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Many questions related to well-posedness/ill-posedness in critical spaces forhydrodynamic equations have been open for many years. In this article we give anew approach to studying norm inflation (in some critical spaces) for a wideclass of equations arising in hydrodynamics. As an application, we prove strongill-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 wholespace, a smooth bounded domain, or the torus. We also apply our method to theOldroyd B, surface quasi-geostrophic, and Boussinesq systems.

Ill-posedness results in critical spaces for some equations arising in hydrodynamics