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. %These solutions are in the set of solutions with small norms. 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.