# Well-Posedness of a kind of the free surface equation of shallow water wave

## TL;DR

This paper establishes local well-posedness and wave-breaking mechanisms for a one-dimensional free surface shallow water wave equation in Sobolev spaces, advancing understanding of its mathematical properties.

## Contribution

It proves local well-posedness and describes wave-breaking for the shallow water wave equation, which are novel results for this specific model.

## Key findings

- Proved local well-posedness in Sobolev spaces.
- Derived a wave-breaking mechanism for strong solutions.
- Enhanced understanding of the equation's mathematical behavior.

## Abstract

This paper is concerned with the Cauchy problem of the one-dimensional free surface equation of shallow water wave, we obtain local well-posedness of the free surface equation of shallow water wave in Sobolev spaces. In addition, we also derive a wave-breaking mechanism for strong solutions.