Nonstationary Navier-Stokes flows in a two-dimensional exterior domain with rotational symmetries

In a 2D exterior domain, we look for Navier-Stokes flows for which the associated total net force to the boundary vanishes (see (1.2)). It is shown that this is the case at each time whenever the initial velocity is summable and possesses some regularity and rotational symmetry. The result shows that this condition of summability, regularity and rotational symmetry is preserved in time. An asymptotic profile is deduced and a lower bound estimate on time-decay rates is found for such solutions. Moreover, existence is shown for flows with higher symmetry for which the time-decay rates exceed the above-mentioned lower bound. The decay rates are directly connected with the orders of groups of symmetry.