### Abstract

It is known that the Stokes operator is not well-defined in L^{q}-spaces for certain unbounded smooth domains unless q = 2. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general un-bounded smooth domains from the three-dimensional case, see [7], to the n-dimensional one, n ≥ 2, replacing the space L^{q}, 1 < q < ∞, by L̃^{q} where L̃^{q} = L̃^{q} ∩ L^{2} for q ≥ 2 and L̃^{q} = L^{q} + L^{2} for 1 < q < 2. In particular, we show that the Stokes operator is well-defined in L^{q} for every unbounded domain of uniform C^{1,1}-type in R^{n}, n ≥ 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

Original language | English |
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Pages (from-to) | 111-136 |

Number of pages | 26 |

Journal | Hokkaido Mathematical Journal |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

### Keywords

- Domains of uniform c-type
- General unbounded domains
- Maximal regularity
- Stokes operator
- Stokes resolvent
- Stokes semigroup

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Hokkaido Mathematical Journal*,

*38*(1), 111-136. https://doi.org/10.14492/hokmj/1248787007