Weakly nonlinear models for hydroelastic water waves

TL;DR

This paper develops weakly nonlinear reduced models for hydroelastic water waves coupled with a viscoelastic plate, deriving bidirectional and unidirectional equations that capture key interface dynamics and dispersive effects.

Contribution

It introduces novel bidirectional and unidirectional weakly nonlinear models for hydroelastic water waves with rigorous well-posedness results.

Findings

Derived bidirectional nonlocal evolution equations up to quadratic order.

Established local well-posedness for the bidirectional model with small data.

Proved global well-posedness for unidirectional models with small data.

Abstract

In this work, we derive reduced interface models for hydroelastic water waves coupled to a nonlinear viscoelastic plate. In a weakly nonlinear small-steepness regime we obtain bidirectional nonlocal evolution equations capturing the interface dynamics up to quadratic order, and we also derive two unidirectional models describing one-way propagation while retaining the leading dispersive and dissipative effects induced by the plate. Remarkably, one of the bidirectional model has a doubly nonlinear structure in the sense that there there is a nonlinear elliptic operator acting on the acceleration of the interface. We prove local well-posedness for the bidirectional model for small data via a two-parameter regularization and nested fixed points. For the unidirectional models, we obtain local well-posedness for arbitrary data and global well-posedness for small data.