On the global well-posedness for the Boussinesq system with horizontal dissipation

TL;DR

This paper proves the global well-posedness of the 3D Boussinesq system with horizontal dissipation for axisymmetric initial data without swirl, overcoming challenges due to lack of vertical dissipation using harmonic analysis techniques.

Contribution

It establishes the first global well-posedness result for this Boussinesq system with only horizontal dissipation and no vertical smoothing effect, utilizing a novel harmonic analysis approach.

Findings

Proved global existence and uniqueness of solutions.

Identified key harmonic analysis relationships for axisymmetric flows.

Demonstrated regularity despite absence of vertical dissipation.

Abstract

In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of \eqref{eq1.1} entails the desired regularity.