Velocity blow-up in $C^1\cap H^2$ for the 2D Euler equations

TL;DR

This paper demonstrates that solutions to the 2D Euler equations can instantaneously lose regularity in both $C^1$ and $H^2$ spaces, revealing a form of ill-posedness in these function spaces.

Contribution

It provides a vorticity-dynamical proof of $C^1\cap H^2$-illposedness, showing solutions can escape these spaces instantly.

Findings

Unique Yudovich solution escapes $C^1$ and $H^2$ instantly.

The proof is based on vorticity dynamics.

Ill-posedness occurs in the intersection of $C^1$ and $H^2$.

Abstract

We give a vorticity-dynamical proof of $C^1\cap H^2$-illposedness of the 2D Euler equations. Our construction shows that the unique Yudovich solution escapes both $C^1$ and $H^2$ instantaneously.