Global Regularity for a 2D Tropical Climate Model with Fractional Dissipation

This paper examines the global regularity problem for a 2D tropical climate model with fractional dissipation. The inviscid version of this model was derived by Frierson, Majda and Pauluis for large-scale dynamics of precipitation fronts in the tropical atmosphere. The model considered here has some very special features. This nonlinear system involves interactions between a divergence-free vector field and a non-divergence-free vector field. In addition, the fractional dissipation not only models long-range interactions but also allows simultaneous investigations of a family of system. Our study leads to the global regularity of solutions when the indices of the fractional Laplacian are in two very broad ranges. In order to establish the global-in-time bounds, we introduce an efficient way to control the gradient of the non-divergence-free vector field and make sharp estimates by controlling the regularity of related quantities simultaneously.