2D TROPICAL CLIMATE MODEL WITH FRACTIONAL DISSIPATION AND WITHOUT THERMAL DIFFUSION

This paper investigates the global existence and regularity problem on 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 fractionally dissipated system studied here is capable of modeling nonlocal and long-range interactions. Mathematically this system involves two parameters controlling the regularization due to the dissipation and our aim is the global regularity for smallest possible parameters. 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. 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. The global estimates on the Sobolev norms of the solutions are extremely involved and lengthy. We take advantage of some of the most recent developments and tools on the fractional Laplacian operators and introduce some new techniques.