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Inviscid fluid interacting with a nonlinear two-dimensional plate

Abhishek Balakrishna, Igor Kukavica, Boris Muha, Amjad Tuffaha

Abstract

We address a moving boundary problem that consists of a system of equations modeling an inviscid fluid interacting with a two-dimensional nonlinear Koiter plate at the boundary. We derive a priori estimates needed to prove the local-in-time existence of solutions. We use the Arbitrary Lagrange Euler (ALE) coordinates to fix the domain and obtain careful estimates for the nonlinear Koiter plate, ALE velocity, and pressure {without any viscoelastic smoothing}. For the nonlinear Koiter plate, higher order energy estimates are obtained, whereas estimates for the ALE pressure are obtained by setting up an elliptic problem. For the ALE velocity, the bounds are obtained through div-curl estimates by estimating the ALE vorticity. We then extend our results in two directions: (1) to include fractional Sobolev spaces and (2) to incorporate the normalized second fundamental form.

Inviscid fluid interacting with a nonlinear two-dimensional plate

Abstract

We address a moving boundary problem that consists of a system of equations modeling an inviscid fluid interacting with a two-dimensional nonlinear Koiter plate at the boundary. We derive a priori estimates needed to prove the local-in-time existence of solutions. We use the Arbitrary Lagrange Euler (ALE) coordinates to fix the domain and obtain careful estimates for the nonlinear Koiter plate, ALE velocity, and pressure {without any viscoelastic smoothing}. For the nonlinear Koiter plate, higher order energy estimates are obtained, whereas estimates for the ALE pressure are obtained by setting up an elliptic problem. For the ALE velocity, the bounds are obtained through div-curl estimates by estimating the ALE vorticity. We then extend our results in two directions: (1) to include fractional Sobolev spaces and (2) to incorporate the normalized second fundamental form.

Paper Structure

This paper contains 15 sections, 9 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Mathematical model and main results
  3. Model
  4. ALE change of variables
  5. The elastic operator
  6. A priori estimates
  7. The plate estimates
  8. Pressure estimates
  9. The velocity estimates and the conclusion
  10. Estimates with normalized curvature tensor
  11. Construction of solutions
  12. Euler equations with given coefficients and non-homogeneous boundary data
  13. Local existence theorem for the plate equation
  14. Local existence of solutions to the coupled system via a fixed point
  15. Passage to the limit as viscosity goes to zero

Key Result

Theorem 2.1

Let $\nu \geq 0$. Assume that $(v,q,w)$ is a $C^{\infty}([0,T_{0}] \times \Omega)$ solution to the initial boundary value problem eulerr--PressureNormalized on a time interval $[0,T_0]$, where $T_0>0$, such that where $M\geq1$. Then $v$, $w$, and $q$ satisfy and where $C_0>0$ is a constant that is independent of the parameter $\nu \geq0$, while $K$ and $T$ are constants that depend on $M$, but

Theorems & Definitions (19)

  • Theorem 2.1
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['L01']}
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof : Proof of Lemma \ref{['L09']}
  • proof : Proof of Lemma \ref{['L03']}
  • proof : Proof of Theorem \ref{['main']}
  • ...and 9 more