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Strong solutions for two-phase free boundary problems for a class of non-Newtonian fluids

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Abstract

Consider the two-phase free boundary problem subject to surface tension and gravitational forces for a class of non-Newtonian fluids with stress tensors T n of the form \({T_n=-qI+\mu_n(|D(v)|^2)D(v)}\) for \({n=1,2}\), respectively, where the viscosity functions \({\mu_n}\) satisfy \({\mu_n\in C^3([0,\infty))}\) and \({\mu_n(0) > 0}\) for \({n=1,2}\). It is shown that for given \({T > 0}\) this problem admits a unique strong solution on (0,T) provided the initial data are sufficiently small in their natural norms.

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Authors and Affiliations

  1. Department of Mathematics, TU Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Matthias Hieber

  2. 607 Benedum Engineering Hall, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Matthias Hieber

  3. Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo, 169-8555, Japan

    Hirokazu Saito

Authors
  1. Matthias Hieber
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  2. Hirokazu Saito
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Correspondence to Matthias Hieber.

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Dedicated to Jan Prüss on the occasion of his 65th Birthday.

This work was supported by the Japanese–German Graduate Externship JGGE. The second author was partly supported by Grant-in-Aid for JSPS Research Fellows (No. 25·5259).

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Hieber, M., Saito, H. Strong solutions for two-phase free boundary problems for a class of non-Newtonian fluids. J. Evol. Equ. 17, 335–358 (2017). https://doi.org/10.1007/s00028-016-0351-5

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