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

2025/10/23 by Cozzi, Elaine, Harrison, Nicholas, Radke, Zachary
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.20773

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 H²(ℝ²). In this work, we extend those results by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in H²(ℝ²) and which contain H^s(ℝ²) 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≤ 1/2, then the 2D Euler equations are strongly ill-posed.

Citations

• Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness
• A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces
• Strong ill-posedness of logarithmically regularized 2D Euler equations\n in the borderline Sobolev Space
• On the Sobolev space of functions with derivative of logarithmic order
• Ill-posedness for the incompressible Euler equations in critical Sobolev spaces
• Lagrangian solutions to the 2D euler system with L¹ vorticity and infinite energy

Related

• Strong ill-posedness of logarithmically regularized 2D Euler equations\n in the borderline Sobolev Space
• Instability of Data-to-Solution Map for the Log-Regularized 2D Euler System
• Note on the Euler equations in C^k spaces
• Ill-posedness for the incompressible Euler equations in critical Sobolev spaces
• A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces