Chiodaroli, ElisabettaFeireisl, EduardKreml, OndrejWiedemann, Emil2015-11-112015-11-112015-11-11201710.1007/s10231-016-0629-9https://infoscience.epfl.ch/handle/20.500.14299/120552WOS:000406034400017Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: Generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which can not be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. This difference between weak and measure-valued solutions in the compressible case is in contrast with the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.text::journal::journal article::research article