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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
Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

The authors would like to thank László Székelyhidi Jr. for helpful discussions. The research of E. F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The work of O. K. was supported by grant of GAČR (Czech Science Foundation) GA13-00522S in the general framework of RVO:67985840.

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Copyright information

© 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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