Existence ‘in the large’ of a solution to the system of equations of large-scale ocean dynamics on a manifold

A theorem is presented proving the unique solvability ‘in the large’ of the system of primitive equations on an arbitrary smooth oriented Riemannian manifold in a cylindrical domain. Namely, it is shown for an arbitrary interval of time $[0,T]$, in the $3$d domain $\Omega\equiv\Omega'\times[-h,0]$, where $h=\mathrm{const}$ and $\Omega'$ is a compactly embedded subdomain of a $2$-manifold $\mathscr{M}$, for any viscosity coefficients $\mu,\nu,\mu_1,\nu_1>0$ and initial conditions $\mathbf{u}_0\in\mathbf{W}_2^2(\Omega)$, $\displaystyle\int_{-h}^0\operatorname{div}\mathbf{u}_0\,dz=0$, and $\rho_0\in W_2^2(\Omega)$, there exists a unique generalized solution such that $\partial_z\mathbf{u} \in\mathbf{W}_2^1(Q_T)$, $\partial_z\rho \in W_2^1(Q_T)$ ($z$ is the vertical variable) and the norms $\|\mathbf{u}\|_{\mathbf{W}^1_2(\Omega)}$ and $\|\rho\|_{W^1_2(\Omega)}$ are continuous in $t$.
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