The existence of a weak solution for a compressible multicomponent fluid structure interaction problem

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 modeled by a system resembling compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. The shell possesses a non-linear, non-convex Koiter energy. Considering that the densities are comparable initially we prove the existence of a weak solution until the degeneracy of the energy or the self-intersection of the structure occurs for two cases. In the first case the adiabatic exponents are assumed to satisfy $\max \{\gamma ,\beta \}>2$, $\min \{\gamma ,\beta \}>0$, and the structure involved is assumed to be non-dissipative. For the second case we assume the critical case $\max \{\gamma ,\beta \}\ge 2$ and $\min \{\gamma ,\beta \}>0$ and the dissipativity of the structure. The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy and the pressure, the almost compactness argument, added structural dissipation and suitable limit passages depending on uniform estimates.