A Liouville-type theorem for the 3D primitive equations

Abstract

The 3D primitive equations are used in most geophysical fluid models to approximate the large scale oceanic and atmospheric dynamics. We prove that there do not exist smooth stationary solutions to the 3D primitive equations with compact support, independently of the presence of the Coriolis rotation term or the viscosity. This result is in strong contrast with the recently established existence of compactly supported smooth solutions to the incompressible 3D Euler equations.