On the Classical Solutions of Two Dimensional Inviscid Rotating Shallow Water System

TL;DR

This paper establishes the global existence and long-term behavior of classical solutions for the 2D inviscid rotating shallow water equations with small initial data, using a reformulation into a Klein-Gordon system.

Contribution

It introduces a novel reformulation of the system into a symmetric Klein-Gordon form and proves global solutions under small initial data with zero vorticity, also analyzing lifespan for general data.

Findings

Global existence of classical solutions for small initial data.

Reformulation into a symmetric Klein-Gordon system facilitates analysis.

Lifespan lower bound inversely proportional to initial vorticity.

Abstract

We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity.