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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Elisabetta Chiodaroli, Ondřej Kreml, Václav Mácha and Sebastian Schwarzacher
Journal: Trans. Amer. Math. Soc. 374 (2021), 2269-2295
MSC (2020): Primary 35L65, 76N10, 35Q31
DOI: https://doi.org/10.1090/tran/8129
Published electronically: February 2, 2021
MathSciNet review: 4223016
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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.


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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.
Article copyright: © Copyright 2021 American Mathematical Society