Mathematical Sciences: Riemann Problems for Nonlinear Conservation Laws in One and Two Space Dimensions

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9403598 Canic This award supports mathematical research focusing on solutions of nonlinear systems of conservation laws. The investigation has two goals. The first is to understand the structure of solutions of Riemann problems for a two-dimensional system of conservation laws that model weak shack reflection. The second is to elucidate the physical admissibility of shockwave solutions of one-dimensional systems. The reflection of weak shack waves has been extensively studied both experimentally and numerically, but the corresponding two-dimensional Riemann problems for the compressible Euler equations are intractable on a theoretical level. This work will consider solutions of the unsteady small disturbance equation that arises in an asymptotic limit of the Euler system. In the study of one-dimensional systems of conservation laws, the physical admissibility of shock waves is a central question. A recent study of stability of viscous-admissible shock waves for quadratic conservation laws under generic two-parameter perturbations. These results use the theory of unfoldings of vector fields within the framework of the fundamental wave manifold. In this project, work will be done to extend these results to perturbations within the three-parameter family that arises naturally in this context. The study of wave phenomena and shocks is represented mathematically by partial differential equations or systems of such equations. The equations reflect certain physical (conservation) laws which govern the solutions. The mathematical and numerical analysis of these solutions is the basis for studies supported by this grant. ***