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Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations

Sebastian Schwarzacher, Pei Su

Abstract

A perfectly elastic beam is situated on top of a two dimensional fluid canister. The beam is deforming in accordance to an interaction with a Navier-Stokes fluid. Hence a hyperbolic equation is coupled to the Navier-Stokes equation. The coupling is partially of geometric nature, as the geometry of the fluid domain is changing in accordance to the motion of the beam. Here the existence of a unique strong solution for large initial data and all times up to geometric degeneracy is shown. For that an a-priori estimate on the time-derivative of the coupled solution is introduced. For the Navier-Stokes part it is a borderline estimate in the spirit of Ladyzhenskaya applied directly to the in-time differentiated system.

Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations

Abstract

A perfectly elastic beam is situated on top of a two dimensional fluid canister. The beam is deforming in accordance to an interaction with a Navier-Stokes fluid. Hence a hyperbolic equation is coupled to the Navier-Stokes equation. The coupling is partially of geometric nature, as the geometry of the fluid domain is changing in accordance to the motion of the beam. Here the existence of a unique strong solution for large initial data and all times up to geometric degeneracy is shown. For that an a-priori estimate on the time-derivative of the coupled solution is introduced. For the Navier-Stokes part it is a borderline estimate in the spirit of Ladyzhenskaya applied directly to the in-time differentiated system.
Paper Structure (20 sections, 8 theorems, 208 equations, 1 figure)

This paper contains 20 sections, 8 theorems, 208 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Main results
  4. Organization of the paper
  5. Preparation and preliminaries
  6. Notation
  7. Compatibility conditions
  8. Change of variables
  9. Energy estimate
  10. Hölder continiuity in time-space
  11. Formal time-derivative estimate
  12. Excluding the pressure
  13. Estimating the right hand side of (\ref{['sum']})
  14. Formally obtaining the initial values for $\partial_t u$ and $\partial_t^2 h$
  15. Construction of a strong solution
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(u_0, h_0, h_1)$ satisfy (initialcond) and $h_0\in H^4(0,L)$, $h_1\in H^2(0,L)$, $u_0\in H^2({\Omega}_{h_0})$, there exists unique strong solution to the system fluideq--initiald that satisfies the following a-priori estimate: as long as $h(t,x)\geqslant h_{\min}$ for all $(t,x)\in (0, T)\times (0, L)$.

Figures (1)

  • Figure 1: 1D beam interacting with a 2D fluid

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3: Minimal interval of existence
  • Lemma 3.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • ...and 8 more