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

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.