  Many questions related to well-posedness/ill-posedness in critical spaces for
hydrodynamic equations have been open for many years. In this article we give a
new approach to studying norm inflation (in some critical spaces) for a wide
class of equations arising in hydrodynamics. As an application, we prove strong
ill-posedness of the $n$-dimensional Euler equations in the class $C^1\cap L^2
(\Omega)$ and also in $C^k \cap L^2(\Omega)$ where $\Omega$ can be the whole
space, a smooth bounded domain, or the torus. We also apply our method to the
Oldroyd B, surface quasi-geostrophic, and Boussinesq systems.
