Stability of a two-dimensional stationary rotating flow in an exterior cylinder

We investigate the stability of an exact stationary flow in an exterior cylinder. The horizontal velocity is the two-dimensional rotating flow in an exterior disk with a critical spatial decay, for which the L-2 stability is known under smallness conditions. We prove its stability property for three-dimensional perturbations although the Hardy type inequalities are absent as in the two-dimensional case. The proof uses a large time estimate for the linearized equations exhibiting different behaviors in the Fourier modes, namely, the standard L-2-Lq decay of the two-dimensional mode and an exponential decay of the three-dimensional modes.