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

TL;DR

This paper demonstrates that the 2D Euler equations are strongly ill-posed in certain logarithmically refined Sobolev spaces, extending previous results from the critical Sobolev space $H^2$ to a broader class of spaces.

Contribution

It extends the ill-posedness results of the 2D Euler equations to new logarithmically regularized spaces that are strictly contained in $H^2$ and include all $H^s$ for $s>2$, for certain parameters.

Findings

Strong ill-posedness in logarithmically refined spaces for $eta ext{ with } eta eq 1/2$

Ill-posedness holds when the logarithmic derivative power $eta ext{ satisfies } eta ext{ with } eta eq 1/2$

Extension of Bourgain and Li's results to a broader class of function spaces.

Abstract

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 $\alpha\leq 1/2$, then the 2D Euler equations are strongly ill-posed.