L p -L q Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

Abstract

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 equation to prove the global existence of a unique solution which goes to a stationary flow as t → ∞ with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L3,∞ (weak-L3 space).