Analysis of a three-dimensional rapidly rotating convection model without thermal diffusion

TL;DR

This paper analyzes a 3D rapidly rotating convection model without thermal diffusion, establishing existence, uniqueness, and well-posedness of solutions, and justifying the limit as thermal diffusion vanishes.

Contribution

It introduces a rigorous analysis of a convection model without thermal diffusion, including the vanishing diffusivity limit, extending previous work.

Findings

Global existence and uniqueness of weak solutions

Well-posedness of strong solutions

Validation of the vanishing thermal diffusion limit

Abstract

We study a three-dimensional rapidly rotating convection model featuring tall columnar structures, in the absence of thermal diffusion. We establish the global existence and uniqueness of weak solutions, as well as the Hadamard well-posedness of global strong solutions to this model. The lack of thermal diffusion introduces significant challenges in the analysis. To overcome these challenges, we first investigate the regularized model with thermal diffusion and establish delicate estimates that are independent of the thermal diffusion coefficient, and consequently justify the vanishing diffusivity limit. This work serves as a continuation of our previous paper [6].