Existence of weak solution for a compressible multicomponent fluid structure interaction problem

TL;DR

This paper proves the existence of weak solutions for a complex fluid-structure interaction system involving two compressible fluids and a deformable shell, extending previous results to time-dependent domains with dissipative structures.

Contribution

It establishes global weak solution existence for a bi-fluid and shell interaction system with realistic pressure laws, in time-dependent domains, using advanced compactness and regularization techniques.

Findings

Global existence for $ ext{max}\{ ext{γ, β}\}>2$ without dissipation.

Extension to critical case $ ext{max}\{ ext{γ, β}\}=2$ with dissipation.

Development of compactness arguments for time-dependent domains.

Abstract

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modelled by a system resembilng compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. In fact in the present article the dependence of the fluid pressure on the densities is analogous to the ones considered in \cite{NovoPoko} (where the authors deal with a bi-fluid system in a time independent smooth domain). The shell constitutes the boundary of the fluid domain and it possesses a non-linear, non-convex Koiter energy (of a quite general form). We are interested in the existence of a weak solution to the system until the time-dependent boundary approaches a self-intersection or the Koiter energy degenerates. We first prove a global existence result when the adiabatic exponents solve $\max\{\gamma, \beta\}>2$ and $\min\{\gamma,\beta\}>0,$ further the densities are comparable and the structure involved is non-dissipative. Next with the assumption that the structure is dissipative we extend our global existence result to the critical case $\max\{\gamma,\beta\}\geq 2$ and $\min\{\gamma,\beta\}>0.$ The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy, added structural dissipation and suitable limit passages depending on uniform estimates. In order to deal with the bi-fluid system we generalize the almost compactness argument developed in \cite{NovoPoko, Vasseur} to the case of time dependent domains with uniform H\"{o}lder continuous boundaries. Moreover, the proof of such a compactness result depends on the existence of renormalized continuity equation in time dependent domains.