  To circumvent the ill-posedness issues present in various models of continuum
fluid mechanics, we present a dynamical systems approach aiming at selection of
physically relevant solutions. Even under the presence of infinitely many
solutions to the full Euler system describing the motion of a compressible
inviscid fluid, our approach permits to select a system of solutions (one
trajectory for every initial condition) satisfying the classical semiflow
property. Moreover, the selection respects the well accepted admissibility
criteria for physical solutions, namely, maximization of the entropy production
rate and the weak--strong uniqueness principle. Consequently, strong solutions
are always selected whenever they exist and stationary states are stable and
included in the selection as well. To this end, we introduce a notion of
dissipative solution, which is given by a triple of density, momentum and total
entropy defined as expectations of a suitable measure--valued solution.
