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Local Solvability of Free Boundary Problems for the Two-phase Navier-Stokes Equations with Surface Tension in the Whole Space

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Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 80))

Abstract

We consider the free boundary problem of the two-phase Navier-Stokes equation with surface tension and gravity in the whole space. We prove a local-in-time unique existence theorem in the space W 2,1 q,p with 2 < p < ∞ and n < q < ∞ for any initial data satisfying certain compatibility conditions. Our theorem is proved by the standard fixed point argument based on the maximal L p -L q regularity theorem for the corresponding linearized equations.

Mathematics Subject Classification (2000). Primary 35R35; Secondary 35Q30, 76D05, 76D03, 76T10.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shizuoka University, Shizuoka, 422-8529, Japan

    Senjo Shimizu

Authors
  1. Senjo Shimizu
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Correspondence to Senjo Shimizu .

Editor information

Editors and Affiliations

  1. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Joachim Escher

  2. School of Physical Sciences, Dept. Mathematics, University of California, Irvine, Irvine, 92697-3875, California, USA

    Patrick Guidotti

  3. , Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr 7, Darmstadt, 64289, Germany

    Matthias Hieber

  4. Inst. Applied Mathematics & Mechanics, Warsaw University, ul. Banacha 2, Warszawa, 02-097, Poland

    Piotr Mucha

  5. , FB Mathematik und Informatik, Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, Halle, 06099, Germany

    Jan W. Prüss

  6. 3-4-1, Ohkubo Shinjuku-ku, Tokyo, 169-8555, Japan

    Yoshihiro Shibata

  7. , Deparment of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

    Gieri Simonett

  8. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Christoph Walker

  9. Fac. Mathematics & Information Sciences, Warsaw University of Technology, pl. Politechniki 1, Warszawa, 00-661, Poland

    Wojciech Zajaczkowski

Additional information

Dedicated to Professor Herbert Amann on the occasion of his 70th birthday.

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Shimizu, S. (2011). Local Solvability of Free Boundary Problems for the Two-phase Navier-Stokes Equations with Surface Tension in the Whole Space. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_32

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