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

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 $(u,\rho )\in L_T^{\infty }(H^{0,s}\times H^{0,1-s})$ with $({\nabla }_{h}u,{\nabla }_h\rho )\in L_T^2(H^{0,s}\times H^{0,1-s})$ 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.