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$$L_{p}-L_{q}$$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. (English) Zbl 1169.76015
Summary: We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the $$L_{p}-L_{q}$$ estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier-Stokes equations to prove the global existence of a unique solution which goes to a stationary flow as $$t \rightarrow \infty$$ with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in $$L_{3,\infty}$$ (weak-$$L_{3}$$ space).

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35Q30 Navier-Stokes equations 76U05 General theory of rotating fluids
##### Keywords:
uniqueness; Lorentz spaces; Stokes semigroup; global existence
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##### References:
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