Global small solutions to a tropical climate model without thermal diffusion

Abstract

We obtain the global well-posedness of classical solutions to a tropical climate model with only the dissipation of the first baroclinic model of the velocity (−ηΔv) under small initial data. This model is a modified version of the original system derived by Frierson-Majda-Pauluis in Frierson et al. [Commun. Math. Sci. 2, 591-626 (2004)]. The main difficulty is the absence of thermal diffusion. To overcome it, we exploit the structure of the equations coming from the coupled terms, dissipation term, and damp term. Then, we find the hidden thermal diffusion. In addition, based on the Littlewood-Paley theory, we establish a generalized commutator estimate, which may be applied to other partial differential equations.