The object of this research is to investigate the long time dynamics of large solutions for nonlinear hyperbolic balance laws with lower order dissipations where the theory is most lacking. The research will concentrate on problems for large solutions and strong waves. This often encounters the investigation of qualitative behavior of weak solutions without any regularity. First, we will verify the Darcy's law for porous media flows as long time asymptotic of damped Euler equations. Next, we plan to establish nonlinear stability of elementary waves for compressible Euler equations with relaxation for traffic flows and then extend the results obtained that far to more general dissipative hyperbolic balance laws. The study of these problems requires new techniques that will be developed in this project.
The work deals with mathematical analysis of a class of partial differential equations describing nonlinear flows, waves and materials. The research seeks better understanding on qualitative behaviors of solutions to systems with important applications to porous media flows, traffic flows and semi-conductor devices. The results of this research are expected to benefit the study of compressible flows in areospace sciences, traffic control systems, semi-conductor devices industries and the energy industries.
