The barotropic quasi-geostrophic equation under a free surface

TL;DR

This paper proves the global existence and uniqueness of weak solutions for the inviscid barotropic quasi-geostrophic equation with a free surface, addressing the challenges posed by non-standard boundary conditions.

Contribution

It establishes the first rigorous proof of global existence and uniqueness for this equation with a free surface boundary condition.

Findings

Global existence and uniqueness of weak solutions

Solutions satisfy initial and boundary conditions classically

Uniform bounds on potential vorticity

Abstract

The inviscid barotropic quasi-geostrophic equation with a free surface is considered. The free surface mandates a non-standard boundary condition. The global existence existence and uniqueness of a weak solution is established, thanks to the uniform in time bounds on the potential vorticity. The solution is also shown to satisfy the initial and boundary conditions in the classical sense.