ABSTRACT

Multidimensional conservation laws is an active research area with fundamental open questions about existence, uniqueness, and stability of properly defined weak solutions, even for classical models such as the compressible Euler system. Understanding of particular classes of weak solutions, such as multidimensional Riemann problems, is crucial in this context. This talk focuses on self-similar solutions to two-dimensional Riemann problems involving transonic shocks for compressible Euler system. Examples include regular shock reflection, Prandtl reflection, and four-shock Riemann problem. We first review the results on existence, regularity, geometric properties and uniqueness of global self-similar solutions of regular reflection structure in the framework of potential flow equation. A major open problem is to extend these results to compressible Euler system, i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions of regular reflection structure have low regularity. We further discuss existence, uniqueness and stability of renormalized solutions to the transport equation for vorticity in this low regularity setting.