In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity $v_0\in H^2\left(\Omega \right)$ and the initial temperature $T_0\in H^1\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$ with ${\nabla }_H{T}_0\in L^q\left(\Omega \right)$, for some $q\in (2,\infty )$. Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to $H^2\left(\Omega \right)$. The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the “ mismatching ” of regularities between the horizontal momentum and temperature equations. To handle this “mismatching” of regularities, we introduce several auxiliary functions, i.e., $\eta ,\theta ,\phi ,$ and $\psi$ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity $v$ or its vertical derivative ${\partial }_{z}v$. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some $L_t^1\left(W_{\mathbf{x}}^{1,\infty }\right)$-type a priori estimates on $v$, we decompose the velocity into the “temperature-independent” and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Brézis–Gallouet–Wainger and Beale–Kato–Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work (Cao et al., 2016), is used, and a new logarithmic type Gronwall inequality is exploited.

在本文中，我们考虑了海洋和大气动力学的3D基本方程，其中水平动量方程中仅具有水平涡流粘度，而温度方程中仅具有垂直扩散率。针对任何初始数据建立强解的全局适定性，以便初始水平速度$v_0\in H^2（\Omega ）$ 和初始温度 ${Ť}_0\in H^{\mathrm{1个}}（\Omega ）\cap {\mathrm{大号}}^{\infty }（\Omega ）$ 与 ${\nabla }_H{Ť}_0\in {\mathrm{大号}}^q（\Omega ）$， 对于一些 $q\in （2，\infty ）$。此外，如果初始温度为$H^2（\Omega ）$。主要困难是缺乏垂直粘度和缺乏水平扩散性，它们相互影响，从而导致水平动量和温度方程之间的规律性“失配”。为了处理这种“不匹配”的规律性，我们引入了一些辅助功能，即$\eta ，\theta ，\phi ，$ 和 $\psi$ 在纸上是水平卷曲或温度与水平速度水平散度的某种适当组合 $v$ 或其垂直导数 ${\partial }_{ž}v$。为了克服由于水平扩散而导致的困难，这导致需要一些${\mathrm{大号}}_{Ť}^{\mathrm{1个}}（{\mathrm{w\; ^}}_{\mathbf{X}}^{\mathrm{1个}，\infty }）$-根据先验估计 $v$，我们分别使用Brézis–Gallouet–Wainger和Beale–Kato–Majda类型的对数Sobolev不等式，将速度分解为“温度无关”和温度相关部分，并以不同的方式处理它们。具体来说，使用了我们之前的工作（Cao等人，2016）中介绍的极限类型的对数Sobolev不等式，并利用了新的对数型Gronwall不等式。