Generic small-scale creation in the two-dimensional Euler equation

TL;DR

This paper demonstrates that for a dense set of initial conditions, solutions to the 2D Euler equations become irregular over infinite time, confirming a longstanding conjecture about the potential for small-scale creation.

Contribution

It proves that generic smooth initial data can lead to loss of regularity in solutions, advancing understanding of turbulence and singularity formation in fluid dynamics.

Findings

Solutions lose regularity for a dense set of initial data

Confirms Yudovich's conjecture in the smooth setting

Highlights the potential for small-scale creation in Euler flows

Abstract

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.