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

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

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
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
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
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
5. St. Petersburg Math. J., 32:1 (2021), 91–137
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
7. Saito H., Shibata Y., Zhang X., “Some Free Boundary Problem For Two-Phase Inhomogeneous Incompressible Flows”, SIAM J. Math. Anal., 52:4 (2020), 3397–3443
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