Vorticity gradients for the Euler equation on the sphere are bounded above by double-exponential growth in time, with this rate achieved by explicit symmetric constructions.
Confinement results near point vortices on the rotating sphere
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abstract
We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.
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2026 1verdicts
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Growth of vorticity gradient for the Euler equation on the sphere
Vorticity gradients for the Euler equation on the sphere are bounded above by double-exponential growth in time, with this rate achieved by explicit symmetric constructions.