A Liouville-type theorem for the 3D primitive equations

TL;DR

This paper proves that smooth stationary solutions with compact support do not exist for the 3D primitive equations, highlighting a fundamental difference from the 3D Euler equations and advancing understanding of geophysical fluid models.

Contribution

It establishes a Liouville-type theorem showing non-existence of compactly supported stationary solutions for the 3D primitive equations, regardless of rotation or viscosity effects.

Findings

No smooth stationary solutions with compact support exist for the 3D primitive equations.

The result contrasts with the existence of such solutions in the 3D Euler equations.

The theorem holds independently of Coriolis rotation and viscosity effects.

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.