Stability of 3D perturbations near a special 2D solution to the rotating Boussinesq equations

Abstract

AbstractThe three‐dimensional (3D) rotating Boussinesq system governs the dynamics of fluid flows in the atmosphere and the ocean. It admits a special two‐dimensional (2D) solution which involves only vertical motion and density changes that both vary only in the horizontal plane. This solution illustrates the important effect of gravity and leads to the renowned Brunt–Väisälä frequency. A natural question with practical applications is whether or not general 3D perturbations near this 2D solution are stable. This paper establishes the stability of any 3D perturbations even when the Boussinesq system involves only horizontal dissipation and horizontal thermal conduction. An anisotropic Sobolev setting is selected to reduce the regularity assumption on the initial data.