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

本研究项目涉及一些非线性偏微分守恒方程，其中一些重要的附加物理效应作为源项出现，如弛豫和电场。项目的第一部分是研究具有松弛等低阶耗散的双曲型守恒律的多维激波前沿解的整体存在性和结构。第二部分是研究一维和多维情况下半导体欧拉-泊松方程平面跨音速激波的非线性稳定性。本研究旨在了解具有一维和多维附加物理效应的非线性守恒律系统冲击波解的整体结构和行为，阐明松弛和电场等附加物理效应对冲击波结构和行为的影响，为非线性偏微分方程的研究提供新的思路和技术。并为具有附加物理效应的非线性守恒系统的激波数值计算提供了新的见解。本项目研究的非线性偏微分方程组出现在应用科学和工程的许多分支中，如气体动力学、浅水波浪、半导体器件和生物物理学。这些方程为广泛的应用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。由于激波是高度非线性的，因此对它们的研究是非常具有挑战性的。这在几个空间维度上尤其如此，当考虑到额外的重要物理效应时更是如此。这项研究将加深对非线性波，特别是激波的认识。同时，新的理论和技术也将得到发展和应用。此外，本研究将发展的理论和方法将增进对许多重要的非线性波动现象的基本认识及其在应用科学和工程中的应用。

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.