Digital Object Identifier (DOI) 10.1007/s00205-013-0652-6 Arch. Rational Mech. Anal. On Time-Periodic Flow of a Viscous Liquid past a Moving Cylinder Giovanni P. Galdi Communicated by V. Šverák Abstract We show existence, uniqueness and spatial asymptotic behavior of a two- dimensional time-periodic flow around a cylinder that moves orthogonal to its axis, with a time-periodic velocity, v. The result is proved if the size of the data is sufficiently small, and the average of v over a period is not zero. 1. Introduction Consider a cylinder, C , moving in an unlimited mass of viscous liquid in a direction perpendicular to its axis a, with prescribed velocity -v ∞ =-v ∞ (t ). It is assumed that the function v ∞ (t ) is periodic, of period T . In a region of flow sufficiently far from the two ends of C and including C , one expects that the velocity field of the liquid is independent of the coordinate parallel to a and, moreover, that there is no flow in the direction of a. Under these conditions, the motion of the liquid will then be two-dimensional and take place in a plane orthogonal to a. The question that we will address in this paper is simply formulated as follows: Will the planar motion of the liquid be periodic as well, with period T ? In order to give a mathematical formulation, let v = v(x , t ), and let p = p(x , t ) be the unknown velocity and pressure fields of the liquid, respectively, whose motion is assumed to be governed by the Navier–Stokes equations. Having to cope with the above question means investigating whether the following set of equations admits a time-periodic solution (v, p) of period T : v t + v ·∇v = νv -∇ p + f ∇· v = 0  in  × (-∞, ∞), v = v ∗ on ∂ × (-∞, ∞), lim |x |→∞ v(x , t ) = v ∞ (t ), t ∈ (-∞, ∞). (1.1)