Loss of regularity for the 2D Euler equations

TL;DR

This paper constructs solutions to the 2D Euler equations that start with certain regularity but lose their regularity over time, highlighting potential limitations of solution regularity persistence.

Contribution

It demonstrates the loss of regularity in solutions to the 2D Euler equations within the Yudovich class, a novel insight into solution behavior.

Findings

Solutions in the Yudovich class can lose $W^{1,p}$ regularity over time.

Regularity loss occurs continuously, not abruptly.

Highlights limitations of regularity persistence in 2D Euler solutions.

Abstract

In this note, we construct solutions to the 2D Euler equations which belong to the Yudovich class but lose $W^{1,p}$ regularity continuously with time.