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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- Tristan Buckmaster, Onsager’s conjecture almost everywhere in time, Comm. Math. Phys. 333 (2015), no. 3, 1175–1198. MR 3302631, DOI https://doi.org/10.1007/s00220-014-2262-z
- Tristan Buckmaster, Camillo De Lellis, Philip Isett, and László Székelyhidi Jr., Anomalous dissipation for $1/5$-Hölder Euler flows, Ann. of Math. (2) 182 (2015), no. 1, 127–172. MR 3374958, DOI https://doi.org/10.4007/annals.2015.182.1.3
- Tristan Buckmaster, Camillo De Lellis, and László Székelyhidi Jr., Dissipative Euler flows with Onsager-critical spatial regularity, Comm. Pure Appl. Math. 69 (2016), no. 9, 1613–1670. MR 3530360, DOI https://doi.org/10.1002/cpa.21586
- Tristan Buckmaster, Camillo de Lellis, László Székelyhidi Jr., and Vlad Vicol, Onsager’s conjecture for admissible weak solutions, Comm. Pure Appl. Math. 72 (2019), no. 2, 229–274. MR 3896021, DOI https://doi.org/10.1002/cpa.21781
- Elisabetta Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ. 11 (2014), no. 3, 493–519. MR 3261301, DOI https://doi.org/10.1142/S0219891614500143
- Elisabetta Chiodaroli, Camillo De Lellis, and Ondřej Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1157–1190. MR 3352460, DOI https://doi.org/10.1002/cpa.21537
- Elisabetta Chiodaroli and Laurent Gosse, A numerical glimpse at some non-standard solutions to compressible Euler equations, Innovative algorithms and analysis, Springer INdAM Ser., vol. 16, Springer, Cham, 2017, pp. 111–140. MR 3643878
- Elisabetta Chiodaroli and Ondrej Kreml, On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 1019–1049. MR 3269641, DOI https://doi.org/10.1007/s00205-014-0771-8
- Peter Constantin, Weinan E, and Edriss S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys. 165 (1994), no. 1, 207–209. MR 1298949
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- Sara Daneri and László Székelyhidi Jr., Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 471–514. MR 3614753, DOI https://doi.org/10.1007/s00205-017-1081-8
- Camillo De Lellis and László Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. MR 2600877, DOI https://doi.org/10.4007/annals.2009.170.1417
- Camillo De Lellis and László Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 225–260. MR 2564474, DOI https://doi.org/10.1007/s00205-008-0201-x
- Camillo De Lellis and László Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math. 193 (2013), no. 2, 377–407. MR 3090182, DOI https://doi.org/10.1007/s00222-012-0429-9
- Gregory L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D 78 (1994), no. 3-4, 222–240. MR 1302409, DOI https://doi.org/10.1016/0167-2789%2894%2990117-1
- Eduard Feireisl, Maximal dissipation and well-posedness for the compressible Euler system, J. Math. Fluid Mech. 16 (2014), no. 3, 447–461. MR 3247361, DOI https://doi.org/10.1007/s00021-014-0163-8
- Eduard Feireisl, Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, and Emil Wiedemann, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1375–1395. MR 3594358, DOI https://doi.org/10.1007/s00205-016-1060-5
- Philip Isett, A proof of Onsager’s conjecture, Ann. of Math. (2) 188 (2018), no. 3, 871–963. MR 3866888, DOI https://doi.org/10.4007/annals.2018.188.3.4
- P. Isett, Nonuniqueness and existence of continuous, globally dissipative Euler flows, arXiv:1710.11186, October 2017.
- László Székelyhidi, Weak solutions to the incompressible Euler equations with vortex sheet initial data, C. R. Math. Acad. Sci. Paris 349 (2011), no. 19-20, 1063–1066 (English, with English and French summaries). MR 2842999, DOI https://doi.org/10.1016/j.crma.2011.09.009
Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 35L65, 76N10, 35Q31
Retrieve articles in all journals with MSC (2020): 35L65, 76N10, 35Q31
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.
Article copyright:
© Copyright 2021
American Mathematical Society