Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body K = R 3 \ 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 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,∞ ( ) and u ∈ L 3,∞ ( ) under the smallness condition on |U | + |ω| + F L 3/2,∞ ( ) . Then the uniqueness is shown for solutions (u, p) satisfying ∇u, p ∈ L 3/2,∞ ( ) ∩ L q,r ( ) and u ∈ L 3,∞ ( ) ∩ L q * ,r ( ) provided that 3/2 < q < 3 and F ∈ L 3/2,∞ ( ) ∩ L q,r ( ). Here L q,r ( ) denotes the well-known Lorentz space and q * = 3q/(3 − q) is the Sobolev exponent to q.
IntroductionLet be an exterior domain in R 3 with smooth boundary ∂ . Consider the motion of an incompressible Navier-Stokes fluid around the rigid body K = R 3 \ which is rotating about an axis with constant angular velocity ω = ce 3 = (0, 0, c) T . We also assume that the body K is moving in the direction of the axis of rotation with constant velocity U = ke 3 . Then with respect to a coordinate system attached to the body, the velocity u = (u 1 , u 2 , u 3 ) T and pressure p of the fluid is governed by the following initial boundary value problem for a variant of the Navier-Stokes equations in (see [7,12,18] for a detailed derivation):