### Abstract

Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body K = ℝ ^{3}\Ω which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U,ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier-Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ∇u, p ∈ L _{3/2,∞}(Ω) and u ∈ L _{3,∞}(Ω) under the smallness condition on {pipe}U{pipe} + {pipe}ω{pipe} + {double pipe}F{double pipe} _{L3/2,∞}(Ω). Then the uniqueness is shown for solutions (u, p) satisfying ∇u, p ∈ L _{3/2,∞}(Ω) ∩ L _{q,r}(Ω) and u ∈ L _{3,∞}(Ω) ∩ L _{q*,r}(Ω) provided that 3/2 < q < 3 and F ∈ L _{3/2,∞}(Ω) ∩ L _{q,r}(Ω). Here L _{q,r}(Ω) denotes the well-known Lorentz space and q* = 3q/(3 - q) is the Sobolev exponent to q.

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

Pages (from-to) | 315-345 |

Number of pages | 31 |

Journal | Manuscripta Mathematica |

Volume | 138 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2012 Jul |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Manuscripta Mathematica*,

*138*(3-4), 315-345. https://doi.org/10.1007/s00229-011-0494-1

**On the stationary Navier-Stokes flows around a rotating body.** / Heck, Horst; Kim, Hyunseok; Kozono, Hideo.

Research output: Contribution to journal › Article

*Manuscripta Mathematica*, vol. 138, no. 3-4, pp. 315-345. https://doi.org/10.1007/s00229-011-0494-1

}

TY - JOUR

T1 - On the stationary Navier-Stokes flows around a rotating body

AU - Heck, Horst

AU - Kim, Hyunseok

AU - Kozono, Hideo

PY - 2012/7

Y1 - 2012/7

N2 - Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body K = ℝ 3\Ω which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U,ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier-Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ∇u, p ∈ L 3/2,∞(Ω) and u ∈ L 3,∞(Ω) under the smallness condition on {pipe}U{pipe} + {pipe}ω{pipe} + {double pipe}F{double pipe} L3/2,∞(Ω). Then the uniqueness is shown for solutions (u, p) satisfying ∇u, p ∈ L 3/2,∞(Ω) ∩ L q,r(Ω) and u ∈ L 3,∞(Ω) ∩ L q*,r(Ω) provided that 3/2 < q < 3 and F ∈ L 3/2,∞(Ω) ∩ L q,r(Ω). Here L q,r(Ω) denotes the well-known Lorentz space and q* = 3q/(3 - q) is the Sobolev exponent to q.

AB - Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body K = ℝ 3\Ω which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U,ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier-Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ∇u, p ∈ L 3/2,∞(Ω) and u ∈ L 3,∞(Ω) under the smallness condition on {pipe}U{pipe} + {pipe}ω{pipe} + {double pipe}F{double pipe} L3/2,∞(Ω). Then the uniqueness is shown for solutions (u, p) satisfying ∇u, p ∈ L 3/2,∞(Ω) ∩ L q,r(Ω) and u ∈ L 3,∞(Ω) ∩ L q*,r(Ω) provided that 3/2 < q < 3 and F ∈ L 3/2,∞(Ω) ∩ L q,r(Ω). Here L q,r(Ω) denotes the well-known Lorentz space and q* = 3q/(3 - q) is the Sobolev exponent to q.

UR - http://www.scopus.com/inward/record.url?scp=84860731886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860731886&partnerID=8YFLogxK

U2 - 10.1007/s00229-011-0494-1

DO - 10.1007/s00229-011-0494-1

M3 - Article

VL - 138

SP - 315

EP - 345

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3-4

ER -