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Zap. Nauchn. Sem. POMI, 2011, Volume 397, Pages 20–52 (Mi znsl4666)  

This article is cited in 6 scientific papers (total in 6 papers)

Global solvability of a problem governing the motion of two incompressible capillary fluids in a container

I. V. Denisovaa, V. A. Solonnikovb

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We deal with the motion of two incompressible fluids in a container, one of which is inside another. We take surface tension into account. We prove that this problem is uniquely solvable in an infinite time interval provided the initial velocity of the liquids is small and an initial configuration of the inner fluid is close to a ball. Moreover, we show that the velocity decays exponentially at infinity with respect to time and that the interface between the fluids tends to a sphere of the certain radius. The proof is based on the exponential estimate of a generalized energy and on a local existence theorem of the problem in anisotropic Hölder spaces. We follow the scheme developed by one of the authors for proving global solvability of a problem governing the motion of one incompressible capillary fluid bounded by a free surface.

Key words and phrases: two-phase problem with unknown interface, incompressible capillary fluid, Lagrangian coordinates, Hölder spaces.

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English version:
Journal of Mathematical Sciences (New York), 2012, 185:5, 668–686

Bibliographic databases:

UDC: 532.526
Received: 03.11.2011

Citation: I. V. Denisova, V. A. Solonnikov, “Global solvability of a problem governing the motion of two incompressible capillary fluids in a container”, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Zap. Nauchn. Sem. POMI, 397, POMI, St. Petersburg, 2011, 20–52; J. Math. Sci. (N. Y.), 185:5 (2012), 668–686

Citation in format AMSBIB
\Bibitem{DenSol11}
\by I.~V.~Denisova, V.~A.~Solonnikov
\paper Global solvability of a~problem governing the motion of two incompressible capillary fluids in a~container
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~42
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 397
\pages 20--52
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl4666}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2870107}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 185
\issue 5
\pages 668--686
\crossref{https://doi.org/10.1007/s10958-012-0951-8}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866924649}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Denisova I.V., “Global l-2-Solvability of a Problem Governing Two-Phase Fluid Motion Without Surface Tension”, Port Math., 71:1 (2014), 1–24  crossref  mathscinet  zmath  isi  elib  scopus
    2. Denisova I.V., “on Energy Inequality For the Problem on the Evolution of Two Fluids of Different Types Without Surface Tension”, J. Math. Fluid Mech., 17:1 (2015), 183–198  crossref  mathscinet  zmath  isi  elib  scopus
    3. Maryani S. Saito H., “on the R-Boundedness of Solution Operator Families For Two-Phase Stokes Resolvent Equations”, Differ. Integral Equ., 30:1-2 (2017), 1–52  mathscinet  zmath  isi
    4. Denisova I.V. Solonnikov V.A., “L-2-Theory For Two Incompressible Fluids Separated By a Free Interface”, Topol. Methods Nonlinear Anal., 52:1 (2018), 213–238  crossref  isi  scopus
    5. V. A. Solonnikov, “$L_2$-theory for two viscous fluids of different type: compressible and incompressible”, Algebra i analiz, 32:1 (2020), 121–186  mathnet
    6. V. A. Solonnikov, “$L_p$-estimates of solution of the free boundary problem for viscous compressible and incompressible fluids in the linear approximation”, Algebra i analiz, 32:3 (2020), 254–291  mathnet
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