Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev Space

Logarithmically regularized 2D Euler equations are active scalar equations with the non-local velocity $u = \nabla^\perp \Delta^{-1}T_\gamma \omega$ for the scalar $\omega$. Two types of the regularizing operator $T_\gamma$ with a parameter $\gamma> 0$ are considered: $T_\gamma = \ln^{-\gamma} (e+|\nabla|)$ and $T_\gamma = \ln^{-\gamma} (e-\Delta)$. These models regularize the 2D Euler equation for the vorticity (conventionally corresponding to the $\gamma=0$ case), which results in their local well-posedness in the borderline Sobolev space $H^1(\mathrm{R}^2)\cap\dot{H}^{-1}(\mathrm{R}^2)$ when $\gamma>\frac 12$. In this paper, we examine the regularized models in the remaining regime $\gamma\leq \frac 12$ and establish the strong ill-posedness in the borderline space. This completely solves the well-posedness problem of the regularized models in the borderline space by closing the gap between the local well-posedness result for $\gamma>\frac 12$ and the strong ill-posedness for $\gamma = 0$.