Degenerate Bifurcation of Two-Fold Doubly Connected Uniformly Rotating Vortex Patches

Abstract

This paper investigates the existence of uniformly rotating, doubly connected vortex patch solutions to the 2D incompressible Euler equation in the plane. These solutions bifurcate from concentric annulus at critical inner-to-outer radius ratios characterized by a high degeneracy in the linearization—the kernel is two-dimensional and the transversality condition fails. Our result settles the existence problem for specific degenerate cases left unresolved in [Adv. Math. 302, 799-850 (2016)].

Keywords

Uniformly rotating vortex patch / Incompressible Euler equation / Degenerate bifurcation / 35Q31 / 37K50 / 76B47

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Funding

National Science Foundation of China(11831009, 12471110)

National Science Foundation of China(12271437,11971498)

National Key Research and Development Program of China(2021YFA1002400)

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Peking University