  In this paper we extend and complement some recent results by Chiodaroli, De
Lellis and Kreml on the well-posedness issue for weak solutions of the
compressible isentropic Euler system in $2$ space dimensions with pressure law
$p(\rho)=\rho^\gamma$, $\gamma \geq 1$. First we show that every Riemann
problem whose one-dimensional self-similar solution consists of two shocks
admits also infinitely many two-dimensional admissible bounded weak solutions
(not containing vacuum) generated by the method of De Lellis and Sz\'ekelyhidi.
Moreover we prove that for some of these Riemann problems and for $1\leq \gamma
< 3$ such solutions have greater energy dissipation rate than the self-similar
solution emanating from the same Riemann data. We therefore show that the
maximal dissipation criterion proposed by Dafermos does not favour the
classical self-similar solutions.
