Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space

In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^2({\mathbb{R}}^2)$. In this work, we extend those results by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in $H^2({\mathbb{R}}^2)$ and which contain $H^s({\mathbb{R}}^2)$ for all $s>2$. These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power $\alpha$ of the logarithmic derivative satisfies $\alpha \le 1\mathrm{/}2$, then the 2D Euler equations are strongly ill-posed.