Instability of Data-to-Solution Map for the Log-Regularized 2D Euler System

In this paper, we study the logarithmically regularized $2$D Euler system \eqref{e1}, which is derived by regularizing the Euler equation for the vorticity. We establish local well-posedness of the logarithmically regularized $2$D Euler equations in the subcritical space $H^s(\mathbb{R}^2)$ with $s>2$ for $\gamma \ge 0$. Furthermore, we show that for $\gamma$ close to $0$, the data-to-solution map is not uniformly continuous in the Sobolev $H^s(\mathbb{R}^2)$ topology for any $s>2$.