Talk Title: Self-Similar Solutions to 2-D Riemann Problems for Hyperbolic Conservation Laws
Abstract: Although multi-dimensional hyperbolic conservation laws arise in many applications (such as aerodynamics, industrial gas processing and environmental science), the development of the general theory of solutions is rather challenging. The main feature of hyperbolic conservation laws is the finite-time formation of singularities (or shock waves) in solutions no matter how smooth the initial data are. This causes major difficulties even in one space dimension. In two or more space dimensions, the added degrees of freedom allow formation of singularities that cannot occur in one space dimension. This is why standard, well-developed, one-dimensional techniques do not generalize to more than one space dimension. In fact, even when self-similar solutions to two-dimensional (Riemann) problems are considered, new techniques, based on the study of mixed, hyperbolic-elliptic systems, need to be employed to understand the structure of solutions. This can be seen in a benchmark problem of weak shock reflection by a wedge. In a collaborative effort with E-H. Kim, B.L. Keyfitz and G. Lieberman we have successfully resolved global existence and structure of self-similar solutions for a class of two-dimensional Riemann problems which includes regular reflection of weak shocks modeled by the unsteady transonic small disturbance equation. An outline of the methods and results will be presented, and a comparison with the related works in this field will be given. Generalization of the techniques to Mach and von Neumann reflection of weak shocks will be outlined. A proposal for the analysis of shock reflection modeled by the isentropic Euler equations and the nonlinear wave system will be outlined.