Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u3 under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\hat{\bf u}}_{0} = (u_1, u_2) \in W_{2}^{2}(\Omega), \quad \int_{0}^{1}(\partial_{1}u_{1} + \partial_{2}u_{2})dz = 0, \quad \rho_{0} \in W_{2}^{2}(\Omega),$$ \end{document} a weak solution exists and is unique and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\hat{\bf u}}_{x_3} \in {\bf W}_{2}^{1}(Q_T), \rho_{x_{3}} \in W_{2}^{1}(Q_T)$$ \end{document} and the norms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\|\nabla{\hat{\bf u}}\|_{\Omega}, \|\nabla \rho \|_{\Omega}$$ \end{document} are continuous in t.