigate several open problems for dissipative hyperbolic balance laws. Hyperbolic balance laws are important partial differential equations modeling the motion of fluids, gas, and waves. The research will concentrate on problems for large solutions, strong waves, and realistic models in multiple spatial dimensions. These situations often involve complicated but interesting phenomena resulting from nonlinearity and resonance. The results are expected to have application to dynamics of compressible fluids, elastic material mechanics, traffic control systems, semi-conductor devices modeling, porous medium flows, and geophysical dynamics. The proposed research will be helpful in the design of effective numerical schemes for scientific computation.