On the stationary Navier–Stokes flows around a rotating body

Abstract

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body \( \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}\) 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 \({\nabla u, p \in L_{3/2, \infty} (\Omega)}\) and \({u \in L_3, \infty (\Omega)}\) under the smallness condition on \({|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}\) . Then the uniqueness is shown for solutions (u, p) satisfying \({\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}\) and \({u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}\) provided that 3/2 < q < 3 and \({{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}\) . Here L q,r (Ω) denotes the well-known Lorentz space and q* = 3q /(3 − q) is the Sobolev exponent to q.