Non-rigid time-periodic vortex patch solutions bifurcating from translating symmetric dipoles are constructed for the 2D Euler equations using Lyapunov-Schmidt reduction and Nash-Moser methods.
Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations
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abstract
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.
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math.AP 4years
2026 4representative citing papers
Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.
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Initial data close to large quasi-periodic traveling waves in the β-plane equation remain close for arbitrary long times independent of wave size, yielding almost global existence for open sets of large initial data.
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Time-periodic vortices near translating symmetric dipole patches
Non-rigid time-periodic vortex patch solutions bifurcating from translating symmetric dipoles are constructed for the 2D Euler equations using Lyapunov-Schmidt reduction and Nash-Moser methods.
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Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.
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Vorticity near point vortices on the rotating sphere shows logarithmic confinement in time, improbability of collisions, and power-law confinement in some cases.
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