Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 4, pp 1557–1572 | Cite as

\({\mathcal {A}}\)-free rigidity and applications to the compressible Euler system

  • Elisabetta Chiodaroli
  • Eduard Feireisl
  • Ondřej Kreml
  • Emil Wiedemann


Can 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 cannot 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. While a priori it is not unexpected that not every measure-valued solution arises from a sequence of weak solutions, it is noteworthy that this observation in the compressible case is in contrast to the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.


\(\mathcal {A}\)-free condition Rigidity Compressible Euler equations Measure-valued solutions 


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Ecole polytechnique fédérale de LausanneLausanneSwitzerland
  2. 2.Institute of Mathematics of the Czech Academy of SciencesPragueCzech Republic
  3. 3.Leibniz University HannoverHannoverGermany

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