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 $\nablaω$ grows exponentially in time: $\|\nablaω(t, \cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$

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