Nonlinear Stability of Taylor Vortices in Infinite Cylinders

We consider the Taylor‐Couette problem in an infinitely extended cylindrical domain in the case when Couette flow is weakly unstable and a family of spatially periodic equilibria, called the Taylor vortices, has bifurcated from this trivial ground state. We show that those Taylor vortices which are not linearly unstable in the sense of Eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations. The main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the Taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis. Bloch‐wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime.