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

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.