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

Chiodaroli, Elisabetta; Kreml, Ondřej; Mácha, Václav; Schwarzacher, Sebastian

doi:10.1090/tran/8129

Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
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by Elisabetta Chiodaroli, Ondřej Kreml, Václav Mácha and Sebastian Schwarzacher PDF

Trans. Amer. Math. Soc. 374 (2021), 2269-2295 Request permission

Abstract:

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ 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.

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