Prototype Systems of Multidimensional Conservation Laws

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

This project studies two-dimensional Riemann problems for systems of conservation laws, including the Euler equations of gas dynamics. The theory of multi-dimensional conservation laws is in its infancy, and this analysis of Riemann problems is a first step that will indicate the types of singularities that arise in multi-dimensional systems. Recent numerical simulations have presented evidence of unexpectedly singular behavior at the formation points of Mach stems. These studies call for analytical confirmation, and insight will be gained from simple prototype examples. In the self-similar approach, the systems change type -- hyperbolic far from the origin ("supersonic flow") and of mixed hyperbolic-elliptic type near the origin ("subsonic region"). Analysis of the subsonic solution gives rise to free boundary problems in mixed-type partial differential equations with mixed boundary conditions (Dirichlet and oblique derivative). Earlier work solved such free boundary problems locally near shock reflection points, but only for one simple interaction (regular reflection) and only for simplified systems (where the elliptic equation decoupled from the hyperbolic system). The current project will develop new techniques for the more complicated problems that arise when the boundary condition is not uniformly oblique, and when the equations in the subsonic system do not decouple. Finally, study of a prototype example of diverging rarefactions problems will shed light on singular Mach stem formation points. Conservation laws model fundamental problems in aerodynamics and continuum mechanics. Better understanding has consequences in real world applications. For example, the large-scale numerical simulations that form the basis of weather and climate prediction and of nuclear reactor safety analysis are based on partial differential equations in the form of conservation laws. Theoretical advances, particularly in the analysis of singular behavior, will find their way into making numerical codes more efficient and more reliable. In carrying out this research, the investigators will build on success in introducing beginning researchers, including members of groups underrepresented in mathematics, to the theory of conservation laws, and in expanding their career horizons.