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

0541167 - MÚ 2022 RIV US eng J - Journal Article
Chiodaroli, E. - Kreml, Ondřej - Mácha, Václav - Schwarzacher, Sebastian
Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data.
American Mathematical Society. Transactions. Roč. 374, č. 4 (2021), s. 2269-2295. ISSN 0002-9947. E-ISSN 1088-6850
R&D Projects: GA ČR(CZ) GJ17-01694Y
Institutional support: RVO:67985840
Keywords : compressible Euler equation * two-dimensional space
OECD category: Pure mathematics
Impact factor: 1.353, year: 2021
Method of publishing: Limited access
https://doi.org/10.1090/tran/8129

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a C∞ 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.
Permanent Link: http://hdl.handle.net/11104/0318774