### Abstract

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal L_{p}-L_{q} regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

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

Pages (from-to) | 1-33 |

Number of pages | 33 |

Journal | Tijdschrift voor Urologie |

Volume | 2014 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 Jan 17 |

### Fingerprint

### Keywords

- compressible-incompressible two phase problem
- generalized resolvent problem
- model problem
- Stokes equations
- ℛ-boundedness

### ASJC Scopus subject areas

- Urology

### Cite this

*Tijdschrift voor Urologie*,

*2014*(1), 1-33. https://doi.org/10.1186/s13661-014-0141-3

**On the ℛ-boundedness for the two phase problem : compressible-incompressible model problem.** / Kubo, Takayuki; Shibata, Yoshihiro; Soga, Kohei.

Research output: Contribution to journal › Article

*Tijdschrift voor Urologie*, vol. 2014, no. 1, pp. 1-33. https://doi.org/10.1186/s13661-014-0141-3

}

TY - JOUR

T1 - On the ℛ-boundedness for the two phase problem

T2 - compressible-incompressible model problem

AU - Kubo, Takayuki

AU - Shibata, Yoshihiro

AU - Soga, Kohei

PY - 2014/1/17

Y1 - 2014/1/17

N2 - The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

AB - The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

KW - compressible-incompressible two phase problem

KW - generalized resolvent problem

KW - model problem

KW - Stokes equations

KW - ℛ-boundedness

UR - http://www.scopus.com/inward/record.url?scp=84919399315&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919399315&partnerID=8YFLogxK

U2 - 10.1186/s13661-014-0141-3

DO - 10.1186/s13661-014-0141-3

M3 - Article

AN - SCOPUS:84919399315

VL - 2014

SP - 1

EP - 33

JO - Tijdschrift voor Urologie

JF - Tijdschrift voor Urologie

SN - 2211-3037

IS - 1

ER -