Derivation and analysis of a new 2D Green-Naghdi system

TL;DR

This paper introduces a new variant of the 2D Green-Naghdi equations for shallow water waves that maintains accuracy, allows control of rotational velocity, and ensures solution regularity and conservation properties.

Contribution

A novel 2D Green-Naghdi model enabling control of rotational velocity and regularity of solutions, with proven conservation of irrotationality.

Findings

Solution can be constructed via Picard iteration without regularity loss.

The new model conserves the almost irrotationality of the velocity.

It retains the same accuracy as standard Green-Naghdi equations.

Abstract

We derive here a variant of the 2D Green-Naghdi equations that model the propagation of two-directional, nonlinear dispersive waves in shallow water. This new model has the same accuracy as the standard $2D $ Green-Naghdi equations. Its mathematical interest is that it allows a control of the rotational part of the (vertically averaged) horizontal velocity, which is not the case for the usual Green-Naghdi equations. Using this property, we show that the solution of these new equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition. Finally, we prove that the new Green-Naghdi equations conserve the almost irrotationality of the vertically averaged horizontal component of the velocity.