The $ 3D $-primitive equations with only horizontal viscosity are considered on a cylindrical domain $ \Omega = (-h,h) \times G $, $ G\subset \mathbb{R}^2 $ smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering a vanishing vertical viscosity limit, we apply a direct approach which in particular avoids unnecessary boundary conditions on top and bottom. For the initial value problem, we obtain existence and uniqueness of local $ z $-weak solutions for initial data in $ H^1((-h,h),L^2(G)) $ and local strong solutions for initial data in $ H^1(\Omega) $. If $ v_0\in H^1((-h,h),L^2(G)) $, $ \partial_z v_0\in L^q(\Omega) $ for $ q>2 $, then the $ z $-weak solution regularizes instantaneously and thus extends to a global strong solution. This goes beyond the global well-posedness result by Cao, Li and Titi (J. Func. Anal. 272(11): 4606-4641, 2017) for initial data near $ H^1 $ in the periodic setting. For the time-periodic problem, existence and uniqueness of $ z $-weak and strong time periodic solutions is proven for small forces. Since this is a model with hyperbolic and parabolic features for which classical results are not directly applicable, such results for the time-periodic problem even for small forces are not self-evident.

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具有水平粘度的本原方程：物理边界条件的初始值和时间周期问题

在圆柱域上考虑仅具有水平粘度的$ 3D $本原方程$ \ Omega =（-h，h）\ times G $，$ G \ subset \ mathbb {R} ^ 2 $光滑，具有物理Dirichlet边界条件。我们没有考虑消失的垂直粘度极限，而是采用了直接方法，特别是避免了顶部和底部的不必要边界条件。对于初始值问题，我们获得$ H ^ 1（（-h，h），L ^ 2（G））$中初始数据的局部$ z $弱解的存在性和唯一性，以及初始数据的局部强解在$ H ^ 1（\ Omega）$中。如果$ v_0 \ in H ^ 1（（-h，h），L ^ 2（G））$，$ \ partial_z v_0 \ in L ^ q（\ Omega）$ for $ q> 2 $，则$ z $-弱解决方案会立即进行规范化，从而扩展到全球性的强解决方案。这超出了Cao，Li和Titi（J. 功能 肛门 272（11）：4606-4641，2017）在周期设置中接近$ H ^ 1 $的初始数据。对于时间周期问题，弱势和强时间周期解的存在和唯一性被证明适用于小型部队。由于这是具有双曲线和抛物线特征的模型，因此无法直接应用经典结果，因此，即使对于较小的力，这种关于时间周期问题的结果也不言而喻。