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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 969–1011 | Cite as

Compressible–Incompressible Two-Phase Flows with Phase Transition: Model Problem

  • Keiichi Watanabe
Article

Abstract

We study the compressible and incompressible two-phase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in \(\mathbb {R}^N\), and the Navier–Stokes–Korteweg equations is used in the upper domain and the Navier–Stokes equations is used in the lower domain. We prove the existence of \(\mathcal {R}\)-bounded solution operator families for a resolvent problem arising from its model problem. According to Göts and Shibata (Asymptot Anal 90(3–4):207–236, 2014), the regularity of \(\rho _+\) is \(W^1_q\) in space, but to solve the kinetic equation: \(\mathbf {u}_\Gamma \cdot \mathbf {n}_t = [[\rho \mathbf {u}]]\cdot \mathbf {n}_t /[[\rho ]]\) on \(\Gamma _t\) we need \(W^{2-1/q}_q\) regularity of \(\rho _+\) on \(\Gamma _t\), which means the regularity loss. Since the regularity of \(\rho _+\) dominated by the Navier–Stokes–Korteweg equations is \(W^3_q\) in space, we eliminate the problem by using the Navier–Stokes–Korteweg equations instead of the compressible Navier–Stokes equations.

Keywords

Two-phase flows Phase transition Surface tension Navier–Stokes–Korteweg equation Compressible and incompressible viscous flow Maximal \(L_p-L_q\) regularity \(\mathcal {R}\)-bounded solution operator 

Mathematics Subject Classification

Primary: 35Q30 Secondary: 76T10 

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Notes

Acknowledgements

The author would like to thank Prof. Yoshihiro Shibata and Dr. Hirokazu Saito for stimulating discussions on the subject of this paper. This research was partly supported by JSPS Japanese-German Graduate Externship and by Top Global University Project, Waseda University.

Compliance with ethical standards

Conflict of interest

The author declares no conflicts of interest associated with this manuscript.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and EngineeringWaseda UniversityTokyoJapan

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