This research program deals with several aspects of the theory of hyperbolic systems of conservation laws and involves projects that can be divided into two categories. The first category is the application of the method of vanishing viscosity to (a) hyperbolic systems with weakly dissipative mechanisms such as friction, fading memory, and relaxation and (b) special systems of hyperbolic conservation laws with large initial data. These projects will establish the existence, stability, and qualitative behavior of solutions to systems with important applications; for example, porous media flows and the motion of viscoelastic materials with fading memory. The second category is the continuous dependence of entropy weak solutions to hyperbolic systems of conservation laws on physical parameters such as the adiabatic exponent and speed of light. The aim of these projects is to initiate an investigation on this issue of dependence on nonlinear flux functions with applications to compressible fluids in gas dynamics and special relativity.
This research project concerns problems lying on the interface between continuum physics and the theory of hyperbolic systems of conservation laws. Most of the partial differential equations arising in the study of elasticity, plasticity, fluid mechanics, semiconductors, gas dynamics, and combustion can be formulated as conservation laws. This project investigates mathematical properties of systems of such conservation laws, making use of the underlying physical structure to direct the analysis, while conversely elucidating properties of the equations to help better understand continuum physics. The results of this program are also expected to suggest computational algorithms that will advance numerical simulation of the systems.
