  We discuss the problem of well-posedness of the compressible (barotropic)
Euler system in the framework of weak solutions. The principle of maximal
dissipation introduced by C.M. Dafermos is adapted and combined with the
concept of admissible weak solutions. We use the method of convex integration
in the spirit of the recent work of C.DeLellis and L.Szekelyhidi to show
various counterexamples to well-posedness. On the other hand, we conjecture
that the principle of maximal dissipation should be retained as a possible
criterion of uniqueness as it is violated by the oscillatory solutions obtained
in the process of convex integration.
