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

Abstract

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 $\gamma >\frac{1}{2}$. In this paper, we examine the regularized models in the remaining regime $\gamma \le \frac{1}{2}$ 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 $\gamma >\frac{1}{2}$ and the strong ill-posedness for $\gamma =0$.