Global small solutions to a tropical climate model without thermal diffusion

TL;DR

This paper proves the global existence of solutions to a tropical climate model without thermal diffusion, using novel techniques to uncover hidden dissipation and establish key estimates.

Contribution

It demonstrates global well-posedness for a tropical climate model lacking thermal diffusion, introducing a generalized commutator estimate and revealing hidden dissipation mechanisms.

Findings

Established global classical solutions under small initial data.

Identified hidden thermal diffusion through equation structure.

Developed a generalized commutator estimate applicable to PDEs.

Abstract

We obtain the global well-posedness of classical solutions to a tropical climate model derived by Feireisl-Majda-Pauluis in \cite{FMP} with only the dissipation of the first baroclinic model of the velocity ($-\eta \Delta v$) under small initial data. The main difficulty is the absence of thermal diffusion as the work by Li-Titi in \cite{LT}. 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-Palay theory, we establish a generalized commutator estimate, which may be applied to other partial differential equations.