Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

TL;DR

This paper constructs solutions to the 2D incompressible Euler equations that exhibit an instantaneous loss of Sobolev regularity, demonstrating a sharp regularity gap in the evolution of these solutions.

Contribution

It provides explicit solutions showing immediate Sobolev regularity loss for the 2D Euler equations, highlighting a new phenomenon in fluid dynamics regularity theory.

Findings

Solutions start in super-critical Sobolev spaces but lose regularity instantly.

Solutions are globally existing and unique within a certain classical framework.

Demonstrates a sharp regularity gap in the evolution of Euler solutions.

Abstract

We construct solutions of the 2D incompressible Euler equations in $\mathds{R}^2\times [0,\infty)$ such that initially the velocity is in the super-critical Sobolev space $H^\beta$ for $1<\beta<2$, but are not in $H^{\beta'}$ for $\beta'>1+\frac{(3-\beta)(\beta-1)}{2 - (\beta-1)^2}$ for $0<t<\infty$. These solutions are not in the Yudovich class, but they exists globally in time and they are unique in a determined family of classical solutions.