Cauchy problem for viscous rotating shallow water equations

TL;DR

This paper proves the global existence of solutions for a viscous rotating shallow water system with surface tension, highlighting the effects of rotation on the mathematical analysis of the model.

Contribution

It establishes the global existence of solutions in Besov spaces for the rotating shallow water equations with surface tension, accounting for the coupling caused by Coriolis forces.

Findings

Global existence of solutions near equilibrium

Coupling effects due to rotation require additional regularity

Analysis extends previous models without Coriolis forces

Abstract

We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.