### 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 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 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 ρ_{+} is Wq1 in space, but to solve the kinetic equation: u_{Γ}· n_{t}= [[ρu]] · n_{t}/ [[ρ]] on Γ _{t} we need Wq2-1/q regularity of ρ_{+} on Γ _{t}, which means the regularity loss. Since the regularity of ρ_{+} dominated by the Navier–Stokes–Korteweg equations is Wq3 in space, we eliminate the problem by using the Navier–Stokes–Korteweg equations instead of the compressible Navier–Stokes equations.

Original language | English |
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Pages (from-to) | 969-1011 |

Number of pages | 43 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Sep 1 |

### Keywords

- Compressible and incompressible viscous flow
- Maximal L- L regularity
- Navier–Stokes–Korteweg equation
- Phase transition
- R-bounded solution operator
- Surface tension
- Two-phase flows

### ASJC Scopus subject areas

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics