On a Lagrangian formulation of the 1D Green-Naghdi system

TL;DR

This paper develops a Lagrangian formulation for the 1D Green-Naghdi water wave system and proves improved local well-posedness results in Sobolev spaces, enhancing understanding of shallow water wave dynamics.

Contribution

It introduces a Lagrangian formulation on a Sobolev diffeomorphism group and establishes local well-posedness for a broader Sobolev space range.

Findings

Lagrangian formulation on Sobolev diffeomorphism group

Proved local well-posedness for (h,u) in (1+H^s) x H^{s+1} with s > 1/2

Improved the well-posedness range for the 1D Green-Naghdi system

Abstract

In this paper we consider the 1D Green-Naghdi system. This system describes the evolution of water waves over a flat bottom in the shallow water regime in terms of the surface height $h$ and the horizontal velocity $u$. We give a Lagrangian formulation of the 1D Green-Naghdi system on a Sobolev type diffeomorphism group. As an application of this formulation we prove local well-posedness for $(h,u)$ in the Sobolev space $(1+H^s(\R)) \times H^{s+1}(\R),\; s > 1/2$. This improves the local well-posedness range for the 1D Green-Naghdi system.