On the well-posedness of the inviscid multi-layer quasi-geostrophic equations

TL;DR

This paper proves the global well-posedness of the inviscid multi-layer quasi-geostrophic equations with non-homogeneous boundary conditions, using mode decomposition and elliptic regularity theories.

Contribution

It establishes the well-posedness of these equations under general boundary conditions, extending previous results to more realistic boundary scenarios.

Findings

Global existence and uniqueness of solutions

Classical satisfaction of initial and boundary conditions

Applicability of 2D elliptic regularity theories

Abstract

The inviscid multi-layer quasi-geostrophic equations are considered over an arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions are imposed to accommodate the free fluctuations of the top and layer interfaces. Using the barotropic and baroclinic modes in the vertical direction, the elliptic system governing the streamfunctions and the potential vorticity is decomposed into a sequence of scalar elliptic boundary value problems, where the regularity theories from the two-dimensional case can be applied. With the initial potential vorticity being essentially bounded, the multi-layer quasi-equations are then shown to be globally well-posed, and the initial and boundary conditions are satisfied in the classical sense.