  In the literature, for semidynamical systems in infinite dimensional phase
spaces, different topological structures are used (Hilbert, Banach, Sobolev,
locally convex, Hausdorf topology etc.). That is because there are neither set
rules nor understanding of the "right way" to choose the phase space and its
topology starting from a system of partial differential equations. The goal of
this paper is to reveal the influence of the choice of the phase space and its
topology as concern the results obtained for the semidynamical system defined
by the same system of partial differential equations. In the paper the linear
3D Euler equations are considered which are obtained by linearizing the non
linear 3D Euler equations at a constant solution. The well posedness of the
instantaneous perturbation propagation problem, that of the permanent source
produced time harmonic perturbation propagation problem as well the stability
of the null solution are analyzed in three different phase spaces. The idea is
to derive explicit solutions for the linear 3D Euler equations, to build up
different phase spaces by using explicit solutions and analyze the
well-posedness as well the stability of the null solution. The obtained results
present significant differences and some of them are surprising. For instance,
in one phase space the stability of the null solution coexists with solutions
having strictly positive exponential growth rate, but in other one the
propagation problem is ill posed. This aspect is extremely important from the
point of view of a real phenomenon modeled by the linear 3D Euler equations.
