Growth of vorticity gradient for the Euler equation on the sphere

TL;DR

This paper establishes that vorticity gradient growth for Euler equations on a sphere is at most double-exponential, and constructs solutions demonstrating this sharp bound, extending to rotating spheres.

Contribution

It provides the first sharp bound on vorticity gradient growth for ideal fluids on a compact manifold with non-trivial geometry, specifically the sphere.

Findings

Vorticity gradient growth is at most double-exponential in time.

Explicit solutions with odd symmetry show double-exponential growth.

Results extend to rotating spheres.

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.