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.
