Formula omitted]-solutions for the stationary Navier–Stokes equations in the exterior of a rotating obstacle

Abstract

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body R3∖Ω which is also moving in the direction of the axis of rotation with nonzero constant velocity −ke1. We assume that the angular velocity ω=|ω|e1 is also constant and the external force is given by f=divF. Then the motion is described by a variant of the stationary Navier–Stokes equations with the velocity ke1 at infinity. Our main result is the existence of at least one solution u satisfying u−ke1∈L3(Ω) for arbitrarily large F∈L3∕2(Ω). The uniqueness is also proved by assuming that |ω|+|k|+‖F‖L3∕2(Ω) is sufficiently small in comparison with the viscosity ν. Moreover, we establish several regularity results to obtain an existence theorem for weak solutions u satisfying ∇u∈L3∕2(Ω) and u−ke1∈L3(Ω).