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 \(s\in [\frac{1}{2},1]\). As a consequence, we improve the conditions stated in the paper Miao and Zheng (Commun Math Phys 321:33–67, 2013) in order to obtain a global well-posedness result in the case of axisymmetric initial data.

在本文中，对于具有水平耗散的3-D Navier–Stokes–Boussinesq系统，其对垂直导数没有平滑作用，我们证明了L ^ {中的解\（（u，\ rho）\）的唯一性结果\ infty} _T \ big（H ^ {0，s} \ times H ^ {0,1-s} \ big）\）与\（（\ nabla _h u，\ nabla _h \ rho）\ in L ^ { 2} _T \ big（H ^ {0，s} \ times H ^ {0,1-s} \ big）\）和\（s \ in [\ frac {1} {2}，1] \）。因此，我们改善了Miao和Zheng（Commun Math Phys 321：33–67，2013）中所述的条件，以便在轴对称初始数据的情况下获得整体的适定性结果。