Global Stability for the 3D Boussinesq Equations With Fractional Dissipation Near the Hydrostatic Equilibrium

ABSTRACT

The global stability problem remains a core research topic in fluid dynamics, especially for those systems with anisotropic and partial dissipation. Previous work by Ji, Yan and Wu (Calc. Var. Partial Differ. Equ., 2022, 61(4): 136) established the global stability of the three‐dimensional (3D) Boussinesq system with horizontal dissipation and horizontal thermal diffusion near hydrostatic equilibrium. In this paper, we extend the dissipation operator from integer order to fractional order, namely, we study the global stability of the 3D Boussinesq system with horizontal fractional dissipation and . To overcome the difficulties caused by fractional dissipation in the stability study, we construct some novel anisotropic estimates involving fractional derivatives . In the framework of , we obtain the global stability for any . While in , the global stability holds under the condition or . These results provide a mathematical framework for analyzing geophysical flows with anisotropic turbulence, thereby advancing climate and ocean modeling.