Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems

TL;DR

This paper proves the existence of global weak solutions for a class of singular 3D quasi-geostrophic systems with boundary degeneracy, using approximation methods combining Galerkin and regularization techniques.

Contribution

It introduces a novel approach to establish global weak solutions for singular geophysical models with boundary degeneracy, expanding the mathematical understanding of such systems.

Findings

Existence of global weak solutions for singular 3D quasi-geostrophic systems.

Development of approximation methods combining Galerkin and regularization.

Handling boundary degeneracy in the density profile.

Abstract

Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+}\times\Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2D$ torus or a bounded convex domain in $\mathbb{R}^2$. In this paper, we show the existence of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary.