Superlinear gradient growth for 2D Euler equation without boundary

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.