  It is nowadays well understood that the multidimensional isentropic Euler
system is desperately ill--posed. Even certain smooth initial data give rise to
infinitely many solutions and all available selection criteria fail to ensure
both global existence and uniqueness. We propose a different approach to
well--posedness of this system based on ideas from the theory of Markov
semigroups: we show the existence of a Borel measurable solution semiflow. To
this end, we introduce a notion of dissipative solution which is understood as
time dependent trajectories of the basic state variables - the mass density,
the linear momentum, and the energy - in a suitable phase space. The underlying
system of PDEs is satisfied in a generalized sense. The solution semiflow
enjoys the standard semigroup property and the solutions coincide with the
strong solutions as long as the latter exist. Moreover, they minimize the
energy (maximize the energy dissipation) among all dissipative solutions.
