Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus

TL;DR

This paper constructs solutions to the 2D Euler equations on a torus where the vorticity gradient grows exponentially over infinite time, demonstrating unbounded growth in a classical fluid dynamics model.

Contribution

It provides explicit solutions exhibiting infinite-time exponential growth of vorticity gradient on the 2D torus, a new phenomenon in Euler dynamics.

Findings

Vorticity gradient grows exponentially for all time

Constructs explicit solutions with unbounded growth

Demonstrates infinite-time exponential growth in Euler equations

Abstract

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: $$\|\nabla\omega(t, \cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$$