On the energy dissipation rate of solutions to the compressible isentropic Euler system

In this paper we extend and complement the results in [4] on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law $p(\rho) = \rho^\gamme, \geq 1$. First we show that every Riemann problem whose one dimensional self-similar solution consists of two shocks admits also in_nitely many two dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and $Sz\’{e}kelyhidi$ [11], [12]. Moreover we prove that for some of these Riemann problems and for $1\leq < 3$ 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 in [7] does not favour the classical self similar solutions.