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

Abstract

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 twodimensional case. The proof uses a large time estimate for the linearized equations exhibiting different behaviors in the Fourier modes, namely, the standard 𝐿 2 -𝐿 𝑞 decay of the two-dimensional mode and an exponential decay of the three-dimensional modes.

Stability of planar exterior stationary flows with suction

“…Also, the nonlinear stability of αU is proved when both |α| and the L 2 -norm of initial data in (NP) are sufficiently small. This stability result is extended by the author in [22] to a certain class of non-symmetric domains where the domains are assumed to be small perturbations of the exterior unit disk, and in [24] for three-dimensional initial disturbances around an infinite cylinder.…”