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 Mat. Sb., 2011, Volume 202, Number 10, Pages 55–86 (Mi msb7769)

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

A. V. Drutsa

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 $t$.
Bibliography: 12 titles.

Keywords: primitive equations, ocean dynamics equations, nonlinear partial differential equations, a priori bounds, existence ‘in the large’.

DOI: https://doi.org/10.4213/sm7769

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English version:
Sbornik: Mathematics, 2011, 202:10, 1463–1492

Bibliographic databases:

Document Type: Article
UDC: 517.958
MSC: Primary 35A01, 35Q35; Secondary 35D30

Citation: A. V. Drutsa, “Existence ‘in the large’ of a solution to the system of equations of large-scale ocean dynamics on a manifold”, Mat. Sb., 202:10 (2011), 55–86; Sb. Math., 202:10 (2011), 1463–1492

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Drutsa A.V., “O poryadke skhodimosti raznostnykh skhem dlya uravnenii dinamiki okeana”, Vychislitelnye metody i programmirovanie: novye vychislitelnye tekhnologii, 13:1 (2012), 398–408
2. A. V. Drutsa, “A difference scheme for equations of ocean dynamics on unstructured grids”, Russian J. Numer. Anal. Math. Modelling, 29:3 (2014), 145–165
3. G. M. Kobel'kov, “On the existence of a global solution of the modified Navier–Stokes equations”, Trans. Moscow Math. Soc., 77 (2016), 177–201
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