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

Abstract

In a fractional Sobolev space $H^s({\mathbb{R}}^2)$ with $s\le \frac{7}{4}$, 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 $H^{\frac{7}{4}}$ regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.

43 pages