Superlinear gradient growth for 2D Euler equation without boundary

TL;DR

This paper demonstrates superlinear growth of vorticity gradients in 2D Euler equations on boundaryless domains, showing such growth occurs for open sets of smooth initial data in both torus and plane settings.

Contribution

It provides the first superlinear growth results for open sets of smooth initial data without symmetry assumptions and extends findings to the plane with compactly supported vorticity.

Findings

Superlinear vorticity gradient growth in the torus with stable steady states.

First superlinear growth result for smooth, compactly supported vorticity in the plane.

Growth occurs for open sets of initial data, not just special symmetric cases.

Abstract

We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady state $\omega^*$ that is orbitally stable up to a translation and has a saddle point, we construct ${\tilde{\omega}}_0 \in C^\infty(\mathbb{T}^2)$ that is arbitrarily close to $\omega^*$ in $L^2$, such that superlinear growth of the vorticity gradient occurs for an open set of smooth initial data around ${\tilde{\omega}}_0$. This seems to be the first superlinear growth result which holds for an open set of smooth initial data (and does not require any symmetry assumptions on the initial vorticity). Furthermore, we obtain the first superlinear growth result for smooth and compactly supported vorticity in the plane, using perturbations of the Lamb-Chaplygin dipole.