Calculus of Variations and Geometric Measure Theory

E. Chiodaroli - C. De Lellis - O. Kreml

Surprising solutions to the isentropic system of gas dynamics

created by delellis on 02 Jul 2013
modified on 24 May 2017


Published Paper

Inserted: 2 jul 2013
Last Updated: 24 may 2017

Journal: Proceedings of HYP2012
Year: 2014

Hyperbolic problems: theory, numerics, applications, 1–10, AIMS Ser. Appl. Math., 8, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2014.


In a recent paper, jointly with Elisabetta Chiodaroli and Ond\v{r}ej Kreml we consider the Cauchy problem for the isentropic compressible Euler system in $2$ space dimensions, with initial data which assume two different constant values and have a discontinuity across a line. If we consider selfsimilar solutions we then encounter a classical $1$-dimensional Riemann problem for the corresponding hyperbolic system of conservation laws. We show that for some suitable choice of the pressure and of the initial data there exist infinitely many bounded admissible solutions which are not selfsimilar and indeed are genuinely $2$-dimensional. We also show that some of these Riemann data are generated by a $1$-dimensional compression wave. Our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. Each of these solutions coincide as long as the classical (Lipschitz) solution exists and they differentiate themselves immediately after the first blow-up time. Our approach is heavily influenced by a work of L\'aszl\'o Sz\'ekelyhidi which provides a similar result in the case of the classical vortex-sheet problem for the incompressible Euler equations.