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

Yadong Liu, Sourav Mitra, Šárka Nečasová

TL;DR

We study a coupled 3D compressible fluid and 2D elastic plate with a moving boundary under Navier-slip conditions, proving the existence of weak solutions for a broad range of the adiabatic exponent $\gamma$ and damping, via domain extension and regularization. A domain-transformation framework and an extended-regularization scheme underpin the construction, with recovery of the boundary coupling in the limit through strong convergence results and effective viscous flux identities. The paper then provides a rigorous singular-limit analysis in the flat reference geometry using a modified relative entropy method to justify the incompressible inviscid limit (low Mach, high Reynolds) and discusses the incompressible limit at fixed Reynolds. These results fill a gap in the theory of compressible FSI with slip boundaries and offer a robust approach to handling moving-boundary interactions in time-dependent domains.

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 $γ> \frac{12}{7}$ without damping and $γ> \frac{3}{2}$ 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.

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

TL;DR

We study a coupled 3D compressible fluid and 2D elastic plate with a moving boundary under Navier-slip conditions, proving the existence of weak solutions for a broad range of the adiabatic exponent and damping, via domain extension and regularization. A domain-transformation framework and an extended-regularization scheme underpin the construction, with recovery of the boundary coupling in the limit through strong convergence results and effective viscous flux identities. The paper then provides a rigorous singular-limit analysis in the flat reference geometry using a modified relative entropy method to justify the incompressible inviscid limit (low Mach, high Reynolds) and discusses the incompressible limit at fixed Reynolds. These results fill a gap in the theory of compressible FSI with slip boundaries and offer a robust approach to handling moving-boundary interactions in time-dependent domains.

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 without damping and 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.
Paper Structure (30 sections, 15 theorems, 167 equations, 3 figures)

This paper contains 30 sections, 15 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Literature Review
  4. Main results
  5. Technical Discussions
  6. Outline
  7. Preliminaries
  8. Function spaces and auxiliary results.
  9. Geometric settings and regularities
  10. Existence of Weak Solutions
  11. Rough ideas of the proof
  12. Extended problem with artificial pressure
  13. Uniform estimates
  14. Existence of weak solutions
  15. Recover the boundary identity
  16. ...and 15 more sections

Key Result

Theorem 1.4

Assume that the initial data satisfies eqs:initial. Let either $\nu_s > 0$, $\gamma > \frac{3}{2}$ or $\Omega \in \mathbbm{2}$, $\nu_s = 0$, $\gamma > \frac{12}{7}$. Then there exists $T > 0$ and a weak solution to eqs:FSI-Model in the sense of Definition def:bounded-weak. Moreover, either $T = + \i

Figures (3)

  • Figure 1: Sketch of two different geometries: general reference domain (left) and flat reference domain (right).
  • Figure 2: A slab-like geometry.
  • Figure 3: Construction of the mapping between two deformed configurations

Theorems & Definitions (47)

  • Definition 1.1: Two geometries, Figure \ref{['fig:two-geometries']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Weak solution, proved in Section \ref{['sec:weak']}
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Incompressible Inviscid limit
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10: Incompressible limit
  • ...and 37 more