### 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 |
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Pages (from-to) | 315-345 |

Number of pages | 31 |

Journal | Manuscripta Mathematica |

Volume | 138 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2012 Jul 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Manuscripta Mathematica*,

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