Analysis of a time-dependent fluid-solid interaction problem above a local rough surface

TL;DR

This paper provides a rigorous mathematical analysis of a time-dependent fluid-solid interaction problem involving an elastic body above a rough surface, including reformulation, well-posedness, and stability results.

Contribution

It introduces a new formulation with transparent boundary conditions and proves well-posedness and stability for the problem.

Findings

Established well-posedness and stability of the problem.

Derived a priori estimates in the time domain.

Reformulated the scattering problem into a bounded domain.

Abstract

This paper is concerned with the mathematical analysis of time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition (TBC) on a hemisphere. Analyzing the reduced problem with Lax-Milgram lemma and abstract inversion theorem of Laplace transform, we prove the well-posedness and stability for the reduced problem. Moreover, an a priori estimate is established directly in the time domain for the acoustic wave and elastic displacement with using the energy method.