### 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 |
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Pages (from-to) | 441-505 |

Number of pages | 65 |

Journal | Funkcialaj Ekvacioj |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Dec 27 |

### 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

- Analysis
- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Funkcialaj Ekvacioj*,

*56*(3), 441-505. https://doi.org/10.1619/fesi.56.441