### Abstract

In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L_{p}-L_{q} regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

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

Pages (from-to) | 441-505 |

Number of pages | 65 |

Journal | Funkcialaj Ekvacioj |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

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### Keywords

- Analytic semigroup
- Compressible viscous fluid
- Exponential stability
- General domain
- Global in time unique existence theorem
- Local in time unique existence theorem
- Maximal L-L regularity
- Navier-Stokes equations
- R-sectoriality
- Stokes equations

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

**On the R-sectoriality and the initial boundary value problem for the viscous compressible fluid flow.** / Enomoto, Yuko; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Funkcialaj Ekvacioj*, vol. 56, no. 3, pp. 441-505. https://doi.org/10.1619/fesi.56.441

}

TY - JOUR

T1 - On the R-sectoriality and the initial boundary value problem for the viscous compressible fluid flow

AU - Enomoto, Yuko

AU - Shibata, Yoshihiro

PY - 2013

Y1 - 2013

N2 - In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal Lp-Lq regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

AB - In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal Lp-Lq regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

KW - Analytic semigroup

KW - Compressible viscous fluid

KW - Exponential stability

KW - General domain

KW - Global in time unique existence theorem

KW - Local in time unique existence theorem

KW - Maximal L-L regularity

KW - Navier-Stokes equations

KW - R-sectoriality

KW - Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=84890886308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890886308&partnerID=8YFLogxK

U2 - 10.1619/fesi.56.441

DO - 10.1619/fesi.56.441

M3 - Article

AN - SCOPUS:84890886308

VL - 56

SP - 441

EP - 505

JO - Funkcialaj Ekvacioj

JF - Funkcialaj Ekvacioj

SN - 0532-8721

IS - 3

ER -