  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 Sz\'ekelyhidi 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.
