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

Chiodaroli, Elisabetta; Kreml, Ondřej; Mácha, Václav; Schwarzacher, Sebastian

doi:10.1090/tran/8129

Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
HTML articles powered by AMS MathViewer

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

Tristan Buckmaster, Onsager’s conjecture almost everywhere in time, Comm. Math. Phys. 333 (2015), no. 3, 1175–1198. MR 3302631, DOI 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 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 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 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 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 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 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, DOI 10.1007/978-3-642-04048-1
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 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 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 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 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 10.1016/0167-2789(94)90117-1
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 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 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 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 10.1016/j.crma.2011.09.009

AMS Website Down

Transactions of the American Mathematical Society

Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
HTML articles powered by AMS MathViewer

Abstract: