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We consider the isentropic Euler equations of gas dynamics in the wholetwo-dimensional space and we prove the existence of a $C^\infty$ initial datumwhich admits infinitely many bounded admissible weak solutions. Takingadvantage of the relation between smooth solutions to the Euler system and tothe Burgers equation we construct a smooth compression wave which collapsesinto a perturbed Riemann state at some time instant $T > 0$. In order tocontinue the solution after the formation of the discontinuity, we adjust andapply the theory developed by De Lellis and Sz\'ekelyhidi and we constructinfinitely many solutions. We introduce the notion of an admissible generalizedfan subsolution to be able to handle data which are not piecewise constant andwe reduce the argument to finding a single generalized subsolution.

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