Some Consequences of the Phase Space Choice

Abstract

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.