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.
