# Singularity formation and global existence of classical solutions for one dimensional rotating shallow water system

## TL;DR

This paper investigates the conditions under which classical solutions to the one-dimensional rotating shallow water system either develop singularities in finite time or exist globally, depending on initial data characteristics.

## Contribution

It establishes finite-time singularity formation for large classes of initial data and proves global existence for small data with constant potential vorticity.

## Key findings

- Finite-time singularities occur for certain initial conditions.
- Global existence is proven for small data with constant potential vorticity.
- Singularity formation and global existence are characterized by initial data properties.

## Abstract

We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of ${C}^1$ initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.