### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Hokkaido Mathematical Journal*,

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

**On the Stokes operator in general unbounded domains.** / Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann.

Research output: Contribution to journal › Article

*Hokkaido Mathematical Journal*, vol. 38, no. 1, pp. 111-136. https://doi.org/10.14492/hokmj/1248787007

}

TY - JOUR

T1 - On the Stokes operator in general unbounded domains

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2009/1/1

Y1 - 2009/1/1

N2 - It is known that the Stokes operator is not well-defined in Lq-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 Lq, 1 < q < ∞, by L̃q where L̃q = L̃q ∩ L2 for q ≥ 2 and L̃q = Lq + L2 for 1 < q < 2. In particular, we show that the Stokes operator is well-defined in Lq for every unbounded domain of uniform C1,1-type in Rn, n ≥ 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

AB - It is known that the Stokes operator is not well-defined in Lq-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 Lq, 1 < q < ∞, by L̃q where L̃q = L̃q ∩ L2 for q ≥ 2 and L̃q = Lq + L2 for 1 < q < 2. In particular, we show that the Stokes operator is well-defined in Lq for every unbounded domain of uniform C1,1-type in Rn, n ≥ 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

KW - Domains of uniform c-type

KW - General unbounded domains

KW - Maximal regularity

KW - Stokes operator

KW - Stokes resolvent

KW - Stokes semigroup

UR - http://www.scopus.com/inward/record.url?scp=78449285512&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78449285512&partnerID=8YFLogxK

U2 - 10.14492/hokmj/1248787007

DO - 10.14492/hokmj/1248787007

M3 - Article

AN - SCOPUS:78449285512

VL - 38

SP - 111

EP - 136

JO - Hokkaido Mathematical Journal

JF - Hokkaido Mathematical Journal

SN - 0385-4035

IS - 1

ER -