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Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions

Kuntal Bhandari, Imene Aicha Djebour, Šárka Nečasová

Abstract

We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a three-dimensional bounded domain, with the structure occupies a part of its boundary. Instead of standard no-slip boundary conditions, we consider the Navier-slip boundary conditions at the fluid-structure interface as well as at the fixed boundary. We establish the local-in-time existence and uniqueness of strong solutions within $L^{p}-L^{q}$ framework. The existence result is obtained for small time by decoupling the linearized system and employing a cascade strategy combined with the Tikhonov fixed point theorem, whereas the uniqueness is shown by deriving weak regularity properties for the associated linear coupled operator in a Hilbert space setting. It is the first result addressing strong solutions for a compressible fluid interacting with a damped plate under Navier-slip boundary conditions.

Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions

Abstract

We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a three-dimensional bounded domain, with the structure occupies a part of its boundary. Instead of standard no-slip boundary conditions, we consider the Navier-slip boundary conditions at the fluid-structure interface as well as at the fixed boundary. We establish the local-in-time existence and uniqueness of strong solutions within framework. The existence result is obtained for small time by decoupling the linearized system and employing a cascade strategy combined with the Tikhonov fixed point theorem, whereas the uniqueness is shown by deriving weak regularity properties for the associated linear coupled operator in a Hilbert space setting. It is the first result addressing strong solutions for a compressible fluid interacting with a damped plate under Navier-slip boundary conditions.

Paper Structure

This paper contains 17 sections, 9 theorems, 236 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Bibliographic comments
  4. Main features of the work
  5. Notations
  6. Main result
  7. Change of variables
  8. Linearized system
  9. The plate equation
  10. The Lamé system
  11. Transport equation
  12. Existence and uniqueness of solution
  13. Local-in-time existence
  14. Uniqueness
  15. Some useful results
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p>2$, $q>3$ with $\frac{1}{p}+\frac{1}{2q}<\frac{1}{2}$ and suppose that the initial data satisfy: Then there exists $T>0$ such that the system fs--ic admits a unique solution in the time interval $(0,T)$ with $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: The fluid domain associated to the plate displacement $\eta$.

Theorems & Definitions (14)

  • Theorem 1.1: Existence and uniqueness
  • Theorem 2.1: Existence and uniqueness---reformulated
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 4.1: Tikhonov
  • Proposition A.1
  • ...and 4 more