### Abstract

This paper is concerned with the stationary problem of the Stokes equation in an infinite layer and provides a condition on the external force sufficient for the existence of the solution. Since the Poiseuille flow is a solution to the homogeneous equation, the solution is not unique when p = ∞. It is also proved that, under some suitable conditions, solutions to the homogeneous equation are limited only to the Poiseuille flow.

Original language | English |
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Pages (from-to) | 61-100 |

Number of pages | 40 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 Mar |

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### Keywords

- Besov space
- Homogeneous Besov space
- Infinite layer
- Poiseuille flow
- Sobolev space
- Stationary problem
- Stokes equation

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematical Physics
- Computational Mathematics
- Condensed Matter Physics

### Cite this

**On a Stationary Problem of the Stokes Equation in an Infinite Layer in Sobolev and Besov Spaces.** / Abe, Takayuki; Yamazaki, Masao.

Research output: Contribution to journal › Article

*Journal of Mathematical Fluid Mechanics*, vol. 12, no. 1, pp. 61-100. https://doi.org/10.1007/s00021-008-0276-z

}

TY - JOUR

T1 - On a Stationary Problem of the Stokes Equation in an Infinite Layer in Sobolev and Besov Spaces

AU - Abe, Takayuki

AU - Yamazaki, Masao

PY - 2010/3

Y1 - 2010/3

N2 - This paper is concerned with the stationary problem of the Stokes equation in an infinite layer and provides a condition on the external force sufficient for the existence of the solution. Since the Poiseuille flow is a solution to the homogeneous equation, the solution is not unique when p = ∞. It is also proved that, under some suitable conditions, solutions to the homogeneous equation are limited only to the Poiseuille flow.

AB - This paper is concerned with the stationary problem of the Stokes equation in an infinite layer and provides a condition on the external force sufficient for the existence of the solution. Since the Poiseuille flow is a solution to the homogeneous equation, the solution is not unique when p = ∞. It is also proved that, under some suitable conditions, solutions to the homogeneous equation are limited only to the Poiseuille flow.

KW - Besov space

KW - Homogeneous Besov space

KW - Infinite layer

KW - Poiseuille flow

KW - Sobolev space

KW - Stationary problem

KW - Stokes equation

UR - http://www.scopus.com/inward/record.url?scp=77950458376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950458376&partnerID=8YFLogxK

U2 - 10.1007/s00021-008-0276-z

DO - 10.1007/s00021-008-0276-z

M3 - Article

VL - 12

SP - 61

EP - 100

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 1

ER -