## 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 Shibata [13], the regularity of ρ_{+} is W_{q}^{1} in space, but to solve the kinetic equation: u_{Γ} · n_{t} = [[ρu]] · n_{t}/[[ρ]] on Γ_{t} we need W_{q}^{2−1/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 W_{q}^{3} in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.

MSC Codes 35Q30 (Primary), 76T10 (Secondary)

Original language | English |
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Journal | Unknown Journal |

Publication status | Published - 2017 May 9 |

## Keywords

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

## ASJC Scopus subject areas

- General