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arXiv · 2604.19358

Growth of vorticity gradient for the Euler equation on the sphere

Daomin Cao·Junhong Fan·Guolin Qin
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Abstract

We prove that for solutions of the Euler equation on the sphere, the vorticity gradient can grow at most double-exponentially in time, and we show that this upper bound is sharp by constructing explicit solutions with odd symmetry that exhibit double-exponential growth in the hemisphere. We also extend the results to the case of a rotating sphere. This seems to be the first result on the growth of the vorticity gradient for ideal fluids on a compact manifold with non-trivial geometry.

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Daomin Cao, Junhong Fan, Guolin Qin. 2026-04-21. Growth of vorticity gradient for the Euler equation on the sphere. https://arxiv.org/abs/2604.19358

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