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Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

Part of: Miscellaneous topics - Partial differential equations Parabolic equations and systems Incompressible viscous fluids

Published online by Cambridge University Press:  02 December 2020

Bogdan–Vasile Matioc Open the ORCID record for Bogdan–Vasile Matioc [Opens in a new window]  and
Georg Prokert
Bogdan–Vasile Matioc
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Deutschland (bogdan.matioc@ur.de)
Georg Prokert
Affiliation:
Faculty of Mathematics and Computer Science, Technical University Eindhoven, The Netherlands (g.prokert@tue.nl)
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Abstract

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We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

Keywords

Stokes problemtwo-phasesingular integralscontour integral formulation

MSC classification

Primary: 35R37: Moving boundary problems
Secondary: 76D07: Stokes and related (Oseen, etc.) flows 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1815 - 1845
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press.

References

Abels, H. and Matioc, B.-V.. Well-posedness of the Muskat problem in subcritical L p-Sobolev spaces. arXiv:2003.07656. 2020.CrossRefGoogle Scholar
Amann, H.. Linear and quasilinear parabolic problems. Vol. I, volume 89 of Monographs in mathematics (Boston, MA: Birkhäuser Boston, Inc., 1995). Abstract linear theory.CrossRefGoogle Scholar
Angenent, S. B.. Nonlinear analytic semiflows. Proc. R. Soc. Edinb. A 115 (1990), 91107.CrossRefGoogle Scholar
Badea, A. and Duchon, J.. Capillary driven evolution of an interface between viscous fluids. Nonlinear Anal. 31 (1998), 385–403.CrossRefGoogle Scholar
Escher, J. and Simonett, G.. Analyticity of the interface in a free boundary problem. Math. Ann. 305 (1996), 439–459.CrossRefGoogle Scholar
Gancedo, F.. A survey for the Muskat problem and a new estimate. SeMA J. 74 (2017), 21–35.CrossRefGoogle Scholar
Granero-Belinchón, R. and Lazar, O.. Growth in the Muskat problem. Math. Model. Nat. Phenom. 15 (2020), 7.CrossRefGoogle Scholar
Ladyzhenskaya, O. A.. The mathematical theory of viscous incompressible flow. Revised English edition. (Translated from the Russian by Richard A. Silverman) (New York-London: Gordon and Breach Science Publishers, 1963).Google Scholar
Lunardi, A.. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, 16 (Basel: Birkhäuser Verlag, 1995).Google Scholar
Matioc, B.-V.. Viscous displacement in porous media: the Muskat problem in 2D. Trans. Am. Math. Soc. 370 (2018), 7511–7556.CrossRefGoogle Scholar
Matioc, B.-V.. The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12 (2019), 281–332.CrossRefGoogle Scholar
Prüss, J., Shao, Y. and Simonett, G.. On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension. Interfaces Free Bound. 17 (2015), 555–600.CrossRefGoogle Scholar