Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas

0539553 - MÚ 2022 RIV CH eng J - Journal Article
Březina, J. - Kreml, Ondřej - Mácha, Václav
Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas.
Nodea-Nonlinear Differential Equations and Applications. Roč. 28, č. 2 (2021), č. článku 13. ISSN 1021-9722. E-ISSN 1420-9004
R&D Projects: GA ČR(CZ) GJ17-01694Y
Grant - others:Neuron Fund for Support of Science(CZ) Neuron Impuls Junior
Institutional support: RVO:67985840
Keywords : admissible weak solution * Chaplygin gas * Delta shock * maximal dissipation
OECD category: Pure mathematics
Impact factor: 1.061, year: 2021
Method of publishing: Limited access
https://doi.org/10.1007/s00030-021-00672-0

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 δ-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.
Permanent Link: http://hdl.handle.net/11104/0317270