An inf-sup approach to semigroup wellposedness for a compressible flow - incompressible fluid interactive PDE system

Abstract

This work presents qualitative and numerical results on a system of partial differential equations (PDEs) which models certain fluid-fluid interaction dynamics. This system models a compressible fluid in a domain $\Omega^+ \subset \mathbb{R}^2$, coupled to an incompressible fluid modeled by Stokes flow in domain $\Omega^- \subset \mathbb{R}^2$, with the strong coupling implemented through certain boundary conditions on the shared interface, $\Gamma$. The wellposedness of this system is established by means of constructing for it a semigroup generator representation. This representation is accomplished by eliminating one of the pressure variables via identifying it as the solution of a certain boundary value problem, while the wellposedness is established via a nonstandard usage of the Babuska-Brezzi Theorem. In later sections, we demonstrate how the earlier constructive proof of wellposedness naturally lends itself to a certain finite element method (FEM), by which to numerically approximate solutions of the given coupled PDE system. This FEM is provided with error estimates and associated rates of convergence.