Well-posedness of a nonlinear shallow water model for an oscillating water column with time-dependent air pressure

TL;DR

This paper develops and analyzes a new nonlinear mathematical model for an oscillating water column, incorporating a time-dependent air pressure, and proves its local well-posedness using an iterative scheme.

Contribution

It introduces a novel transmission problem with a time-dependent boundary condition for the OWC model and establishes its well-posedness.

Findings

The model accounts for variable air pressure inside the chamber.

The transmission problem reduces to a quasilinear hyperbolic PDE with a semi-linear boundary condition.

Local well-posedness is proven using linear and nonlinear estimates.

Abstract

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device is reformulated as a transmission problem related to the interaction between waves and a fixed partially-immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in \cite{bocchihevergara2021}. We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE.