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

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We deal with the motion of two incompressible fluids in a container one of which is inside another, taking into account the surface tension. We prove that this problem is uniquely solvable in an infinite time interval provided that the initial velocity of the fluids is small and the 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 a certain radius. The proof is based on an exponential estimate of a generalized energy and on a local existence theorem for the problem in anisotropic Hölder spaces. We follow the scheme developed by one of the authors for proving the global solvability of a problem governing the motion of one incompressible capillary fluid bounded by a free surface. Bibliography: 18 titles.

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Authors and Affiliations

  1. Institute of Problems of Mechanical Engineering, St. Petersburg, Russia

    I. V. Denisova

  2. St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg, Russia

    V. A. Solonnikov

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  1. I. V. Denisova
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  2. V. A. Solonnikov
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Correspondence to I. V. Denisova.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 397, 2011, pp. 20–52.

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Denisova, I.V., Solonnikov, V.A. Global solvability of a problem governing the motion of two incompressible capillary fluids in a container. J Math Sci 185, 668–686 (2012). https://doi.org/10.1007/s10958-012-0951-8

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  • DOI: https://doi.org/10.1007/s10958-012-0951-8

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