A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain

TL;DR

This paper proves the global well-posedness of the three-dimensional inviscid quasi-geostrophic equation in a cylindrical domain with specific boundary conditions, ensuring existence and uniqueness of solutions under bounded initial potential vorticity.

Contribution

It establishes the first rigorous global existence and uniqueness results for this equation with these boundary conditions in a cylindrical setting.

Findings

Global existence and uniqueness of solutions for bounded initial PV.

Classical solutions when initial PV is differentiable.

Applicable to cylindrical domains with specific boundary conditions.

Abstract

The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existence and uniqueness of a generalized solution is proven, provided that the initial potential vorticity (PV) field is essentially bounded. If the initial PV field is differentiable, then the solution is shown to satisfy the system in the classical sense.