### Abstract

We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension n ≥ 3. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-L^{n} space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-L^{n} space with time-independent external force.

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

Pages (from-to) | 635-675 |

Number of pages | 41 |

Journal | Mathematische Annalen |

Volume | 317 |

Issue number | 4 |

Publication status | Published - 2000 Aug |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The Navier-Stokes equations in the weak-L ^{n} space with time-dependent external force.** / Yamazaki, Masao.

Research output: Contribution to journal › Article

^{n}space with time-dependent external force',

*Mathematische Annalen*, vol. 317, no. 4, pp. 635-675.

^{n}space with time-dependent external force. Mathematische Annalen. 2000 Aug;317(4):635-675.

}

TY - JOUR

T1 - The Navier-Stokes equations in the weak-Ln space with time-dependent external force

AU - Yamazaki, Masao

PY - 2000/8

Y1 - 2000/8

N2 - We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension n ≥ 3. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-Ln space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-Ln space with time-independent external force.

AB - We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension n ≥ 3. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-Ln space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-Ln space with time-independent external force.

UR - http://www.scopus.com/inward/record.url?scp=0040790106&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040790106&partnerID=8YFLogxK

M3 - Article

VL - 317

SP - 635

EP - 675

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -