### Abstract

We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R^{3} in a solution class L^{s}(0,T; L ^{q}(Ω)) of very low regularity in space and time, satisfying Serrin's condition 2/s + 3/q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|_{∂Ω} = g ε L^{s}(0,T; W^{-1/q,q}(∂Ω)), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = div u ε L^{s} (0,T; L^{r}(Ω)), where 1/3 + 1/q = 1/r.

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

Pages (from-to) | 127-150 |

Number of pages | 24 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jan |

Externally published | Yes |

### Fingerprint

### Keywords

- Nonhomogeneous data
- Serrin's class
- Stokes and navier-stokes equations
- Very weak solutions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the Mathematical Society of Japan*,

*59*(1), 127-150. https://doi.org/10.2969/jmsj/1180135504

**Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data.** / Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 59, no. 1, pp. 127-150. https://doi.org/10.2969/jmsj/1180135504

}

TY - JOUR

T1 - Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2007/1

Y1 - 2007/1

N2 - We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R3 in a solution class Ls(0,T; L q(Ω)) of very low regularity in space and time, satisfying Serrin's condition 2/s + 3/q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|∂Ω = g ε Ls(0,T; W-1/q,q(∂Ω)), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = div u ε Ls (0,T; Lr(Ω)), where 1/3 + 1/q = 1/r.

AB - We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R3 in a solution class Ls(0,T; L q(Ω)) of very low regularity in space and time, satisfying Serrin's condition 2/s + 3/q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|∂Ω = g ε Ls(0,T; W-1/q,q(∂Ω)), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = div u ε Ls (0,T; Lr(Ω)), where 1/3 + 1/q = 1/r.

KW - Nonhomogeneous data

KW - Serrin's class

KW - Stokes and navier-stokes equations

KW - Very weak solutions

UR - http://www.scopus.com/inward/record.url?scp=34248594458&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248594458&partnerID=8YFLogxK

U2 - 10.2969/jmsj/1180135504

DO - 10.2969/jmsj/1180135504

M3 - Article

VL - 59

SP - 127

EP - 150

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -