# Uniqueness result for the 3-D Navier-Stokes-Boussinesq Equations with Horizontal Dissipation

## TL;DR

This paper establishes a uniqueness result for solutions to the 3-D Navier-Stokes-Boussinesq equations with horizontal dissipation, improving conditions for global well-posedness in axisymmetric cases.

## Contribution

It proves a new uniqueness theorem for solutions with specific regularity, enhancing previous results and enabling global well-posedness for axisymmetric initial data.

## Key findings

- Proved uniqueness of solutions under new regularity conditions.
- Improved conditions for global well-posedness in axisymmetric cases.
- Extended the understanding of horizontal dissipation effects.

## Abstract

In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions $ (u,\rho)\in L^{\infty}_T\big( H^{0,s}\times H^{0,1-s}\big)$ with $ (\nabla_h u,\nabla_h\rho)\in L^{2}_T\big( H^{0,s}\times H^{0,1-s}\big)$ and $s\in [1/2,1]$. As a consequence, we improve the conditions stated in the paper \cite{Miao} in order to obtain a global well-posedness result in the case of axisymmetric initial data.