### Abstract

In an exterior domain Ω⊂R^{3} and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L^{2}(0, T;L^{2}(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

Original language | English |
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Pages (from-to) | 2633-2658 |

Number of pages | 26 |

Journal | Journal of Differential Equations |

Volume | 256 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2014 Apr 1 |

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### Keywords

- Exterior domain
- Instationary Navier-Stokes equations
- Non-zero boundary values
- Strong energy inequality
- Time-dependent data
- Weak solutions

### ASJC Scopus subject areas

- Analysis

### Cite this

**Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality.** / Farwig, Reinhard; Kozono, Hideo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

AU - Farwig, Reinhard

AU - Kozono, Hideo

PY - 2014/4/1

Y1 - 2014/4/1

N2 - In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

AB - In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

KW - Exterior domain

KW - Instationary Navier-Stokes equations

KW - Non-zero boundary values

KW - Strong energy inequality

KW - Time-dependent data

KW - Weak solutions

UR - http://www.scopus.com/inward/record.url?scp=84895908566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84895908566&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2014.01.029

DO - 10.1016/j.jde.2014.01.029

M3 - Article

AN - SCOPUS:84895908566

VL - 256

SP - 2633

EP - 2658

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -