Double exponential growth of the vorticity gradient for the two-dimensional Euler equation

TL;DR

This paper proves that for the 2D Euler equation on a torus, the vorticity gradient can grow double exponentially in finite time if initially large enough, indicating unbounded Lipschitz norm growth.

Contribution

It establishes the double exponential growth of the vorticity gradient for the 2D Euler equation, revealing linear unboundedness in Lipschitz norm over finite time.

Findings

Vorticity gradient can grow double exponentially

Growth occurs for sufficiently large initial vorticity gradient

Euler evolution is linearly unbounded in Lipschitz norm

Abstract

For the two-dimensional Euler equation on the torus, we prove that the uniform norm of the vorticity gradient can grow as double exponential over arbitrarily long but finite time provided that at time zero it is already sufficiently large. Our result is equivalent to the statement that the Euler evolutions is linearly unbounded in Lipschitz norm for any time t>0.