In the past years, results based on a technique called convex integration have drawn lots of interest within the community of mathematical fluid mechanics. Among other fascinating results, this technique allows to prove existence of infinitely many solutions for the multi-dimensional compressible Euler equations. All these solutions satisfy the energy inequality which is commonly used in the literature to identify physically relevant solutions. On the other hand, intuitively at least some of the infinitely many solutions still seem to be non-physical. For this reason one has studied additional admissibility criteria like the maximal energy dissipation criterion or the least action criterion -- to no avail: such criteria do not select the solution which is expected to be the physical one.
In this talk we give an overview on the aforementioned non-uniqueness results, and we explain why maximal dissipation as well as the least action criterion fail to single out the solution which is presumably the physical solution.
