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

TL;DR

This paper proves that logarithmically regularized 2D Euler equations are strongly ill-posed in the borderline Sobolev space when the regularization parameter is less than or equal to 1/2, completing the understanding of their well-posedness.

Contribution

It establishes the strong ill-posedness of the models in the borderline space for all regularization parameters  , closing the gap in the well-posedness theory.

Findings

Proves strong ill-posedness for   in the borderline Sobolev space.

Completes the classification of well-posedness for the regularized 2D Euler equations.

Identifies the critical threshold   for the regularization parameter.

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 $\gamma>\frac 12$. 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 $\gamma = 0$.