## Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data

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- by Elisabetta Chiodaroli, Ondřej Kreml, Václav Mácha and Sebastian Schwarzacher PDF
- Trans. Amer. Math. Soc.
**374**(2021), 2269-2295 Request permission

## Abstract:

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi [Ann. of Math. (2) 170 (2009), no. 3, pp. 1417–1436; Arch. Ration. Mech. Anal. 195 (2010), no. 1, pp. 225–260] and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.## References

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## Additional Information

**Elisabetta Chiodaroli**- Affiliation: Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
- MR Author ID: 935797
- Email: elisabetta.chiodaroli@unipi.it
**Ondřej Kreml**- Affiliation: Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
- Email: kreml@math.cas.cz
**Václav Mácha**- Affiliation: Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
- Email: macha@math.cas.cz
**Sebastian Schwarzacher**- Affiliation: Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
- Email: schwarz@math.cas.cz
- Received by editor(s): April 9, 2019
- Published electronically: February 2, 2021
- Additional Notes: The first author was supported by the Italian National Grant FFABR 2017.

The second, third, and fourth authors were supported by the GAČR (Czech Science Foundation) project GJ17-01694Y in the general framework of RVO: 67985840. - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2269-2295 - MSC (2020): Primary 35L65, 76N10, 35Q31
- DOI: https://doi.org/10.1090/tran/8129
- MathSciNet review: 4223016