In this talk, we discuss our recent work concerning a nonstandard implementation of the Babuska-Brezzi Theorem, by way of ascertaining strongly continuous semigroup generation for a certain coupled compressible flow -- incompressible fluid dynamics. The coupling between these two disparate dynamics is enacted via a boundary interface. The modus operandi, entailed in this (continuous) mixed variational formulation, allows for the derivation of a (discrete) finite element method (FEM) with which to numerically approximate the fluid-flow solution variables. This is joint work with Paula Egging.