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 mu(3) under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function p is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions mu(0) = (mu(1), mu(2)) epsilon W-2(2) (Omega), integral(1)(0)(partial derivative(1)mu(1) + partial derivative(2)mu(2))dz =0, rho(0) epsilon W-2(2)(Omega), a weak solution exists and is unique and mu(x3) epsilon W-2(1)(QT), rho(x3) (QT) and the norms parallel to del(alpha)parallel to(Omega), parallel to del(rho)parallel to(Omega)are continuous in t.