Existence, Uniqueness and Asymptotic Behavior of Regular Time-Periodic Viscous Flow around a Moving Body: Rotational Case

We show existence and uniqueness for small data of regular time-periodic solutions to the Navier-Stokes problem in the exterior of a rigid body, $\mathcal{B}$, that moves by time-periodic translational motion of the same period along a constant direction, ${\text{bfe}}_1$, and spins with constant angular velocity $\text{bfomega}$ parallel to ${\text{bfe}}_1$. We also study the spatial asymptotic behavior of such solutions and show, in particular, that if $\mathcal{B}$ has a net motion characterized by a non-zero average translational velocity $\overline{\text{bfxi}}$, then the solution exhibit a wake-like behavior in the direction $-\overline{\text{bfxi}}$ entirely analogous to that of a steady-state flow around a body that moves with velocity $\overline{\text{bfxi}}$ and angular velocity $\text{bfomega}$.