Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions

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.