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 WatanabeEmail author


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.


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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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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Authors and Affiliations

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

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