Existence of steady Navier-Stokes flows exterior to an infinite cylinder

Abstract

We consider the three-dimensional steady Navier-Stokes system in the exterior of an infinite cylinder under the action of an external force. We construct solutions in the class of vertically uniform flows which vanish at horizontal infinity. More precisely, for a boundary datum determined by a rotating flow and a suction flow, and for a small force of the form $f=g+\operatorname{div} F$ with suitable decay, we prove the existence of a weak solution asymptotic to the corresponding Hamel-type flow. Although all data are independent of the vertical variable, the problem is not reduced to the planar exterior Navier-Stokes system: the vertical component satisfies a separate transport-diffusion equation involving the two-dimensional Laplacian, whose fundamental solution has logarithmic growth. The proof is based on a mode-by-mode analysis of the linearized three-dimensional problem around the Hamel-type flow and a contraction argument.