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Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches

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arxiv 2212.01869 v4 pith:D6NSCJZR submitted 2022-12-04 math.AP

Yuchen Wang , Xin Xu , Maolin Zhou This is my paper · ORCID
classification math.AP
keywords doubly-connectedpatchesproblemrotatingtwo-folduniformlyvortexaddition
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In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem, neither the kernel of linearization is one-dimensional nor the transeversallity condition holds. To this end, we make a detailed analysis on the nonlinear functional and the bifurcation curves are obtained by perturbing real algebraic varieties defined by truncated polynomials. In addition, our result partially answers an problem proposed by Hmidi and Mateu in \cite{Hmidi2016a} (\emph{Adv.Math.302 (2016), 799-850}).

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uniformly Rotating Euler Configurations with Multiple Vorticity Holes

    math.AP 2026-08 accept novelty 7.0 of 10

    For every m at least 2, the paper constructs uniformly rotating Euler vortex-patch solutions with one outer patch and m small negative inner patches that collapse to a central point vortex as epsilon tends to zero.

  2. On stationary Quasi-Geostrophic Shallow-Water flows

    math.AP 2026-07 accept novelty 7.0 of 10

    Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.

  3. Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

    math.AP 2025-06 conditional novelty 6.0 of 10

    Uniformly rotating 2D Euler solutions with compactly supported vorticity are forced to be radially symmetric whenever the angular velocity lies outside half the range of the vorticity, including irregular vortex patches.

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