On the local well-posedness of the 1D Green-Naghdi system over a nonflat bottom

TL;DR

This paper proves local well-posedness for the 1D Green-Naghdi water wave system over uneven bottoms, extending the range of initial conditions for which solutions exist and are unique.

Contribution

It establishes local well-posedness of the 1D Green-Naghdi system over a nonflat bottom in a broader Sobolev space range using a Lagrangian formulation.

Findings

Proves local well-posedness for (h,u) in Sobolev spaces with s > 1/2.

Uses a Lagrangian formulation on a Sobolev diffeomorphism group.

Improves the existing well-posedness results for the system.

Abstract

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