  For any $A > 2$, we construct solutions to the two-dimensional incompressible
Euler equations on the torus $\mathbb{T}^2$ whose vorticity gradient
$\nabla\omega$ grows exponentially in time: $$\|\nabla\omega(t,
\cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$$
