We consider the isentropic Euler equations of gas dynamics in the whole
two-dimensional space and we prove the existence of a smooth 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
Szekelyhidi 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.