  We study the Riemann problem for the isentropic compressible Euler equations
in two space dimensions with the pressure law describing the Chaplygin gas. It
is well known that there are Riemann initial data for which the 1D Riemann
problem does not have a classical $BV$ solution, instead a $\delta$-shock
appears, which can be viewed as a generalized measure-valued solution with a
concentration measure in the density component. We prove that in the case of
two space dimensions there exists infinitely many bounded admissible weak
solutions starting from the same initial data. Moreover, we show the same
property also for a subset of initial data for which the classical 1D Riemann
solution consists of two contact discontinuities. As a consequence of the
latter result we observe that any criterion based on the principle of maximal
dissipation of energy will not pick the classical 1D solution as the physical
one.
