Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space
Elaine Cozzi, Nicholas Harrison, Zachary Radke
TL;DR
The paper establishes strong ill-posedness for the 2D Euler equations in logarithmically refined critical Sobolev spaces $H^{2,\alpha}(\mathbb{R}^2)$ when $\alpha\le \tfrac{1}{2}$, extending Bourgain–Li’s critical-space ill-posedness results. It defines $H^{s,\alpha}$ via the operator $\log^{\alpha}(D+1)$, proves a sharp threshold $\alpha=\tfrac{1}{2}$ separating ill-posedness from global well-posedness (proved in prior work for $\alpha>\tfrac{1}{2}$), and develops a two-pronged construction: first, a local norm inflation in $H^{1,\alpha}$ through large Lagrangian deformation and high-frequency perturbations; then, an interaction framework that patches infinitely many such localized configurations to yield global ill-posedness. A key methodological element is a quantitative nonstationary-phase oscillatory integral bound that tracks phase-dependent behavior under the flow, enabling precise control of the trajectory map and vorticity transport. The results delineate the sharp boundary of well-posedness versus ill-posedness at the logarithmically refined critical scale and underscore the delicate role of logarithmic regularity in 2D fluid dynamics.
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 $α$ of the logarithmic derivative satisfies $α\leq 1/2$, then the 2D Euler equations are strongly ill-posed.