The Cauchy problems for the 2D compressible Euler equations and ideal MHD system are ill-posed in $H^\frac{7}{4}(\mathbb{R}^2)$

TL;DR

This paper demonstrates that the 2D compressible Euler equations and ideal MHD system are ill-posed in fractional Sobolev spaces with regularity below or equal to 7/4, highlighting a critical threshold for well-posedness.

Contribution

It establishes low-regularity ill-posedness results for these systems in $H^s(R^2)$ with $s o 7/4$, identifying the precise regularity threshold.

Findings

Ill-posedness in $H^s$ for $s o 7/4$

Matching the regularity threshold for Euler system

Results extend to ideal MHD system

Abstract

In a fractional Sobolev space $H^s(\mathbb{R}^2)$ with $s\leq\frac74$, we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the $H^\frac74$ regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.