Large Time existence For 1D Green-Naghdi equations

TL;DR

This paper proves the existence of solutions for the 1D Green-Naghdi equations, demonstrating that solutions can be constructed iteratively without loss of regularity, which is significant for modeling large amplitude surface waves.

Contribution

It establishes a rigorous existence result for solutions of the 1D Green-Naghdi equations using a Picard iterative scheme, ensuring regularity preservation.

Findings

Solutions can be constructed via Picard iteration.

No loss of regularity in solutions.

Applicable to large amplitude surface waves.

Abstract

We consider here the $1D $ Green-Naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves. We show that the solution of the Green-Naghdi 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.