Qualitative Behavior of Solutions for Systems of Conservation Laws With Additional Physical Effects

本研究计画系针对某些非线性守恒律偏微分方程，其中包含一些重要的附加物理效应，例如松弛与电场。第一部分是研究具有低阶耗散（如松弛）的双曲型守恒律方程的多维激波阵面解的整体存在性和结构。第二部分是研究半导体Euler-Poisson方程平面跨音速激波的非线性稳定性，包括一维和多维情形。 本研究的目的是在一维和多维空间中理解具有某些附加物理效应的非线性守恒律方程组的激波解的整体结构和行为，阐明诸如弛豫和电场等附加物理效应对激波结构和行为的影响，为非线性偏微分方程的研究发展新的思想和技术，本文所研究的非线性偏微分方程组是一个新的非线性方程组，它的数值解是一个非线性方程组的数值解。这一课题出现在应用科学和工程的许多分支中，如气体动力学、浅水波、半导体器件和生物物理学。这些方程在广泛的应用中提供了重要的基本模型。对于这些方程，激波是非常重要的波型。冲击波的研究是非常具有挑战性的，因为它们是高度非线性的。这在若干空间维度中尤其如此，并且当考虑到额外的重要物理效应时。这一研究将加深对非线性波，特别是冲击波的理解。此外，新的理论和技术将被开发用于应用。此外，在这项研究中发展的理论和方法将加强对许多重要的非线性波动现象及其在应用科学和工程中的应用的基本理解。

This research project deals with some nonlinear partial differential equations of conservation laws with some important additional physical effects appearing as source terms, such as relaxation and electric fields. The first part of the project is to study the global existence and structure of multi-dimensional shock fronts solutions for the hyperbolic conservation laws with lower order dissipations, such as relaxation. The second part is to study the nonlinear stability of planar transonic shocks for the Euler-Poisson equations of semiconductors, both in one-dimensional and multi-dimensional cases. This research aims at understanding the global structure and behavior of solutions with shock waves for the nonlinear systems of conservation laws with some additional physical effects, both in one space dimension and several space dimensions, elucidating the influence of the additional physical effects such as relaxations and electric fields on the structure and behavior of shock waves, developing new ideas and techniques for the study of nonlinear partial differential equations, and providing new insight to the numerical computation of shock waves for the nonlinear systems of conservation laws with additional physical effects.The systems of nonlinear partial differential equations to be studied in this project arise in many branches of applied sciences and engineering, such as gas dynamics, shallow water waves, semiconductor devices and biophysics. These equations provide basic models of importance in a wide range of applications. For those equations, shock waves are very important wave patterns. The study of shock waves is very challenging because they are highly nonlinear. This is particular so in several space dimensions and when the additional important physical effects are taken into account. This research will deepen the understanding of nonlinear waves, particularly for shock waves. Also, new theories and techniques will be developed for applications. Moreover, the theories and methods to be developed in this research will enhance basic understanding of many important nonlinear wave phenomena and their applications to applied sciences and engineering.