Abstract.
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 u 3 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
a weak solution exists and is unique and \({\hat{\bf u}}_{x_3} \in {\bf W}_{2}^{1}(Q_T), \rho_{x_{3}} \in W_{2}^{1}(Q_T)\) and the norms \(\|\nabla{\hat{\bf u}}\|_{\Omega}, \|\nabla \rho \|_{\Omega}\) are continuous in t.
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Communicated by A. V. Fursikov
The work was carried out under partial support of Russian Foundation for Basic Research (project 05-01-00864).
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Kobelkov, G.M. Existence of a Solution “in the Large” for Ocean Dynamics Equations. J. math. fluid mech. 9, 588–610 (2007). https://doi.org/10.1007/s00021-006-0228-4
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DOI: https://doi.org/10.1007/s00021-006-0228-4