Progress on the well-posedness theory of the multi-dimensional Euler equations

Abstract: We consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states only, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if is possible to show that there are infinitely many entropy solutions. In this lecture we will show that the solution of this Riemann problem for the 2-d isentropic Euler equations usually is non-unique. Next we are able to show that there exist Lipshitz data that may lead to infinitely many solutions even for the full system of Euler equations. This is joint work with Simon Markfelder.