Growth of vorticity gradient for the Euler equation on the sphere

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.