### Abstract

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the L_{p} in time and the L_{q} in space framework with 2 <p <1 and N <q <∞ under the assumption that the initial domain is a uniform W_{q} ^{2-1/q} domain in ℝ^{N}(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L_{p}-L_{q} regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L_{p}-L_{q} regularity theorem.

Original language | English |
---|---|

Pages (from-to) | 3741-3774 |

Number of pages | 34 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 36 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2016 Jul 1 |

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### Keywords

- Compressible viscous fluid
- Free boundary problem
- Local well-posedness theorem
- Maximal L-L regularity
- R-bounded solution operator
- Two phase problem
- Uniform W domain

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

*Discrete and Continuous Dynamical Systems*,

*36*(7), 3741-3774. https://doi.org/10.3934/dcds.2016.36.3741

**On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface.** / Kubo, Takayuki; Shibata, Yoshihiro; Soga, Kohei.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 36, no. 7, pp. 3741-3774. https://doi.org/10.3934/dcds.2016.36.3741

}

TY - JOUR

T1 - On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

AU - Kubo, Takayuki

AU - Shibata, Yoshihiro

AU - Soga, Kohei

PY - 2016/7/1

Y1 - 2016/7/1

N2 - In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the Lp in time and the Lq in space framework with 2 q 2-1/q domain in ℝN(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal Lp-Lq regularity theorem.

AB - In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the Lp in time and the Lq in space framework with 2 q 2-1/q domain in ℝN(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal Lp-Lq regularity theorem.

KW - Compressible viscous fluid

KW - Free boundary problem

KW - Local well-posedness theorem

KW - Maximal L-L regularity

KW - R-bounded solution operator

KW - Two phase problem

KW - Uniform W domain

UR - http://www.scopus.com/inward/record.url?scp=84962528821&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962528821&partnerID=8YFLogxK

U2 - 10.3934/dcds.2016.36.3741

DO - 10.3934/dcds.2016.36.3741

M3 - Article

VL - 36

SP - 3741

EP - 3774

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 7

ER -