Infinite superlinear growth of the gradient for the two-dimensional Euler equation

TL;DR

This paper proves that the gradient of solutions to the 2D Euler equation on a torus can grow faster than linearly over time, including exponential growth in finite time, for smooth initial conditions.

Contribution

It demonstrates superlinear and exponential growth of the gradient in the 2D Euler equation, revealing new insights into the equation's solution behavior.

Findings

Gradient can grow superlinearly over time.

Gradient exhibits exponential growth in finite time.

Results apply to infinitely smooth initial data.

Abstract

For two-dimensional Euler equation on the torus, we prove that the uniform norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for the finite time.