Inviscid fluid interacting with a nonlinear two-dimensional plate

TL;DR

This paper proves local-in-time existence of solutions for an inviscid fluid interacting with a nonlinear 2D plate, using ALE coordinates and advanced energy estimates without smoothing.

Contribution

It introduces a novel approach to handle inviscid fluid-structure interaction with nonlinear plates using ALE coordinates and fractional Sobolev spaces.

Findings

Established a priori estimates for nonlinear plate, ALE velocity, and pressure.

Proved local-in-time existence of solutions for the coupled system.

Extended analysis to fractional Sobolev spaces and normalized second fundamental form.

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.