Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions

TL;DR

This paper establishes the existence of weak solutions for a compressible fluid interacting with elastic structures under slip boundary conditions and rigorously justifies the incompressible limit in various regimes.

Contribution

It provides the first rigorous analysis of weak solutions and singular limits for compressible fluid-structure interaction problems with slip boundary conditions.

Findings

Existence of weak solutions for $\gamma > 12/7$ and $\gamma > 3/2$ with damping.

Rigorous derivation of the incompressible inviscid limit in low Mach and high Reynolds regimes.

First known results on singular limits for compressible fluids interacting with elastic structures.

Abstract

We study a system describing the compressible barotropic fluids interacting with (visco) elastic solid shell/plate. In particular, the elastic structure is part of the moving boundary of the fluid, and the Navier-slip type boundary condition is taken into account. Depending on the reference geometry (flat or not), we show the existence of weak solutions to the coupled system provided the adiabatic exponent satisfies $\gamma > \frac{12}{7}$ without damping and $\gamma > \frac{3}{2}$ with structure damping, utilizing the domain extension and regularization approximation. Moreover, via a modified relative entropy method in time-dependent domains, we give a rigorous justification of the incompressible inviscid limit of the compressible fluid-structure interaction problem with a flat reference geometry, in the regime of low Mach number, high Reynolds number, and well-prepared initial data. As a byproduct, with a fixed Reynolds number, we derive the incompressible limit without extra assumption. To the best of our knowledge, this is the first result concerning the singular limit problem for compressible fluids interacting with elastic structures.