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

This article is cited in 3 scientific papers (total in 3 papers)

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
Received: 17.05.1993

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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  mathscinet  zmath  isi
    3. Gorshkov A.V., “Boundary Stabilization of Stokes System in Exterior Domains”, J. Math. Fluid Mech., 18:4 (2016), 679–697  crossref  mathscinet  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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