The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

TL;DR

This paper proves that vortex patch boundaries remain smooth over time in the quasi-geostrophic shallow-water equations and shows these solutions converge to Euler solutions as the Rossby radius parameter approaches zero.

Contribution

It establishes boundary regularity persistence for vortex patches in QGSW equations and demonstrates convergence to Euler solutions as the Rossby radius diminishes.

Findings

Boundary smoothness persists over time for vortex patches.

Solutions converge to Euler solutions as Rossby radius tends to zero.

Provides mathematical foundation for QGSW and Euler equation relationship.

Abstract

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little H\"older spaces.