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 Mat. Sb., 1994, Volume 185, Number 3, Pages 41–68 (Mi msb885)

On uniform stabilization of solutions of the exterior problem for the Navier–Stokes equations

F. Kh. Mukminov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The first mixed problem with homogeneous boundary conditions for the system of Stokes and Navier–Stokes equations is considered in a cylinder $D=(0,\infty)\times\Omega$, where $\Omega$ is the complement of the closure of a bounded domain in $R^3$. For solutions of both problems uniform decay with rate $t^{-3/2}$ is proved under certain smoothness conditions on the boundary under the assumption that the initial vector belongs to $\mathbf{L}_2$. Here in the case of the nonlinear problem it is additionally assumed that a weak solution satisfies the strong energy inequality.
A result on the decay of a solution of the linearized system of Navier–Stokes equations is used in the proof of the main assertion on stabilization of a solution of the problem with a bounded initial vector-valued function: existence of a uniform zero spherical limit mean of the initial function is necessary and sufficient for uniform stabilization of the solution to zero.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 81:2, 297–320

Bibliographic databases:

UDC: 517.9
MSC: 35Q30, 35B40, 76D05

Citation: F. Kh. Mukminov, “On uniform stabilization of solutions of the exterior problem for the Navier–Stokes equations”, Mat. Sb., 185:3 (1994), 41–68; Russian Acad. Sci. Sb. Math., 81:2 (1995), 297–320

Citation in format AMSBIB
\Bibitem{Muk94} \by F.~Kh.~Mukminov \paper On uniform stabilization of solutions of the~exterior problem for the~Navier--Stokes equations \jour Mat. Sb. \yr 1994 \vol 185 \issue 3 \pages 41--68 \mathnet{http://mi.mathnet.ru/msb885} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1268797} \zmath{https://zbmath.org/?q=an:0836.35123} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 81 \issue 2 \pages 297--320 \crossref{https://doi.org/10.1070/SM1995v081n02ABEH003540} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RB51300003} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. M. Asadullin, F. Kh. Mukminov, “Uniqueness classes for a non-stationary system of Stokes equations in unbounded domains”, Sb. Math., 187:3 (1996), 315–333
2. Mukminov F., “On the Set of Limit Functions for the Solutions of the Nonstationary Exterior Stokes Problem”, Differ. Equ., 33:8 (1997), 1108–1112
3. Gorshkov A.V., “Boundary Stabilization of Stokes System in Exterior Domains”, J. Math. Fluid Mech., 18:4 (2016), 679–697
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