Steady weak solutions to an inflow/outflow driven compressible fluid-structure interaction problem

TL;DR

This paper establishes the existence of steady weak solutions for a complex 3D/2D fluid-structure interaction model involving compressible Navier-Stokes equations and an elastic plate, addressing significant analytical challenges.

Contribution

It introduces a novel domain-correction approach to handle domain degeneration and volume growth issues in compressible fluid-structure interaction problems.

Findings

Existence of weak solutions under large plate stiffness.

Development of a Lipschitz domain-correction mechanism.

Overcoming non-quantifiable pressure estimates in analysis.

Abstract

We study a stationary 3D/2D fluid-structure interaction problem between an elastic structure described by the linear plate equation and a fluid described by the compressible Navier-Stokes equations with hard-sphere pressure and inflow/outflow boundary data. This problem is motivated by wind-tunnel configuration and by the need for physically relevant steady states about which compressible flow-plate dynamics can be linearized. The main difficulty in the analysis is the lack of uniform estimates, both for approximate and weak solutions. In particular, the fixed-point construction for approximate solution yields a density estimate depending on approximate parameter, while the pressure estimate for the weak solution is only finite and non-quantifiable. As a result, large pressure loads can drive outward volume growth, while low pressure regions may lead to contact and therefore domain degeneration. This necessitates a novel approach based on a Lipschitz \emph{domain-correction} (barrier) mechanism that provides a framework in which solutions can be constructed without volume blow-up or degeneration of the domain. Constrained by the possibly very large fluid pressure load, our main result is the existence of a weak solution for a sufficiently large plate stiffness. Keywords: fluid-structure interaction, compressible Navier-Stokes, stationary weak solutions, hard-sphere pressure, inflow/outflow, linear plate, mathematical aeroelasticity