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 Tr. Mosk. Mat. Obs., 2016, Volume 77, Issue 2, Pages 219–249 (Mi mmo588)

On the existence of a global solution of the modified Navier–Stokes equations

G. M. Kobel'kovab

a Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia
b Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

Abstract: We prove global existence theorems for initial-boundary value problems for the modified Navier–Stokes equations used when modeling ocean dynamic processes. First, the case of distinct vertical and horizontal viscosities for the Navier–Stokes equations is considered. Then a result due to Ladyzhenskaya for the modified Navier–Stokes equations is improved, whereby the elliptic operator is strengthened with respect to the horizontal variables alone and only for the horizontal momentum equations. Finally, the global existence and uniqueness of a solution is proved for the primitive equations describing the large-scale ocean dynamics.

Key words and phrases: Navier–Stokes equations, primitive equations, large-scale ocean dynamics, modification of Navier–Stokes equations, global existence.

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English version:
Transactions of the Moscow Mathematical Society, 2016, 77, 177–201

UDC: 517.955.2
MSC: 76D05, 35Q30

Citation: G. M. Kobel'kov, “On the existence of a global solution of the modified Navier–Stokes equations”, Tr. Mosk. Mat. Obs., 77, no. 2, MCCME, M., 2016, 219–249; Trans. Moscow Math. Soc., 77 (2016), 177–201

Citation in format AMSBIB
\Bibitem{Kob16} \by G.~M.~Kobel'kov \paper On the existence of a global solution of the modified Navier--Stokes equations \serial Tr. Mosk. Mat. Obs. \yr 2016 \vol 77 \issue 2 \pages 219--249 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo588} \elib{http://elibrary.ru/item.asp?id=28931389} \transl \jour Trans. Moscow Math. Soc. \yr 2016 \vol 77 \pages 177--201 \crossref{https://doi.org/10.1090/mosc/258} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85002251319}