Continuation and reduction of steady vortex sheets under metric deformations on a disk
Abstract
We study the local continuation of a steady vortex sheet under a prescribed deformation gs= e2σsge of a conformal metric on the unit disk. Representing the moving sheet by a normal graph, we formulate the streamline and Bernoulli conditions as a nonlinear residual whose two components have different Sobolev orders. If the two one-sided tangential velocities of the reference sheet do not vanish simultaneously, the derivative of the unnormalized residual is Fredholm of index two. Fixing the mean normal displacement and the total circulation gives an index-zero problem; when the normalization differential restricted to the unnormalized kernel is onto ℝ2, we obtain a parameter-dependent Lyapunov-Schmidt reduction. This yields a locally unique normalized branch in the nondegenerate case and explicit first- and second-order necessary conditions in the degenerate case. For a rotationally symmetric reference metric and a concentric circular sheet, the linearized problem separates into Fourier blocks and only finitely many modes can be resonant. In the constant-curvature disk family, every mode k≥2 is nonresonant, whereas the first mode produces a two-dimensional reduced problem. The two first-mode kernel profiles are normal traces of ambient Killing fields, but they do not generate fixed-boundary symmetries or, by themselves, a nonconcentric branch.
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BibTeX
@misc{shimizu2026continuation,
title = {{Continuation and reduction of steady vortex sheets under metric deformations on a disk}},
author = {Yuuki Shimizu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08361},
eprint = {2609.08361},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08361},
}
RIS
TY - UNPB
TI - Continuation and reduction of steady vortex sheets under metric deformations on a disk
AU - Yuuki Shimizu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08361
UR - https://arxiv.org/abs/2609.08361
AB - We study the local continuation of a steady vortex sheet under a prescribed deformation g_s= e^2σ_sg_e of a conformal metric on the unit disk. Representing the moving sheet by a normal graph, we formulate the streamline and Bernoulli conditions as a nonlinear residual whose two components have different Sobolev orders. If the two one-sided tangential velocities of the reference sheet do not vanish simultaneously, the derivative of the unnormalized residual is Fredholm of index two. Fixing the mean normal displacement and the total circulation gives an index-zero problem; when the normalization differential restricted to the unnormalized kernel is onto ℝ^2, we obtain a parameter-dependent Lyapunov-Schmidt reduction. This yields a locally unique normalized branch in the nondegenerate case and explicit first- and second-order necessary conditions in the degenerate case. For a rotationally symmetric reference metric and a concentric circular sheet, the linearized problem separates into Fourier blocks and only finitely many modes can be resonant. In the constant-curvature disk family, every mode k≥2 is nonresonant, whereas the first mode produces a two-dimensional reduced problem. The two first-mode kernel profiles are normal traces of ambient Killing fields, but they do not generate fixed-boundary symmetries or, by themselves, a nonconcentric branch.
ER -