  In a fractional Sobolev space $H^\frac{7}{4}(\mathbb{R}^2)$, we prove the
desired low-regularity ill-posedness results for the 2D compressible Euler
equations and the 2D ideal compressible MHD system. For the Euler equations, it
is sharp with respect to the regularity of the fluid velocity and density. The
mechanism behind the obtained ill-posedness is the instantaneous shock
formation.
