Global in Time Weak Solutions to Singular Three-Dimensional Quasi-Geostrophic Systems

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5881-5914, October 2024.
Abstract. Geophysicists have studied three-dimensional quasi-geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large timescale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in [math] with an equation on the boundary satisfied by the trace, where [math] is either a two-dimensional torus or a bounded convex domain in [math]. In this paper, we show the existence of global in time weak solutions to a family of singular three-dimensional 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.