Abstract
We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical BV solution, instead a \(\delta \)-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exist infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one. In particular, not only the criterion based on comparing dissipation rates of total energy but also a stronger version based on comparing dissipation measures fails to pick the 1D solution.
Similar content being viewed by others
References
Al Baba, H., Klingenberg, C., Kreml, O., Mácha, V., Markfelder, S.: Nonuniqueness of admissible weak solutions to the Riemann problem for the full Euler system in two dimensions. SIAM J. Math. Anal. 52(2), 1729–1760 (2020)
Březina, J., Chioradoli, E., Kreml, O.: Contact discontinuities in multi-dimensional isentropic Euler equations. Electron J. Differ. Equ. 2018(94), 1–11 (2018)
Chaplygin, S.: On gas jets. Sci. Mem. Moscow Univ. Math. Phys. 21, 1–121 (1904)
Chen, G.-Q., Chen, J.: Stability of rarefaction waves and vacuum states for the multidimensional Euler equations. J. Hyperbolic Differ. Equ. 4(1), 105–122 (2007)
Chiodaroli, E.: A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11(3), 493–519 (2014)
Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. 68(7), 1157–1190 (2015)
Chiodaroli, E., Kreml, O.: On the energy dissipation rate of solutions to the compressible isentropic Euler system. Arch. Rational Mech. Anal. 214(3), 1019–1049 (2014)
Chiodaroli, E., Kreml, O.: Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations. Nonlinearity 31(4), 1441–1460 (2018)
Dafermos, C.M.: The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differ. Equ. 14, 202–212 (1973)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundleheren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 3rd edn. Springer-Verlag, Berlin (2010)
De Lellis, C., Székelyhidi, L.: The Euler equations as a differential inclusion. Ann. of Math. 170(3), 1417–1436 (2009)
De Lellis, C., Székelyhidi, L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Rational Mech. Anal. 195(1), 225–260 (2010)
Feireisl, E.: Maximal dissipation and well-posedness for the compressible Euler system. J. Math. Fluid Mech. 16(3), 447–461 (2014)
Feireisl, E., Kreml, O.: Uniqueness of rarefaction waves in multidimensional compressible Euler system. J. Hyperbolic Differ. Equ. 12(3), 489–499 (2015)
Gorini, V., Kamenshchik, A., Moschella, U., Pasquier, V.: The Chaplygin gas as a model for dark energy. In: The 10th Marcel Grossmann Meeting, pp. 840–859 (2006)
Guo, L., Sheng, W., Zhang, T.: The two-dimensional Riemann problem for isentropic Chaplygin gas dynamical system. Commun. Pure Appl. Anal. 9(2), 431–458 (2010)
Klingenberg, C., Markfelder, S.: The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock. Arch. Rational Mech. Anal. 227(3), 967–994 (2018)
Klingenberg, C., Markfelder, S.: Non-uniqueness of energy-conservative solutions to the isentropic compressible two-dimensional Euler equations. J. Hyperbolic Differ. Equ. 15(4), 721–730 (2018)
Klingenberg, C., Kreml, O., Mácha, V., Markfelder, S.: Shocks make the Riemann problem for the full Euler system in multiple spacedimensions ill-posed. Nonlinearity 33(12), 6517–6540 (2020)
Székelyhidi, L.: Weak solutions to the incompressible Euler equations with vortex sheet initial data. C. R. Math. Acad. Sci. Paris 349(19–20), 1063–1066 (2011)
Tsien, H.S.: Two dimensional subsonic flows of compressible fluids. J. Aeron. Sci. 6, 399–407 (1939)
von Karman, T.: Compressibility effects in aerodynamics. J. Aeron. Sci. 8, 337–365 (1941)
Acknowledgements
O. Kreml and V. Mácha were supported by the GAČR (Czech Science Foundation) project GJ17-01694Y in the general framework of RVO: 67985840. O. Kreml acknowledges the support of the Neuron Impuls Junior project 18/2016.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Březina, J., Kreml, O. & Mácha, V. Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas. Nonlinear Differ. Equ. Appl. 28, 13 (2021). https://doi.org/10.1007/s00030-021-00672-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-021-00672-0