Spectral Stability of the Gallay-Wayne Gap under Three-Dimensional Vortex Tube Perturbations

Gallay and Wayne (2005) proved that the Lamb-Oseen vortex is globally asymptotically stable as a solution of the two-dimensional Navier-Stokes equation, with the linearized operator possessing a spectral gap \(\gamma > 0\) in Gaussian-weighted \(L^2\) spaces. We prove that this spectral gap is stable under the perturbations that arise when the 2D cross-section dynamics are embedded in a three-dimensional curved vortex tube. Specifically, if the tube has centerline curvature \(\kappa\) and core radius \(r_0\) with \(\varepsilon = \kappa r_0 \ll 1\), the perturbed operator retains a spectral gap \(\gamma_\varepsilon \geq \gamma - C\varepsilon\) and generates an analytic semigroup with the corresponding exponential decay. The perturbation has four components — metric curvature, self-consistent strain deviation, axial coupling, and anisotropic compression — each shown to be relatively compact with respect to the Oseen operator. The result closes a technical gap in recent approaches to Navier-Stokes regularity that employ Burgers-vortex comparison in tube geometries.

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