Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
Amru Hussein, Martin Saal, Marc Wrona · Discrete and Continuous Dynamical Systems · 2020
The \begin{document}$ 3D $\end{document} -primitive equations with only horizontal viscosity are considered on a cylindrical domain \begin{document}$ \Omega = (-h,h) \times G $\end{document} , \begin{document}$ G\subset \mathbb{R}^2 $\end{document} smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering a vanishing vertical viscosity limit, we apply a direct approach which in particular avoids unnecessary boundary conditions on top and bottom. For the initial value problem, we obtain existence and uniqueness of local \begin{document}$ z $\end{document} -weak solutions for initial data in \begin{document}$ H^1((-h,h),L^2(G)) $\end{document} and local strong solutions for initial data in \begin{document}$ H^1(\Omega) $\end{document} . If \begin{document}$ v_0\in H^1((-h,h),L^2(G)) $\end{document} , \begin{document}$ \partial_z v_0\in L^q(\Omega) $\end{document} for \begin{document}$ q>2 $\end{document} , then the \begin{document}$ z $\end{document} -weak solution regularizes instantaneously and thus extends to a global strong solution. This goes beyond the global well-posedness result by Cao, Li and Titi (J. Func. Anal. 272(11): 4606-4641, 2017) for initial data near \begin{document}$ H^1 $\end{document} in the periodic setting. For the time-periodic problem, existence and uniqueness of \begin{document}$ z $\end{document} -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.