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

Elisabetta Chiodaroli, Ondřej Kreml, Václav Mácha, Sebastian Schwarzacher

Published 2018-12-24Version 1

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^1$ initial data which admit infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to 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 apply the theory developed by De Lellis and Sz\'ekelyhidi in order to 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 the finding of a single generalized subsolution.