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.

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U2 - 10.1007/s00229-011-0494-1

DO - 10.1007/s00229-011-0494-1

M3 - Article

AN - SCOPUS:84860731886

SN - 0025-2611

VL - 138

SP - 315

EP - 345

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 3-4

ER -