Self-similar solutions to two-dimensional Riemann problems with transonic shocks

Multidimensional conservation laws present fundamental open questions concerning the existence, uniqueness, and stability of weak solutions, even for basic models such as the compressible Euler system. Riemann problems provide an important class of solutions for exploring these questions.

In this talk, we discuss self-similar solutions to two-dimensional Riemann problems with transonic shocks, focusing on regular shock reflection. We first discuss global self-similar solutions in the potential flow framework, including their existence, uniqueness, regularity, and geometric properties. We then turn to the isentropic Euler system, where the presence of vorticity introduces new challenges. In particular, regular reflection solutions exhibit low regularity, leading naturally to questions about the transport of vorticity by nonsmooth velocity fields. This motivates the study of stationary transport equations in bounded domains, with low-regularity vector fields and without restrictive assumptions on the geometry of the flow, allowing in particular for stagnation points. We discuss a theory of renormalized solutions adapted to this setting and its application to multidimensional Riemann problems.