### Abstract

This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L _{3/2,∞}(Ω)∩L _{p,q}(Ω) with 3/2<p<3 and 1≤q≤∞ provided ||F||L3/2,∞(Ω) is sufficiently small. Here L _{p,q}(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L _{3/2,∞}(Ω)∩L _{2}(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L _{3/2}(Ω)+L _{2}(Ω), which generalizes Leray's existence result.

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

Pages (from-to) | 486-495 |

Number of pages | 10 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 395 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Nov 15 |

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### Keywords

- Energy inequality
- Exterior problem
- Lorentz space
- Navier-Stokes equations
- Regularity
- Uniqueness

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**On the stationary Navier-Stokes equations in exterior domains.** / Kim, Hyunseok; Kozono, Hideo.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 395, no. 2, pp. 486-495. https://doi.org/10.1016/j.jmaa.2012.05.039

}

TY - JOUR

T1 - On the stationary Navier-Stokes equations in exterior domains

AU - Kim, Hyunseok

AU - Kozono, Hideo

PY - 2012/11/15

Y1 - 2012/11/15

N2 - This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

AB - This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier-Stokes equations in an exterior domain Ω in R3, where the external force is given by divF with F=F(x)=(Fji(x))i,j=1,2,3. First, we prove the existence and uniqueness of a weak solution for F∈L 3/2,∞(Ω)∩L p,q(Ω) with 3/2p,q(Ω) denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for F∈L 3/2,∞(Ω)∩L 2(Ω) under the same smallness condition on ||F||L3/2,∞(Ω). This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data F in a very large class, for instance, in L 3/2(Ω)+L 2(Ω), which generalizes Leray's existence result.

KW - Energy inequality

KW - Exterior problem

KW - Lorentz space

KW - Navier-Stokes equations

KW - Regularity

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=84864016579&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864016579&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2012.05.039

DO - 10.1016/j.jmaa.2012.05.039

M3 - Article

VL - 395

SP - 486

EP - 495

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -