Global ill-posedness of the compressible Euler system of gas dynamics

In this talk we will discuss a counterexample to the well-posedness of entropy solutions to the Cauchy problem for the compressible Euler equations in two space dimensions. In particular we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many entropy weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded entropy weak solutions. Our methods rely heavily on the new analysis of the incompressible Euler equations recently carried out by De Lellis and Székelyhidi and based on a revisited "h-principle".