On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect

TL;DR

This paper derives and justifies a shallow-water approximation model incorporating Earth's rotation effects, demonstrating its validity and well-posedness for describing long-crested shallow-water waves in equatorial regions.

Contribution

It introduces a new rotation-Green-Naghdi model, rigorously justifies its approximation by the rotation-Camassa-Holm equation, and proves local well-posedness in Sobolev spaces.

Findings

Solutions of R-GN tend to R-CH solutions in the specified regime.

The R-GN model is locally well-posed in Sobolev spaces.

The model captures wave-breaking phenomena similar to the Camassa-Holm equation.

Abstract

We consider an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation. This model equation has similar wave-breaking phenomena as the Camassa-Holm equation. It is analogous to the rotation-Green-Naghdi (R-GN) equations with the weak Earth's rotation effect, modeling the propagation of wave allowing large amplitude in shallow water. We provide here a rigorous justification showing that solutions of the R-GN equations tend to associated solution of the R-CH model equation in the Camassa-Holm regime with the small amplitude and the larger wavelength. Furthermore, we demonstrate that the R-GN model equations are locally well-posed in a Sobolev space by the refined energy estimates.