Infinite-time Exponential Growth of the Euler Equation on   Two-dimensional Torus
  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: $$\|\nabla\omega(t, \cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$$ 
Zhen Lei and Jia Shi
math.AP