Asymptotic stability of small Oseen-type vortex under three-dimensional large perturbation

We consider the three-dimensional Navier-Stokes equations whose initial data may have infinite kinetic energy. We establish unique existence of the mild solution to the Navier-Stokes equations for small initial data in the whole space R3 and a vertically periodic product space R2 × T1 which may be constant in vertical direction so that it includes the Oseen vortex. We further discuss its asymptotic stability under arbitrarily large three-dimensional perturbation in R2 × T 1.